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Interpolation Operators on Some Triangles with Curved Sides

Interpolation Operators on Some Triangles with Curved Sides is paper contains a survey regarding interpolation Bernstein-type operators defined on triangles having one or all curved sides; we consider as well some of e product Boolean sum operators. We study e interpolation properties, e orders of accuracy e remainders of e generated approximation formulas. Maematics Subject Classification 2010: 41A05, 41A25, 41A80. Key words: product boolean sum operators, triangles tetrahedrons wi curved sides, interpolation operators, Bernstein-type operators, remainders. 1. Introduction e aim of is survey is to present some interpolation Bernsteintype operators for functions defined on triangles wi one or all curved sides (see [11], [12], [14], [15]). ey come as an extension of e corresponding operators for functions defined on triangles wi all straight sides (see, e.g., [3]-[6], [8]-[10], [13], [23], [24], [26]-[29], [32]). e operators defined on domains wi curved sides permit essential boundary conditions to be satisfied exactly. Such operators can be used in construction of surfaces which satisfy some given conditions (see, e.g., [17], [18]), in finite element meod for differential equation problems (Lagrange operators for Dirichlet boundary conditions, Birkhoff operators for Neumann boundary conditions Hermite operators for Robin boundary conditions) (see, e.g., [19], [25], [26], [35]) in numerical integration of functions (see, e.g., [16]). We study ese operators especially from e eoretical point of view. e idea came from e paper of Barnhill Gregory [4], where ere TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ is considered a triangle wi one curved side ere are used Lagrange projectors on e straight sides, Taylor projector on e curved side. Such operators were also studied in many oer papers in connection wi eir applications in computer aided geometric design (see, e.g., [1], [2], [5]) in finite element analysis (see, e.g., [1], [7], [20], [22], [23], [24], [35]). We study ree main aspects of e constructed operators: 1) e interpolation properties; 2) e orders of accuracy; 3) e remainders of e corresponding interpolation formulas. e order of accuracy of an interpolation operator P is given by e degree of exactness (dex(P )), respectively by e precision set (pres(P )). Recall at dex(P ) = r if P f = f, for f Pr ere exists g Pr+1 such at P g = g, where Pm denotes e space of e polynomials in 2 variables of global degree at most m. e precision set of an interpolation operator is e set of monomials for which e interpolant is exact ([4]). e characteristics 1) 2) can be verified by a straightforward computation. e remainders of e interpolation formulas will be studied using e Peano's eorem for e functions from a Sard-type space (see, e.g., [31]). e Sard-type space, denoted by Bpq (a, c), (p, q N, p + q = m), is e space of e functions f : D R, D = [a, b] × [c, d] satisfying f (p,q) C (D) ; f (m-j,j)(·, c) C [a, b] , j < q; f (i,) (a, ·) C [c, d] , i < p. ~ Given h > 0, denote by e triangle having e vertices V1 = (h, 0), V2 = (0, h) V3 = (0, 0), two straight sides 1 , 2 , along e coordinate axes, e ird side 3 (opposite to e vertex V3 ), which is defined by e one-to-one functions f g, where g is e inverse of e function f, i.e., y = f (x) x = , wi f (0) = g(0) = h (see Figure 1). V2 1 (x,y) (x,f(x)) (0,y) V3 V1 ~ Figure 1: Triangle . ~ ere is no restriction in considering is stard triangle , since any triangle wi one curved side can be obtained from is stard triangle by an affine transformation which preserves e form order of accuracy of e interpolant ([4]). INTERPOLATION OPERATORS In Section 2 we study Lagrange, Hermite Birkhoff interpolation ~ operators, as well as eir product Boolean sum on . In Section 3 we present some Bernstein-type operators togeer wi eir product ~ Boolean sum for e same triangle . Section 4 is dedicated to Bernstein~ type operators defined on a triangle wi all curved sides, denoted by . is triangle has e vertices V1 = (0, h), V2 = (h, 0) V3 = (0, 0), e ree curved sides 1 , 2 (along e coordinate axes) 3 (opposite to e vertex V3 ). We define 1 by (x, f1 (x)), wi f1 (0) = f1 (h) = 0, f1 (x) 0, for x [0, h]; 2 defined by (g2 (y), y), wi g2 (0) = g2 (h) = 0, g2 (y) 0, for y [0, h] 3 defined by e one-to-one functions f3 g3 , where g3 is e inverse of e function f3 , i.e., y = f3 (x) x = g3 (y), wi x, y [0, h] f3 (0) = g3 (0) = h, h R+ , (see Figure 2). For example, f1 g2 can be convex functions. V1 (x,f3(x)) (g2(y),y) (g (y),y) V3 (x,f (x)) V2 ~ Figure 2. Triangle . 2. Interpolation operators on a triangle wi one curved side 2.1. Lagrange-type operators ~ Suppose at F is a real-valued function defined on . Let L1 , L2 L3 be e interpolation operators defined by - x x F (0, y) + F (, y), f (x) - y y (L2 F )(x, y) = F (x, 0) + F (x, f (x)), f (x) f (x) x y (L3 F )(x, y) = F (x + y, 0) + F (0, x + y). x+y x+y (L1 F )(x, y) = (1) TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ 1) Each of e operators L1 , L2 L3 interpolates e function F along ~ two sides of e triangle : (L1 F )(0, y) = F (0, y), (L1 F )(, y) = F (, y), y [0, h], (L2 F )(x, 0) = F (x, 0), (L2 F )(x, f (x)) = F (x, f (x)), x [0, h], (L3 F )(x + y, 0) = F (x + y, 0), (L3 F )(0, x + y) = F (0, x + y), x, y [0, h], properties illustrated in Figure 3. e bold sides points indicate e interpolation domains. V2 V2 (x,f(x) V2 (0,y) L1 (0,x+y) L3 V3 V1 V1 V3 (x+y,0) Figure 3. e interpolation domains for L1 , L2 L3 . 2) e orders of accuracy: (2) dex(Li ) = 1, i = 1, 2, 3, pres(L2 ) = {1, xi , y, i N }, pres(L1 ) = {1, x, y j , j N }, pres(L3 ) = {1, x, y}. L 3) Regarding e remainders Ri F, i = 1, 2, 3, of e interpolation forL F, mulas F = Li F + Ri i = 1, 2, 3, we have: eorem 1 ([14]). If F B11 (0, 0) en L (R1 F )(x, y) = x[x - ] (2,0) F (, 0) 2 xy[ - x] (1 , 1 ) - F (1,1) (2 , 2 ) , wi [0, h], (1 , 1 ) [0, x] × [0, y] (2 , 2 ) [x, ] × [0, y], respectively (3) where · L (R1 F )(x, y) h2 8 F (2,0) (·, 0) denotes e Chebyshev norm. INTERPOLATION OPERATORS Proof. From (2) we have dex(L1 ) = 1 applying e Peano's eorem we get L (R1 F )(x, y) = h 0 ~ K20 (x, y, s)F (2,0) (s, 0)ds K11 (x, y, s, t)F (1,1) (s, t)dsdt, (4) wi e Peano's kernels given by (5) K20 (x, y, s) = (x - s)+ - x ( - s)+ , x K11 (x, y, s, t) = (y - t)0 [(x - s)0 - ( - s)0 ]. + + + As, K20 (x, y, s) 0, K11 (x, y, s, t) 0, K11 (x, y, s, t) 0, K11 (x, y, s, t) = 0, s [0, h], (s, t) [0, x] × [0, y], (s, t) [x, ] × [0, y], (s, t) D1 × D2 , (6) wi D1 D2 illustrated in Figure 4, by e Mean Value eorem we obtain L (R1 F )(x, y) = 20 (x, y)F (2,0) (, 0) + 1 (x, y)F (1,1) (1 , 1 ) 11 + 2 (x, y)F (1,1) (2 , 2 ), 11 wi [0, h], (1 , 1 ) [0, x] × [0, y], (2 , 2 ) [x, ] × [0, y], 20 (x, y) = (7) 1 (x, y) = 11 2 (x, y) = 11 0 y 0 x(x - ) , 2 x xy[ - x] K11 (x, y, s, t)dsdt = , 0 K11 (x, y, s, t)dsdt = - xy[ - x] . relation (3) follows. As |20 (x, y)| h2 8 , 1 (x, y) 11 2 (x, y) 11 TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ D2 (0,y) 0.6 D1 0.8 0.9 (,0) 1 Figure 4: e sign of e kernel K11 . Remark 2. Analogous formulas can be obtained for e remainders L L R2 F R3 F. L Let Pij be e product of e operators Li Lj , i.e., Pij = Li Lj , i, j = 1, 2, 3, i = j. We have h - y - x y - x x F (0, 0) + F (0, h) + F (, y), h h - x L (P13 F )(x, y) = F (0, y) x + [F (y + , 0) + yF (0, y + )], [y + ] f (x) - y L (P23 F )(x, y) = F (x, 0) f (x) y + [xF (x + f (x), 0) + f (x)F (0, x + f (x))]. f (x)[x + f (x)] L (P12 F )(x, y) = L L 1) e interpolation properties: P12 F = F, on 3 V3 ; P13 F = F, on L 1 V1 ; P23 F = F, on 2 V2 . L Remark 3. e operator Pij has e same interpolation properties as L e operator Pji , i, j = 1, 2, 3, i = j. ese properties are illustrated in Figure 5. V2 V2 V2 L2 L L L V3 V1 V3 V3 V1 L L L Figure 5. e interpolation domains for P12 , P13 P23 . INTERPOLATION OPERATORS L L 2) e orders of accuracy: dex(Pij ) = 1, pres(Pij ) = {1, x, y}, i, j = 1, 2, 3, i = j. P L 3) For e remainders Rij F, of e interpolation formulas F = Pij F + LP Rij F, i, j = 1, 2, 3, i = j, we have: eorem 4 ([14]). If F B11 (0, 0) en LP (R12 F )(x, y) = x[x - ] (2,0) y(y - h)[ - x] (0,2) F (, 0) + F (0, ) 2 2 xy[ - x] (1,1) [F (1 , 1 ) - F (1,1) (2 , 2 )], + wi , [0, h], (1 , 1 ) [0, x] × [0, y] (2 , 2 ) [x, ] × [0, y], respectively (8) LP (R12 F )(x, y) h2 8 F (2,0) (·, 0) + F (0,2) (0, ·) L Proof. By dex(P12 ) = 1, applying Peano's eorem we get at LP (R12 F )(x, y) = h 0 K20 (x, y, s)F (2,0) (s, 0)ds + 0 ~ K02 (x, y, t)F (0,2) (0, t)dt (9) K11 (x, y, s, t)F (1,1) (s, t)dsdt, wi e Peano's kernels x [ - s]+ , - x y(h - t) K02 (x, y, t) = (y - t)+ - , h x K11 (x, y, s, t) = (y - t)0 {(x - s)0 - [ - s]0 }. + + + K20 (x, y, s) = (x - s)+ - We notice at e Peano's kernels K20 K11 are e same as e kernels given in (5). erefore, eir sign is given in (6) we have K02 (x, y, t) 0, for t [0, h]. By e Mean Value eorem we obtain LP (R12 F )(x, y) =20 (x, y)F (2,0) (, 0) + 02 (x, y)F (0,2) (0, ) + 1 (x, y)F (1,1) (1 , 1 ) + 2 (x, y)F (1,1) (2 , 2 ), 11 11 TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ for , [0, h], (1 , 1 ) [0, x] × [0, y], (2 , 2 ) [x, ] × [0, y], wi 20 , 1 , 2 given in (7) 11 11 02 (x, y) = 0 h2 K02 (x, y, t)dt = h2 8 , y(y - h))[ - x] . 2 1 (x, y) 11 We have |20 (x, y)| 8 , |02 (x, y)| so e relation (8) follows. 2 (x, y) 11 Remark 5. Analogous formulas can be obtained for e remainders LP LP R23 F R13 F. L L Let Sij be e Boolean sum of e operators Li Lj , i.e., Sij = Li Lj = Li + Lj - Li Lj , i, j = 1, 2, 3, i < j (see, e.g., [21]). We have L S12 F (x, y) = - x f (x) - y y F (0, y) + F (x, 0) + F (x, f (x)) f (x) f (x) - x h - y y - F (0, 0) + F (0, h) , h h x x y L S13 F (x, y) = F (g (y) , y) + F (x + y, 0) + F (0, x + y)- x+y x+y y - F (y + , 0) + F (0, y + ) , y + y + y x y F (x, f (x)) + F (x + y, 0) + F (0, x + y)- f (x) x+y x+y y y f (x) - F (x + f (x), 0) + F (0, x + f (x)) . f (x) x + f (x) x + f (x) L S23 F (x, y) = L ~ 1) e interpolation properties: Sij F = F, i, j = 1, 2, 3, i < j, on . 2) e orders of accuracy: L L L dex(S12 ) = 1, dex(S13 ) = dex(S23 ) = 2, (10) L pres(S12 ) = {1, y, xy, xk , k N }, L pres(S23 ) = {1, x, y, x2 , y 2 , xk y, k N }. LS L 3) For e remainders Rij F, of e interpolation formulas F = Sij F + LS Rij F, i, j = 1, 2, 3, i < j, we have: L pres(S13 ) = {1, x, y, x2 , y 2 , xy k , k N }, INTERPOLATION OPERATORS eorem 6 ([14]). If F B11 (0, 0) en LS (R12 F )(x, y) = h 0 ~ K02 (x, y, t)F (0,2) (0, t)dt K11 (x, y, s, t)F (1,1) (s, t)dsdt, (11) wi e Peano's kernels x [ - x]y x (y - t)+ - [f (x) - t]+ + (h - t), f (x) h y K11 (x, y, s, t) = (x - s)0 {(y - t)0 - [f (x) - t]0 }. + + + f (x) K02 (x, y, t) = Furermore, LS (R12 F )(x, y) F (2,0) (·, 0) h 0 ~ |K02 (x, y, t)|dt |K11 (x, y, s, t)|dsdt. Proof. e proof follows directly by Peano's eorem, taking into L account at dex(S12 ) = 1. eorem 7 ([14]). If F B12 (0, 0) en LS (R13 F )(x, y) = h h 0 (12) ~ wi e Peano's kernels K30 (x, y, s) = x[ - s]2 x(x + y - s)2 x[y + - s]2 (x - s)2 + + + + - - + , 2 2 2(x + y) 2(y + ) xy[ - s]+ K21 (x, y, s) = y(x - s)+ - , [ - x](y - t)2 y(x + y - t)2 xy(y + - t)2 + + + K03 (x, y, t) = - + 2 , 2 2(x + y) g (y)[y + ] x K12 (x, y, s, t) = (y - t)+ {(x - s)0 - [ - s]0 }. + + TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ Furermore, LS (R13 F )(x, y) F (3,0) (·, 0) h 0 h 0 h 0 |K30 (x, y, s)|ds |K21 (x, y, s)|ds |K03 (x, y, t)|dt + F (2,1) (·, 0) (13) + F (0,3) (0, ·) + F (1,2) ~ |K12 (x, y, s, t)|dsdt. L Proof. From (10) it follows at dex(S13 ) = 2 applying Peano's eorem we get (12) e inequality (13). LS Remark 8. An analogous formula can be obtained for R23 F. 2.2. Hermite-type operators ~ Suppose at e real valued function F is defined on e triangle (1,0) F (0,1) on e side . We it possesses e partial derivatives F 3 consider e operators H1 H2 defined by (H1 F )(x, y) = (14) x[2 - x] [x - ]2 F (0, y) + F (, y) 2 (y) g g 2 (y) x[x - ] (1,0) + F (, y), [y - f (x)]2 y[2f (x) - y] (H2 F )(x, y) = F (x, 0) + F (x, f (x)) 2 (x) f f 2 (x) y[y - f (x)] (0,1) + F (x, f (x)). f (x) 1) e interpolation properties: H1 F = F, on 1 3 ; (H1 F )(1,0) = F (1,0) , on 3 ese interpolation properties are illustrated in Figure 6. H2 F = F, on 2 3 ; (H2 F )(0,1) = F (0,1) , on 3 . V2 INTERPOLATION OPERATORS H1 V1 V3 Figure 6. e interpolation domains for H1 H2 . 2) e orders of accuracy: dex(H1 ) = dex(H2 ) = 2, (15) pres(H1 ) = {1, x, y, x2 , y 2 , xy n , n N }, pres(H2 ) = {1, x, y, x2 , y 2 , xn y, n N }. H Ri F , H 3) e interpolation formulas are F = Hi F + Ri F, i = 1, 2, where i = 1, 2 are e remainder terms, for which we have: eorem 9 ([14]). If F B12 (0, 0) en e following inequality holds H (R1 F )(x, y) (16) furer, x[ - x]2 (3,0) xy[ - x]2 (2,1) F (·, 0) + F (·, 0) 6 2 - x xy 2 [ - x][3 - 2x] (1,2) + F (·, ·) , 2g 2 (y) H (R1 F )(x, y) (17) 2h3 (3,0) xy[ - x]2 (2,1) F (·, 0) + F (·, 0) 81 2 - x xy 2 [ - x][3 - 2x] (1,2) + F (·, ·) . 2g 2 (y) Proof. By (15) it follows at dex(H1 ) = 2, erefore by Peano's eorem we get H (R1 F )(x, y) = h 0 0 ~ (18) TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ wi (x - s)2 x[2 - x] ( - s)2 + + - 2 g 2 (y) 2 x[x - ] ( - s)+ , - xy[2 - x] K21 (x, y, s) = y(x - s)+ - ( - s)+ g 2 (y) xy[x - ] - ( - s)0 , + K12 (x, y, s, t) = (y - t)+ [(x - s)0 + x[2 - x] - ( - s)0 ]. + g 2 (y) K30 (x, y, s) = We have 2 2 s [ - x] 0, 2g 2 (y) K30 (x, y, s) = x[ - s] [(s - x) + s[ - x] 0, 2g 2 (y) 0, - 1 sy[ - x]2 0, g 2 (y) xy - [g 2 (y) - s(2 - x)] 0, g 2 (y) K21 (x, y, s) = xy - 2 [g 2 (y) - s(2 - x)] 0, g (y) 0, 2 [ - x] 0, g 2 (y) K12 (x, y, s, t) = (y - t)+ x[x - 2] 0, g 2 (y) 0, s [0, x) s [x, ) s [, h], s [0, x) g (y) s [x, 2-x ) g (y) s [ 2-x , ) s [, h], (s, t) [0, x) × [0, y) (s, t) [x, ) × [0, y) (s, t) D1 D2 , INTERPOLATION OPERATORS wi domains D1 , D2 e sign of K12 as in Figure 4. We obtain at H (R1 F )(x, y) h 0 ~ h 0 F (3,0) (·, 0) + F (1,2) (·, ·) K30 (x, y, s)ds + F (2,1) (·, 0) |K12 (x, y, s, t)| dsdt, |K21 (x, y, s)| ds whence, after some computation we get (16), furer we obtain (17). Remark 10. An analogous formula can be obtained for e remainder H R2 F. e product of e operators H1 H2 is given by H (P12 F )(x, y) = [x - ]2 (y - h)2 F (0, 0) g 2 (y) h2 y(2h - y) y(y - h) (0,1) + F (0, h) + F (0, h) 2 h h x[2 - x] x[x - ] (1,0) + F (, y) + F (, y). 2 (y) g 1) e interpolation properties: H P12 F = F, on V3 3 , H H (P12 F )(1,0) = F (1,0) , (P12 F )(0,1) = F (0,1) , on 3 . e interpolation properties are illustrated in Figure 7. V2 V3 V1 H Figure 7. e interpolation domain for P12 F . 2) e orders of accuracy: (19) H P12 F H H dex(P12 ) = 2, pres(P12 ) = {1, x, y, x2 , xy, y 2 , x2 y, xy 2 }. For e remainder of e corresponding interpolation formula, F = HP + R12 F, we have: TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ eorem 11 ([14]). If F B12 (0, 0) en e following inequality holds HP (R12 F )(x, y) (20) x[ - x]2 (3,0) F (·, 0) 6 xy[ - x]2 (2,1) F (·, 0) + 2 - x y[ - x]2 (h - y) (0,3) + F (0, ·) 6g 2 (y) xy[ - x][3 - 2x] (1,2) F (·, ·) + g 2 (y) H Proof. By (19) it follows at dex(P12 ) = 2 applying Peano's eorem we get HP (R12 F )(x, y) = h h 0 ~ wi (x - s)2 x[2 - x] ( - s)2 x[x - ] + + - - ( - s)+ , 2 (y) 2 g 2 xy[2 - x] xy[x - ] K21 (x, y, s) =y(x - s)+ - ( - s)+ - , 2 (y) g (y - t)2 x[2 - x] (y - t)2 + + K03 (x, y, t) = - 2 g 2 (y) 2 [x - ]2 y(2h - y) (h - t)2 y(y - h)(h - t) - [ + ] g 2 (y) h2 2 h x[2 - x] K12 (x, y, s, t) =(y - t)+ [(x - s)0 - ]. + g 2 (y) K30 (x, y, s)= We have 2 2 s [ - x] 0, 2g 2 (y) K30 (x, y, s) = - x[ - s] [(x - s)+(x - )s]0, 2g 2 (y) 0, s [0, x) s [x, ) s [, h], INTERPOLATION OPERATORS We obtain at sy s [0, x) - 2 [ - x]2 0, g (y) xy g 2 (y) - [( - s) + s(x - )] 0, s [x, 2-x ) g 2 (y) K21 (x, y, s)= g 2 (y) - xy [(-s)+s(x-)]0, s [ 2-x , ) 2 g (y) 0, s [, h], 2 [ - x] 2 t (h - y)2 0, t [0, y) 2g 2 (y) K03 (x, y, t) = [x - ]2 (h - t)y - [h(y - t) + t(y - h)] 0, t [y, h] 2h2 g 2 (y) 2 (y - t) [ - x] 0, (s, t) [0, x) × [0, y) g 2 (y) K12 (x, y, s, t) = (y - t) x[x - 2] 0, (s, t) [x, ) × [0, y) g 2 (y) ~ 0, (s, t) ([0, h] × [y, h]) . HP (R12 F )(x, y) F (3,0) (·, 0) h K30 (x, y, s)ds 0 h + F (2,1) (·, 0) + F (1,2) (·, ·) h 0 |K21 (x, y, s)| ds + F (0,3) (0, ·) |K12 (x, y, s, t)| dsdt, K03 (x, y, t)dt ~ whence, after some computation, we get (20). e Boolean sum of e operators H1 H2 is given by H (S12 F )(x, y) = [x - ]2 [y - f (x)]2 F (0, y) + F (x, 0) g 2 (y) f 2 (x) y[2f (x) - y] y[y - f (x)] (0,1) + F (x, f (x)) + F (x, f (x)) 2 (x) f f (x) [x - ]2 (y - h)2 y(2h - y) y(y - h) (0,1) - F (0, 0) + F (0, h) + F (0, h) . 2 (y) 2 2 g h h h 1) e interpolation properties: H ~ S12 F = F, on H F )(1,0) = F (1,0) , H (S12 (S12 F )(0,1) = F (0,1) , on 3 . TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ xn y, n H H 2) e orders of accuracy: dex(S12 ) = 2, pres(S12 ) = {1, x, y, x2 , y 2 , }. N H HS For e remainder of e interpolation formula, F = S12 F + R12 F, we have: eorem 12 ([14]). If F B12 (0, 0) en e following inequality holds HS (R12 F )(x, y) F (0,3) (0, ·) h 0 |K03 (x, y, t)|dt ~ + F (1,2) (·, ·) |K12 (x, y, s, t)| dsdt. H Proof. By (15) it follows at dex(S12 ) = 2 erefore by Peano's eorem we get h 0 h 0 HS (R12 F )(x, y) = h ~ wi K30 (x, y, s) =K21 (x, y, s) = 0, (y - t)2 g 2 (y) - (x - )2 + 2 g 2 (y) y(2f (x) - y) (f (x) - t)2 y(y - f (x))(f (x) - t)+ + - - f 2 (x) 2 f (x) 2 y(2h - y)(h - t)2 (x - ) y(y - h)(h - t) + + , 2 (y) 2 g 2h h y[2f (x) - y] K12 (x, y, s, t) =(x - s)0 [(y - t)+ - (f (x) - t)+ + f 2 (x) y(y - f (x)) - (f (x) - t)0 ]. + f (x) K03 (x, y, t) = 17 We have INTERPOLATION OPERATORS y(2f (x) - y) (f (x) - t)2 (y - t)2 g 2 (y) - (x - )2 - 2 g 2 (y) f 2 (x) 2 (x - )2 y(2h - y)(h - t)2 y(y - h)(h - t) + + 0, g 2 (y) 2h2 h t [0, y), y(2f (x) - y) (f (x) - t)2 - f 2 (x) 2 K03 (x, y, t) = (x - )2 y(2h - y)(h - t)2 y(y - h)(h - t) + + , g 2 (y) 2h2 h t [y, f (x)) (x - )2 y(2h - y)(h - t)2 y(y - h)(h - t) + 0, g 2 (y) 2h2 h t [f (x), h], K12 (x, y, s, t) (y - t) - y[2f (x) - y] (f (x) - t) - y(y - f (x)) 0, f 2 (x) f (x) (s, t) [0, x) × [0, y), y[2f (x) - y] y(y - f (x)) - (f (x) - t) - 0, f 2 (x) f (x) f 2 (x) (s, t) [0, x) × [y, ), 2f (x) - y = (y - t)+ · y[2f (x) - y] y(y - f (x)) - (f (x) - t) - 0, 2 (x) f f (x) f 2 (x) (s, t) [0, x) × [ , f (x)) 2f (x) - y 0, ~ (s, t) ([x, h] × [0, f (x)] [0, x] × [f (x), h]) . HS (R12 F )(x, y) F (0,3) (0, ·) h 0 We obtain at |K03 (x, y, t)|dt ~ + F (1,2) (·, ·) |K12 (x, y, s, t)| dsdt. TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ 2.3. Birkhoff-type operators In is section we give some examples of operators which interpolate e ~ given function F : R on a side of e triangle its first partial ~ derivatives on anoer side of , respectively. ~ First, we suppose at e function F : R has e partial deriva(1,0) F (0,1) on e side . We consider e Birkhoff-type operatives F 3 tors B1 B2 defined respectively by (B1 F ) (x, y) = F (0, y) + xF (1,0) (g (y) , y) , (B2 F ) (x, y) = F (x, 0) + yF (0,1) (x, f (x)) . 1) e interpolation properties: B1 F = F on 1 (B1 F )(1,0) = F (1,0) on 3 , B2 F = F on 2 (B2 F )(0,1) = F (0,1) on 3 . ese interpolation properties are illustrated in Figure 8. V2 V2 B1 B2 V1 V3 V1 Figure 8. e interpolation domains for B1 B2 . 2) e orders of accuracy: (21) dex (B1 ) = dex (B2 ) = 1, pres (B2 ) = 1, y, xi , i N . pres (B1 ) = 1, x, y j , j N , B 3) For e remainders of e interpolation formulas F = B1 F + R1 F B F we have: F = B2 F + R2 eorem 13 ([14]). If F B11 (0, 0) en x[x - 2] (2,0) F (, 0) + xyF (1,1) (1 , ), 2 ~ wi [0, h], (1 , ) , respectively B (R1 F )(x, y) = (22) B (R1 F )(x, y) h2 (2,0) F (·, 0) 2 h2 (1,1) F (·, ·) 4 INTERPOLATION OPERATORS Proof. By (21) it follows at dex(B1 ) = 1, applying Peano's eorem we obtain B R1 F (x, y) = h 0 ~ K20 (x, y, s) F (2,0) (s, 0) ds K11 (x, y, s, t) F (1,1) (s, t) dsdt + wi K11 (x, y, s, t) = (x - s)0 (y - t)0 . + + We have K20 (x, y, s) = (x - s)+ - x [ - s]0 , + K20 (x, y, s) 0, s [0, h] K11 (x, y, s, t) > 0, (s, t) [0, x] × [0, y] ~ K11 (x, y, s, t) = 0, (s, t) (([x, h] × [0, y]) ([0, x] × [y, h])) , by e Mean Value eorem we obtain B (R1 F )(x, y) = 20 (x, y)F (2,0) (, 0) + 11 (x, y)F (1,1) (1 , ), ~ wi [0, h], (1 , ) , 20 (x, y) = 0 y x 0 x[x - 2] , 2 11 (x, y) = As, |20 (x, y)| B R2 F. h2 2 , K11 (x, y, s, t)dsdt = xy. |11 (x, y)| e relation (22) follows. Remark 14. Analogous formula can be obtained for e remainder ~ Next, we suppose at e function F : R admits e partial (1,0) on F (0,1) on . derivatives F 1 2 We consider e Birkhoff-type operators B3 B4 defined by (B3 F ) (x, y) = F (, y) + [x - ]F (1,0) (0, y) , (B4 F ) (x, y) = F (x, f (x)) + [y - f (x)]F (0,1) (x, 0) . TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ 1) e interpolation properties: B3 F = F on 3 (B3 F )(1,0) = F (1,0) on 1 , B4 F = F on 3 (B2 F )(0,1) = F (0,1) on 2 . ese properties are illustrated in Figure 9. V2 V2 B3 B4 V3 V1 V3 V1 Figure 9. e interpolation domains for B3 B4 . 2) e orders of accuracy: dex (B3 ) = dex (B4 ) = 1, pres (B3 ) = 1, x, y j , j N , pres (B4 ) = 1, y, xi , i N . B 3) For e remainders of e interpolation formulas F = B3 F + R3 F B F, we have: F = B4 F + R4 eorem 15 ([14]). If F B11 (0, 0) en (23) B (R3 F )(x, y) = x2 - g 2 (y) (2,0) F (, 0) + y[x - ]F (1,1) (1 , ), 2 ~ wi [0, ], (1 , ) , respectively (24) B (R3 F )(x, y) h2 (2,0) F (·, 0) 2 h2 (1,1) F (·, ·) 4 Proof. As dex(B1 ) = 1, by Peano's eorem, we obtain B R3 F (x, y) = 0 h 0 K20 (x, y, s) F (2,0) (s, 0) ds K11 (x, y, s, t) F (1,1) (s, t) dsdt 21 where INTERPOLATION OPERATORS K20 (x, y, s) = (x - s)+ -[ - s] , K11 (x, y, s, t) = [(x - s)0 -1] (y - t)0 . + + Taking into account at K20 (x, y, s) 0, s [0, ] K11 (x, y, s, t) = -1, (s, t) D = [0, - x] × [0, y] ~ K11 (x, y, s, t) = 0, (s, t) D, 0 0 0 h x2 - g 2 (y) , 2 K11 (x, y, s, t)dsdt = y[x - ], 2 -g 2 (y) e relation (23) follows. Furer, as | x e inequality (24) follows. h2 2 , |y[x - ]| B Remark 16. Analogous formula can be obtained for e remainder R4 F. Example 17. We consider e following test function (see, e.g., [30]): (25) F1 (x, y) = exp[- 81 ((x - 0.5)2 + (y - 0.5)2 )]/3. 16 (Gentle) ~ We take triangle wi one curved side T1 , (h = 1), wi f : [0, 1] [0, 1], e 2 . In Figure 10 we plot e graphs of L F H F . f (x) = 1 - x 1 1 1 1 L1 F1 H1 F1 Figure 10. Graphs of L1 F1 H1 F1 . TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ Example 18. Table 1 contains some maximum approximation errors ~ for F1 , defined on T1 . Maximum error x Bm F1 0.0525 y Bn F1 0.0452 Pmn F1 0.0858 Qnm F1 0.0857 Smn F1 0.0095 Tnm F1 0.0095 Table 1. Maximum approximation errors for F1 . L1 F1 P13 F1 S12 F1 H1 F1 B1 F1 3. Bernstein-type operators on a triangle wi one curved side Since e Bernstein-type operators interpolate a given function at e endpoints of e interval, ese operators can also be used as interpolation operators bo on triangles wi straight sides (see, e.g., [9], [33], [34]) wi curved sides. Let F be a real-valued function defined on (0, y), (, y), respectively, (x, 0), (x, f (x)) be e points where e parallel lines to e coordinate axes, passing rough e point (x, y) , intersect e sides i , i = 1, 2, 3, (see Figure 1). One considers e Bernstein-type operators y x Bm Bn defined by m x (Bm F ) (x, y) = i=0 n y (Bn F ) (x, y) = j=0 F qn,j (x, y) F ,y , m f (x) n x, j wi = qn,j (x, y) = m i n j y f (x) i j y f (x) n-j , , x+y , 0 x + y f (x), are uniform where x = i i = 0, m y = j f (x) j = 0, n n m m n partitions of e intervals [0, ] [0, f (x)]. INTERPOLATION OPERATORS eorem 19 ([11]). If F is a real-valued function defined on en: y x (i) Bm F = F on 2 3 , Bn F = F on 1 3 , x x (ii) (Bm eij ) (x, y) = xi y j , i = 0, 1; (Bm e2j ) (x, y) = x2 + j N, x(-x) m yj , y y (iii) (Bn eij ) (x, y) = xi y j , j = 0, 1; (Bn ei2 ) (x, y) = xi y 2 + i N. y(f (x)-y) n Proof. e interpolation properties (i) follow from e relations: pm,i (0, y) = respectively by qn,j (x, 0) = 1, 0, for j = 0, for j > 0, qn,j (x, f (x)) = 0, for j < n, 1, for j = n. 1, 0, for i = 0, for i > 0, pm,i (, y) = 0, for i < m, 1, for i = m, Regarding e properties (ii), we have x x (Bm eij ) (x, y) = y j (Bm ei0 )(x, y), x (Bm e00 ) (x, y) = m x Bm e10 (x, y) = m i i=0 m-1 i=0 m x Bm e20 (x, y) = m i i=0 x x + jN = 1, i m = x, =x m-1 i 2 m i=0 m i i2 m = = m i(i - 1) +x m m-1 2 x[ - x] x +x = x2 + . m m m TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ Properties (iii) are proved in e same way. x x Now, we consider e approximation formula F = Bm F + Rm F. eorem 20 ([11]). If F (·, y) C[0, ] en 1+ 2 m y [0, h], where (F (·, y); ) is e modulus of continuity of e function F wi regard to e variable x. Moreover, if = 1/ m en (26) x (Rm F ) (x, y) 1+ h 2 1 (F (·, y); ), m x Proof. From e property (Bm e00 )(x, y) = 1, it follows at m x (Rm F ) (x, y) i=0 F (x, y) - F (i , y) . m Using e inequality F (x, y) - F (i one obtains m x |(Rm F )(x, y)| , y) m 1 x-i + 1 (F (·, y); ) m i=0 m i=0 1 x-i + 1 (F (·, y); ) m m 2 1/2 1 1+ = 1+ 1 x - i (F (·, y); ) x( - x) (F (·, y); ). m (y) Since max0x x(- x) = g 4 max0yh g 2 (y) = h2 , it follows 2 at max x( - x) = h , hence 4 1+ 2 m (F (·, y); ). Now, for = 1/ m, one obtains (26). INTERPOLATION OPERATORS eorem 21 ([11]). If F (·, y) C 2 [0, h] en = x[x - ] (2,0) F (, y), 2m h2 M20 F, 8m for [0, ] (x, y) , where Mij F = max F (i,j) (x, y) . x Proof. Taking into account at dex(Bm ) = 1, by Peano's eorem, it follows = K20 (x, y; s)F (2,0) (s, y)ds, where K20 (x, y; s) = (x - s)+ - i i=0 kernel K20 (x, y; ·) to e interval ( - 1) , , i.e., m m m K20 (x, y; ) = (x - s)+ - For a given {1, ..., m} one denotes by K20 (x, y; ·) e restriction of e i i= -s , m whence, x - s - - m i= m (i i= K20 (x, y; s) - s), m s<x s x. (i - s), m It follows at K20 (x, y; s) 0, for s x. For s < x we have m K20 (x, y; s) = x-s- i=0 i -s + m -1 i i=0 -s . m As, i i=0 = x - s, TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ it follows at -1 K20 (x, y; s) i=0 i So, K20 (x, y; ·)0 for any {1, . . . , m}, i.e., K20 (x, y; s)0, for s[0, ]. By e Mean Value eorem, one obtains = F (2,0) (, y) 0 K20 (x, y; s)ds, 0 . x(x - ) 2m Since, K20 (x, y; s)ds = max0x |x(x-)| 2m g 2 (y) 8m h2 8m , y [0, h] e conclusion follows. Remark 22. Analogous results are obtained for e remainder of e y y formula F = Bn F + Rn F. y x x y Let Pmn = Bm Bn , respectively, Qnm = Bn Bm be e products of e y x operators Bm Bn , i.e., m n (Pmn F ) (x, y) = qn,j i j i i , y F , f m m n m (Qnm F ) (x, y) = j i j j pm,i x, f (x) qn,j (x, y) F g f (x) , f (x) . n m n n Remark 23. e nodes of e operator Pmn , respectively, Qnm are given in Figure 11. Figure 11. e nodes for Pmn Qnm , for m = n = 4. INTERPOLATION OPERATORS eorem 24 ([11]). If F is a real-valued function defined on en: (i) (Pmn F )(V3 ) = F (V3 ), Pmn F = F, on 3 (ii) (Qnm F )(V3 ) = F (V3 ), Qnm F = F, on 3 . Proof. e proof follows from e properties x (Pmn F )(x, 0) = (Bm F )(x, 0), y (Pmn F )(0, y) = (Bn F )(0, y), (Pmn F )(x, f (x)) = F (x, f (x)), x, y [0, h] x y (Qnm F )(x, 0) = (Bm F )(x, 0), (Qnm F )(0, y) = (Bn F )(0, y), (Qnm F )(, y) = F (, y), x, y [0, h], which can be verified by a straightforward computation. Remark 25. e product operators Pmn Qnm interpolate e function F at e vertex (0, 0) on e side y = f (x) (or x = ). P Let us consider now e approximation formula F = Pmn F + Rmn F, P where Rmn is e corresponding remainder operator. eorem 26 ([11]). If F C( ) en 1 1 P Rmn F (x, y) (1 + h) F ; , , m n Proof. We have (x, y) 1 1 + + i , y m i , y m x- i m j f n i m n y- i , y m (F ; 1 , 2 ). TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ Since m n i , y m i , y m i , y m x- y- = 1, i m j i f n m x( - x) , m y(f (x) - y) , n it follows at 1+ h2 4 x( - x) 1 + m 2 y(f (x) - y) n (F ; 1 , 2 ). But - x y f (x) - y 1+ whence, (F ; 1 , 2 ) 1 h 1 h + 1 2 m 2 2 n 1 1 (1 + h) F ; , m n y x Next we consider e Boolean sums of e operators Bm Bn , i.e., x y x y x y Smn := Bm Bn = Bm + Bn - Bm Bn , y x y x y x Tnm := Bn Bm = Bn + Bm - Bn Bm . eorem 27 ([11]). If F is a real-valued function defined on en Smn F Proof. As, x (Pmn F ) (x, 0) = (Bm F ) (x, 0) , y (Pmn F ) (0, y) = (Bn F ) (0, y) , x y (Bm F ) (x, h - x) = (Bn F ) (x, h - x) = (Pmn F )(x, h - x) = F (x, h - x) Tnm F e conclusion follows. For e remainder of e Boolean sum approximation formula, F = S Smn F + Rmn F, we have e following result: INTERPOLATION OPERATORS eorem 28 ([11]). If F C( ) en S (Rmn F )(x, y) (1 + h 1 h 1 ) F (·, y); + (1 + ) F (x, ·); 2 2 m n 1 1 (x, y) . + (1 + h) F ; , , m n y x Proof. e identity F - Smn F = F - Bm F + F - Bn F - (F - Pmn F ) implies at S x y (Rm F ) (x, y) + (Rn F ( x, y) + (Rmn F )(x, y) e conclusion follows. Example 29. Consider e test function e triangle from Example x 17. In Figure 12 we plot e graphs of Bm F1 Pmn F1 , for m = 5, n = 6. x Bm F1 Pmn F1 x Figure 12. Graphs of Bm F1 Pmn F1 . Example 30. Table 2 contains some maximum approximation errors for F1 . y x Bm F1 Bn F1 Pmn F1 Qnm F1 Smn F1 Tnm F1 Max error 0.0525 0.0452 0.0858 0.0857 0.0095 0.0095 Table 2. e maximum approximation error for F1 . TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ 4. Bernstein-type operators on a triangle wi all curved sides ~ Let F be a real-valued function defined on (g2 (y), y), (g3 (y), y), respectively, (x, f1 (x)), (x, f3 (x)) be e points where e parallel lines to e coordinate axes, passing rough e point (x, y) , intersect e sides 1 , 2 3 . We consider e uniform partitions of e intervals [g2 (y), g3 (y)] [f1 (x), f3 (x)], x, y [0, h], x = {g2 (y) + i g3 (y)-g2 (y) i = m m y f3 (x)-f1 (x) j = 0, n} e Bernstein0, m}, respectively, n = {f1 (x) + j n y x type operators Bm Bn defined by (27) (28) wi x (Bm F ) (x, y) = i=0 n y (Bn F ) (x, y) = j=0 F qn,j (x, y) F g2 (y) + i g3 (y) - g2 (y) ,y , m f3 (x) - f1 (x) n , x, f1 (x) + j = qn,j (x, y) = m i n j y - f1 (x) f3 (x) - f1 (x) y - f1 (x) f3 (x) - f1 (x) n-j Remark 31. In Figures 13 14 we plot e points g2 (y) + i g3 (y) - g2 (y) , y , i = 0, m m respectively, x, f1 (x) + j f3 (x)-f1 (x) , j = 0, n, for x, n V1 V3 V2 V2 Figure 13. Points of x , for m = 4. m Figure 14. Points of n , for n = 4. y ~ eorem 32 ([12]). If F is a real-valued function defined on en: INTERPOLATION OPERATORS y x (i) Bm F = F on 2 3 , Bn F = F on 1 3 , x x (ii) (Bm ei0 ) (x, y) = xi , i = 0, 1; (Bm e20 ) (x, y) = x2 + x e ) (x, y) = y j (B x e )(x, y), i = 0, 1, 2; j N; (Bm ij m i0 y y (iii) (Bn e0j ) (x, y) = y j , j = 0, 1; (Bn e02 ) (x, y) = y 2 + y y (Bn eij ) (x, y) = xi (Bn e0j )(x, y), j = 0, 1, 2; i N. [x-g2 (y)][g3 (y)-x] ; m [y-f1 (x)][f3 (x)-y] ; n Proof. e interpolation properties (i) follow by e relations: pm,i (g2 (y), y) = respectively by qn,j (x, f1 (x)) = 1, for j = 0, 0, for j > 0, qn,j (x, f3 (x)) = 0, for j < n, 1, for j = n. 1, for i = 0, 0, for i > 0, pm,i (g3 (y), y) = 0, 1, for i < m, for i = m, Regarding e properties (ii), we have x x (Bm eij ) (x, y) = y j (Bm ei0 )(x, y), m x (Bm e00 ) (x, y) = i=0 m x Bm e10 (x, y) = jN = 1, g2 (y) + i i=0 g3 (y) - g2 (y) m m! x - g2 (y) i!(m - i)! g3 (y) - g2 (y) = g2 (y) + [g3 (y) - g2 (y)] x - g2 (y) · g3 (y) - g2 (y) = g2 (y) + [x - g2 (y)] · i=0 i m m-1 i m-1 i=0 -1 = x, TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ m x Bm e20 (x, y) i=0 g3 (y) - g2 (y) g2 (y) + i m 2 =g2 (y) + 2g2 (y)[x - g2 (y)] + + · (m - 1)[g3 (y) - g2 (y)]2 m i=2 x - g2 (y) x - g2 (y) g3 (y) - g2 (y) g3 (y) - g2 (y) m-1 (2m - 1)g2 (y) + g3 (y) [x - g2 (y)]2 + = m m [x - g2 (y)][g3 (y) - x] 2 2 · [x - g2 (y)] + g2 (y) = x + . m Properties (iii) are proved in e same way. [g3 (y) - g2 (y)][x - g2 (y)] m m (m - 2)! (i - 2)!(m - i)! y x Remark 33. e interpolation properties of Bm F Bn F are illustrated in Figures 15 16. V1 V1 V3 V2 Figure 15. Interpolation x set of Bm F Figure 16. Interpolation y set of Bn F. x x Now, we consider e approximation formula F = Bm F + Rm F. eorem 34 ([12]). If F (·, y) C[g2 (y), g3 (y)], y [0, h], en 1+ g3 (y) - g2 (y) 2 m y [0, h], if M = max0yh |g3 (y) - g2 (y)| en we have 1+ M 2 m INTERPOLATION OPERATORS Moreover, if = 1/ m en (29) x (Rm F ) (x, y) 1+ M 2 1 (F (·, y); ), m x Proof. As (Bm e00 )(x, y) = 1, it follows at m x (Rm F ) (x, y) m i=0 F (x, y) - F g2 (y) + i g3 (y) - g2 (y) ,y m i=0 g3 (y) - g2 (y) 1 ] + 1 (F (·, y); ) x - [g2 (y) + i m g3 (y) - g2 (y) ) m 2 1/2 1+ 1+ m i=0 x - (g2 (y) + i (F (·, y); ) [x - g2 (y)][g3 (y) - x] (F (·, y); ). m [g3 (y)-g2 (y)]2 , 4 Since maxg2 (y)xg3 (y) [x - g2 (y)][g3 (y) - x] = 1 + one obtains g3 (y) - g2 (y) (F (·, y); ) 2 m e proof follows. Remark 35. Analogous results can be obtained for e remainder of y y e formula F = Bn F + Rn F. y x x y Let Pmn = Bm Bn , respectively, Qnm = Bn Bm be e products of e y x operators Bm Bn , i.e., m n (Pmn F ) (x, y)= qn,j (xi , y) F xi , f1 (xi )+j pm,i (x, yj ) qn,j (x, y) F g2 (yj )+i f3 (xi ) - f1 (xi ) , n (Qnm F ) (x, y)= g3 (yj ) - g2 (yj ) , yj , m wi xi = g2 (y) + i g3 (y)-g2 (y) , yj = f1 (x) + j f3 (x)-f1 (x) . m n Remark 36. e nodes of e operators Pmn , respectively Qnm are given in Figures 17 18, where f1 g2 f3 are arcs of e circles (C1) , 15 2 15 1 2 (x - 2 ) + (y + 2 ) = 4, (C2) (x + 2 )2 + (y - 1 )2 = 4, respectively, (C3) 2 x2 + y 2 = 1. TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ V1 V3 V2 Figure 17. e nodes of Pmn , for m = n = 4. Figure 18. e nodes of Qnm , for m = n = 4. ~ eorem 37 ([12]). If F is a real-valued function defined on en: (i) (Pmn F )(V3 ) = F (V3 ), Pmn F = F, on 3 (ii) (Qnm F )(V3 ) = F (V3 ), Qnm F = F, on 3 . Proof. e proof follows from e properties: x (Pmn F )(x, f1 (x)) = (Bm F )(x, f1 (x)), y (Pmn F )(g2 (y), y) = (Bn F )(g2 (y), y), (Pmn F )(x, f3 (x)) = F (x, f3 (x)), x (Qnm F )(x, f1 (x)) = (Bm F )(x, f1 (x)), y (Qnm F )(g2 (y), y) = (Bn F )(g2 (y), y), x, y [0, h] (Qnm F )(g3 (y), y) = F (g3 (y), y), x, y [0, h], which can be verified by a straightforward computation. e interpolation properties of Pmn F Qnm F are illustrated in Figure 19. V3 V2 Figure 19. Interpolation set for Pmn F Qnm F INTERPOLATION OPERATORS For e remainder of e product approximation formula F = Pmn F + P Rmn F we have: ~ eorem 38 ([12]). If F C( ) en P Rmn F (x, y) 1+ 21 m 22 n (F ; 1 , 2 ) , ~ (x, y) , respectively P Rmn F (x, y) 1+ N M + 2 2 1 1 F; , m n ~ (x, y) , where M = max0yh |g3 (y) - g2 (y)| N = max0xh |f3 (x) - f2 (x)|. Proof. We have (xi , y) |x - xi | f3 (xi ) - f1 (xi ) ) n + + n (xi , y) y - (f1 (xi ) + j (xi , y) (F ; 1 , 2 ) n qn,j (xi , y) j=0 m [x - g2 (y)][g3 (y) - x] m [y - f1 (x)][f3 (x) - y] + 1 (F ; 1 , 2 ), n i=0 furer, [x - g2 (y)][g3 (y) - x] m [x - f1 (x)][f3 (x) - y] + 1 (F ; 1 , 2 ) n 1 g3 (y) - g2 (y) 1 f3 (x) - f1 (x) 1+ + + 1 (F ; 1 , 2 ) 1 2 m 2 2 n 1 M 1 N + + 1 (F ; 1 , 2 ), 1+ 1 2 m 2 2 n TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ e proof follows. Q For e remainder Rnm F = F - Qnm F we can also obtain Q Rnm F (x, y) 1+ N M + 2 2 1 1 F; , m n ~ (x, y) . y x We consider e Boolean sums of e operators Bm Bn , i.e., x y x y x y Smn := Bm Bn = Bm + Bn - Bm Bn , y x y x y x Tnm := Bn Bm = Bn + Bm - Bn Bm . ~ eorem 39 ([12]). If F is a real-valued function defined on en Smn F Proof. As x (Pmn F ) (x, f1 (x)) = (Bm F ) (x, f1 (x)) , y (Pmn F ) (g2 (y), y) = (Bn F ) (g2 (y), y) , Tnm F (Pmn F ) (x, f3 (x)) = F (x, f3 (x)) , e proof follows. For e remainder of e Boolean sum approximation formula, F = S Smn F + Rmn F, we have e following result: ~ eorem 40 ([12]). If F C( ) en S (Rmn F )(x, y) (1 + 1 N 1 M ) F (·, y); + (1 + ) F (x, ·); 2 m 2 n M N 1 1 + (1 + + ) F ; , , (x, y) . 2 2 m n y x Proof. e identity F -Smn F = F -Bm F +F -Bn F -(F -Pmn F ) imS F )(x, y) (Rx F ) (x, y) + (Ry F ( x, y) + (RP F )(x, y) plies at (Rmn n m mn e proof follows. T An analogous inequality can be obtained for e error Rnm F -Tnm F. Example 41. Consider e test function from Example 17. In Figure 20 x we plot e graphs of Bm F1 Smn F1 , wi h = 1, m = 5, n = 6, 15 15 f1 , g2 , f3 : [0, 1] [0, 1], f1 (x) = - 2 - 4 - (x - 0.5)2 , g2 (y) = - 2 - 4 - (y - 0.5)2 f3 (x) = 1 - x2 . INTERPOLATION OPERATORS x Figure 20. Graphs of Bm F1 Smn F1 . Example 42. Table 3 contains e maximum approximation errors. y x Bm F1 Bn F1 Pmn F1 Qnm F1 Smn F1 Tnm F1 Max error 0.0847 0.0684 0.1269 0.1266 0.0239 0.0240 Table 3. e maximum approximation error for F1 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Alexandru Ioan Cuza University - Mathematics de Gruyter

Interpolation Operators on Some Triangles with Curved Sides

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de Gruyter
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10.2478/aicu-2013-0028
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Abstract

is paper contains a survey regarding interpolation Bernstein-type operators defined on triangles having one or all curved sides; we consider as well some of e product Boolean sum operators. We study e interpolation properties, e orders of accuracy e remainders of e generated approximation formulas. Maematics Subject Classification 2010: 41A05, 41A25, 41A80. Key words: product boolean sum operators, triangles tetrahedrons wi curved sides, interpolation operators, Bernstein-type operators, remainders. 1. Introduction e aim of is survey is to present some interpolation Bernsteintype operators for functions defined on triangles wi one or all curved sides (see [11], [12], [14], [15]). ey come as an extension of e corresponding operators for functions defined on triangles wi all straight sides (see, e.g., [3]-[6], [8]-[10], [13], [23], [24], [26]-[29], [32]). e operators defined on domains wi curved sides permit essential boundary conditions to be satisfied exactly. Such operators can be used in construction of surfaces which satisfy some given conditions (see, e.g., [17], [18]), in finite element meod for differential equation problems (Lagrange operators for Dirichlet boundary conditions, Birkhoff operators for Neumann boundary conditions Hermite operators for Robin boundary conditions) (see, e.g., [19], [25], [26], [35]) in numerical integration of functions (see, e.g., [16]). We study ese operators especially from e eoretical point of view. e idea came from e paper of Barnhill Gregory [4], where ere TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ is considered a triangle wi one curved side ere are used Lagrange projectors on e straight sides, Taylor projector on e curved side. Such operators were also studied in many oer papers in connection wi eir applications in computer aided geometric design (see, e.g., [1], [2], [5]) in finite element analysis (see, e.g., [1], [7], [20], [22], [23], [24], [35]). We study ree main aspects of e constructed operators: 1) e interpolation properties; 2) e orders of accuracy; 3) e remainders of e corresponding interpolation formulas. e order of accuracy of an interpolation operator P is given by e degree of exactness (dex(P )), respectively by e precision set (pres(P )). Recall at dex(P ) = r if P f = f, for f Pr ere exists g Pr+1 such at P g = g, where Pm denotes e space of e polynomials in 2 variables of global degree at most m. e precision set of an interpolation operator is e set of monomials for which e interpolant is exact ([4]). e characteristics 1) 2) can be verified by a straightforward computation. e remainders of e interpolation formulas will be studied using e Peano's eorem for e functions from a Sard-type space (see, e.g., [31]). e Sard-type space, denoted by Bpq (a, c), (p, q N, p + q = m), is e space of e functions f : D R, D = [a, b] × [c, d] satisfying f (p,q) C (D) ; f (m-j,j)(·, c) C [a, b] , j < q; f (i,) (a, ·) C [c, d] , i < p. ~ Given h > 0, denote by e triangle having e vertices V1 = (h, 0), V2 = (0, h) V3 = (0, 0), two straight sides 1 , 2 , along e coordinate axes, e ird side 3 (opposite to e vertex V3 ), which is defined by e one-to-one functions f g, where g is e inverse of e function f, i.e., y = f (x) x = , wi f (0) = g(0) = h (see Figure 1). V2 1 (x,y) (x,f(x)) (0,y) V3 V1 ~ Figure 1: Triangle . ~ ere is no restriction in considering is stard triangle , since any triangle wi one curved side can be obtained from is stard triangle by an affine transformation which preserves e form order of accuracy of e interpolant ([4]). INTERPOLATION OPERATORS In Section 2 we study Lagrange, Hermite Birkhoff interpolation ~ operators, as well as eir product Boolean sum on . In Section 3 we present some Bernstein-type operators togeer wi eir product ~ Boolean sum for e same triangle . Section 4 is dedicated to Bernstein~ type operators defined on a triangle wi all curved sides, denoted by . is triangle has e vertices V1 = (0, h), V2 = (h, 0) V3 = (0, 0), e ree curved sides 1 , 2 (along e coordinate axes) 3 (opposite to e vertex V3 ). We define 1 by (x, f1 (x)), wi f1 (0) = f1 (h) = 0, f1 (x) 0, for x [0, h]; 2 defined by (g2 (y), y), wi g2 (0) = g2 (h) = 0, g2 (y) 0, for y [0, h] 3 defined by e one-to-one functions f3 g3 , where g3 is e inverse of e function f3 , i.e., y = f3 (x) x = g3 (y), wi x, y [0, h] f3 (0) = g3 (0) = h, h R+ , (see Figure 2). For example, f1 g2 can be convex functions. V1 (x,f3(x)) (g2(y),y) (g (y),y) V3 (x,f (x)) V2 ~ Figure 2. Triangle . 2. Interpolation operators on a triangle wi one curved side 2.1. Lagrange-type operators ~ Suppose at F is a real-valued function defined on . Let L1 , L2 L3 be e interpolation operators defined by - x x F (0, y) + F (, y), f (x) - y y (L2 F )(x, y) = F (x, 0) + F (x, f (x)), f (x) f (x) x y (L3 F )(x, y) = F (x + y, 0) + F (0, x + y). x+y x+y (L1 F )(x, y) = (1) TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ 1) Each of e operators L1 , L2 L3 interpolates e function F along ~ two sides of e triangle : (L1 F )(0, y) = F (0, y), (L1 F )(, y) = F (, y), y [0, h], (L2 F )(x, 0) = F (x, 0), (L2 F )(x, f (x)) = F (x, f (x)), x [0, h], (L3 F )(x + y, 0) = F (x + y, 0), (L3 F )(0, x + y) = F (0, x + y), x, y [0, h], properties illustrated in Figure 3. e bold sides points indicate e interpolation domains. V2 V2 (x,f(x) V2 (0,y) L1 (0,x+y) L3 V3 V1 V1 V3 (x+y,0) Figure 3. e interpolation domains for L1 , L2 L3 . 2) e orders of accuracy: (2) dex(Li ) = 1, i = 1, 2, 3, pres(L2 ) = {1, xi , y, i N }, pres(L1 ) = {1, x, y j , j N }, pres(L3 ) = {1, x, y}. L 3) Regarding e remainders Ri F, i = 1, 2, 3, of e interpolation forL F, mulas F = Li F + Ri i = 1, 2, 3, we have: eorem 1 ([14]). If F B11 (0, 0) en L (R1 F )(x, y) = x[x - ] (2,0) F (, 0) 2 xy[ - x] (1 , 1 ) - F (1,1) (2 , 2 ) , wi [0, h], (1 , 1 ) [0, x] × [0, y] (2 , 2 ) [x, ] × [0, y], respectively (3) where · L (R1 F )(x, y) h2 8 F (2,0) (·, 0) denotes e Chebyshev norm. INTERPOLATION OPERATORS Proof. From (2) we have dex(L1 ) = 1 applying e Peano's eorem we get L (R1 F )(x, y) = h 0 ~ K20 (x, y, s)F (2,0) (s, 0)ds K11 (x, y, s, t)F (1,1) (s, t)dsdt, (4) wi e Peano's kernels given by (5) K20 (x, y, s) = (x - s)+ - x ( - s)+ , x K11 (x, y, s, t) = (y - t)0 [(x - s)0 - ( - s)0 ]. + + + As, K20 (x, y, s) 0, K11 (x, y, s, t) 0, K11 (x, y, s, t) 0, K11 (x, y, s, t) = 0, s [0, h], (s, t) [0, x] × [0, y], (s, t) [x, ] × [0, y], (s, t) D1 × D2 , (6) wi D1 D2 illustrated in Figure 4, by e Mean Value eorem we obtain L (R1 F )(x, y) = 20 (x, y)F (2,0) (, 0) + 1 (x, y)F (1,1) (1 , 1 ) 11 + 2 (x, y)F (1,1) (2 , 2 ), 11 wi [0, h], (1 , 1 ) [0, x] × [0, y], (2 , 2 ) [x, ] × [0, y], 20 (x, y) = (7) 1 (x, y) = 11 2 (x, y) = 11 0 y 0 x(x - ) , 2 x xy[ - x] K11 (x, y, s, t)dsdt = , 0 K11 (x, y, s, t)dsdt = - xy[ - x] . relation (3) follows. As |20 (x, y)| h2 8 , 1 (x, y) 11 2 (x, y) 11 TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ D2 (0,y) 0.6 D1 0.8 0.9 (,0) 1 Figure 4: e sign of e kernel K11 . Remark 2. Analogous formulas can be obtained for e remainders L L R2 F R3 F. L Let Pij be e product of e operators Li Lj , i.e., Pij = Li Lj , i, j = 1, 2, 3, i = j. We have h - y - x y - x x F (0, 0) + F (0, h) + F (, y), h h - x L (P13 F )(x, y) = F (0, y) x + [F (y + , 0) + yF (0, y + )], [y + ] f (x) - y L (P23 F )(x, y) = F (x, 0) f (x) y + [xF (x + f (x), 0) + f (x)F (0, x + f (x))]. f (x)[x + f (x)] L (P12 F )(x, y) = L L 1) e interpolation properties: P12 F = F, on 3 V3 ; P13 F = F, on L 1 V1 ; P23 F = F, on 2 V2 . L Remark 3. e operator Pij has e same interpolation properties as L e operator Pji , i, j = 1, 2, 3, i = j. ese properties are illustrated in Figure 5. V2 V2 V2 L2 L L L V3 V1 V3 V3 V1 L L L Figure 5. e interpolation domains for P12 , P13 P23 . INTERPOLATION OPERATORS L L 2) e orders of accuracy: dex(Pij ) = 1, pres(Pij ) = {1, x, y}, i, j = 1, 2, 3, i = j. P L 3) For e remainders Rij F, of e interpolation formulas F = Pij F + LP Rij F, i, j = 1, 2, 3, i = j, we have: eorem 4 ([14]). If F B11 (0, 0) en LP (R12 F )(x, y) = x[x - ] (2,0) y(y - h)[ - x] (0,2) F (, 0) + F (0, ) 2 2 xy[ - x] (1,1) [F (1 , 1 ) - F (1,1) (2 , 2 )], + wi , [0, h], (1 , 1 ) [0, x] × [0, y] (2 , 2 ) [x, ] × [0, y], respectively (8) LP (R12 F )(x, y) h2 8 F (2,0) (·, 0) + F (0,2) (0, ·) L Proof. By dex(P12 ) = 1, applying Peano's eorem we get at LP (R12 F )(x, y) = h 0 K20 (x, y, s)F (2,0) (s, 0)ds + 0 ~ K02 (x, y, t)F (0,2) (0, t)dt (9) K11 (x, y, s, t)F (1,1) (s, t)dsdt, wi e Peano's kernels x [ - s]+ , - x y(h - t) K02 (x, y, t) = (y - t)+ - , h x K11 (x, y, s, t) = (y - t)0 {(x - s)0 - [ - s]0 }. + + + K20 (x, y, s) = (x - s)+ - We notice at e Peano's kernels K20 K11 are e same as e kernels given in (5). erefore, eir sign is given in (6) we have K02 (x, y, t) 0, for t [0, h]. By e Mean Value eorem we obtain LP (R12 F )(x, y) =20 (x, y)F (2,0) (, 0) + 02 (x, y)F (0,2) (0, ) + 1 (x, y)F (1,1) (1 , 1 ) + 2 (x, y)F (1,1) (2 , 2 ), 11 11 TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ for , [0, h], (1 , 1 ) [0, x] × [0, y], (2 , 2 ) [x, ] × [0, y], wi 20 , 1 , 2 given in (7) 11 11 02 (x, y) = 0 h2 K02 (x, y, t)dt = h2 8 , y(y - h))[ - x] . 2 1 (x, y) 11 We have |20 (x, y)| 8 , |02 (x, y)| so e relation (8) follows. 2 (x, y) 11 Remark 5. Analogous formulas can be obtained for e remainders LP LP R23 F R13 F. L L Let Sij be e Boolean sum of e operators Li Lj , i.e., Sij = Li Lj = Li + Lj - Li Lj , i, j = 1, 2, 3, i < j (see, e.g., [21]). We have L S12 F (x, y) = - x f (x) - y y F (0, y) + F (x, 0) + F (x, f (x)) f (x) f (x) - x h - y y - F (0, 0) + F (0, h) , h h x x y L S13 F (x, y) = F (g (y) , y) + F (x + y, 0) + F (0, x + y)- x+y x+y y - F (y + , 0) + F (0, y + ) , y + y + y x y F (x, f (x)) + F (x + y, 0) + F (0, x + y)- f (x) x+y x+y y y f (x) - F (x + f (x), 0) + F (0, x + f (x)) . f (x) x + f (x) x + f (x) L S23 F (x, y) = L ~ 1) e interpolation properties: Sij F = F, i, j = 1, 2, 3, i < j, on . 2) e orders of accuracy: L L L dex(S12 ) = 1, dex(S13 ) = dex(S23 ) = 2, (10) L pres(S12 ) = {1, y, xy, xk , k N }, L pres(S23 ) = {1, x, y, x2 , y 2 , xk y, k N }. LS L 3) For e remainders Rij F, of e interpolation formulas F = Sij F + LS Rij F, i, j = 1, 2, 3, i < j, we have: L pres(S13 ) = {1, x, y, x2 , y 2 , xy k , k N }, INTERPOLATION OPERATORS eorem 6 ([14]). If F B11 (0, 0) en LS (R12 F )(x, y) = h 0 ~ K02 (x, y, t)F (0,2) (0, t)dt K11 (x, y, s, t)F (1,1) (s, t)dsdt, (11) wi e Peano's kernels x [ - x]y x (y - t)+ - [f (x) - t]+ + (h - t), f (x) h y K11 (x, y, s, t) = (x - s)0 {(y - t)0 - [f (x) - t]0 }. + + + f (x) K02 (x, y, t) = Furermore, LS (R12 F )(x, y) F (2,0) (·, 0) h 0 ~ |K02 (x, y, t)|dt |K11 (x, y, s, t)|dsdt. Proof. e proof follows directly by Peano's eorem, taking into L account at dex(S12 ) = 1. eorem 7 ([14]). If F B12 (0, 0) en LS (R13 F )(x, y) = h h 0 (12) ~ wi e Peano's kernels K30 (x, y, s) = x[ - s]2 x(x + y - s)2 x[y + - s]2 (x - s)2 + + + + - - + , 2 2 2(x + y) 2(y + ) xy[ - s]+ K21 (x, y, s) = y(x - s)+ - , [ - x](y - t)2 y(x + y - t)2 xy(y + - t)2 + + + K03 (x, y, t) = - + 2 , 2 2(x + y) g (y)[y + ] x K12 (x, y, s, t) = (y - t)+ {(x - s)0 - [ - s]0 }. + + TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ Furermore, LS (R13 F )(x, y) F (3,0) (·, 0) h 0 h 0 h 0 |K30 (x, y, s)|ds |K21 (x, y, s)|ds |K03 (x, y, t)|dt + F (2,1) (·, 0) (13) + F (0,3) (0, ·) + F (1,2) ~ |K12 (x, y, s, t)|dsdt. L Proof. From (10) it follows at dex(S13 ) = 2 applying Peano's eorem we get (12) e inequality (13). LS Remark 8. An analogous formula can be obtained for R23 F. 2.2. Hermite-type operators ~ Suppose at e real valued function F is defined on e triangle (1,0) F (0,1) on e side . We it possesses e partial derivatives F 3 consider e operators H1 H2 defined by (H1 F )(x, y) = (14) x[2 - x] [x - ]2 F (0, y) + F (, y) 2 (y) g g 2 (y) x[x - ] (1,0) + F (, y), [y - f (x)]2 y[2f (x) - y] (H2 F )(x, y) = F (x, 0) + F (x, f (x)) 2 (x) f f 2 (x) y[y - f (x)] (0,1) + F (x, f (x)). f (x) 1) e interpolation properties: H1 F = F, on 1 3 ; (H1 F )(1,0) = F (1,0) , on 3 ese interpolation properties are illustrated in Figure 6. H2 F = F, on 2 3 ; (H2 F )(0,1) = F (0,1) , on 3 . V2 INTERPOLATION OPERATORS H1 V1 V3 Figure 6. e interpolation domains for H1 H2 . 2) e orders of accuracy: dex(H1 ) = dex(H2 ) = 2, (15) pres(H1 ) = {1, x, y, x2 , y 2 , xy n , n N }, pres(H2 ) = {1, x, y, x2 , y 2 , xn y, n N }. H Ri F , H 3) e interpolation formulas are F = Hi F + Ri F, i = 1, 2, where i = 1, 2 are e remainder terms, for which we have: eorem 9 ([14]). If F B12 (0, 0) en e following inequality holds H (R1 F )(x, y) (16) furer, x[ - x]2 (3,0) xy[ - x]2 (2,1) F (·, 0) + F (·, 0) 6 2 - x xy 2 [ - x][3 - 2x] (1,2) + F (·, ·) , 2g 2 (y) H (R1 F )(x, y) (17) 2h3 (3,0) xy[ - x]2 (2,1) F (·, 0) + F (·, 0) 81 2 - x xy 2 [ - x][3 - 2x] (1,2) + F (·, ·) . 2g 2 (y) Proof. By (15) it follows at dex(H1 ) = 2, erefore by Peano's eorem we get H (R1 F )(x, y) = h 0 0 ~ (18) TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ wi (x - s)2 x[2 - x] ( - s)2 + + - 2 g 2 (y) 2 x[x - ] ( - s)+ , - xy[2 - x] K21 (x, y, s) = y(x - s)+ - ( - s)+ g 2 (y) xy[x - ] - ( - s)0 , + K12 (x, y, s, t) = (y - t)+ [(x - s)0 + x[2 - x] - ( - s)0 ]. + g 2 (y) K30 (x, y, s) = We have 2 2 s [ - x] 0, 2g 2 (y) K30 (x, y, s) = x[ - s] [(s - x) + s[ - x] 0, 2g 2 (y) 0, - 1 sy[ - x]2 0, g 2 (y) xy - [g 2 (y) - s(2 - x)] 0, g 2 (y) K21 (x, y, s) = xy - 2 [g 2 (y) - s(2 - x)] 0, g (y) 0, 2 [ - x] 0, g 2 (y) K12 (x, y, s, t) = (y - t)+ x[x - 2] 0, g 2 (y) 0, s [0, x) s [x, ) s [, h], s [0, x) g (y) s [x, 2-x ) g (y) s [ 2-x , ) s [, h], (s, t) [0, x) × [0, y) (s, t) [x, ) × [0, y) (s, t) D1 D2 , INTERPOLATION OPERATORS wi domains D1 , D2 e sign of K12 as in Figure 4. We obtain at H (R1 F )(x, y) h 0 ~ h 0 F (3,0) (·, 0) + F (1,2) (·, ·) K30 (x, y, s)ds + F (2,1) (·, 0) |K12 (x, y, s, t)| dsdt, |K21 (x, y, s)| ds whence, after some computation we get (16), furer we obtain (17). Remark 10. An analogous formula can be obtained for e remainder H R2 F. e product of e operators H1 H2 is given by H (P12 F )(x, y) = [x - ]2 (y - h)2 F (0, 0) g 2 (y) h2 y(2h - y) y(y - h) (0,1) + F (0, h) + F (0, h) 2 h h x[2 - x] x[x - ] (1,0) + F (, y) + F (, y). 2 (y) g 1) e interpolation properties: H P12 F = F, on V3 3 , H H (P12 F )(1,0) = F (1,0) , (P12 F )(0,1) = F (0,1) , on 3 . e interpolation properties are illustrated in Figure 7. V2 V3 V1 H Figure 7. e interpolation domain for P12 F . 2) e orders of accuracy: (19) H P12 F H H dex(P12 ) = 2, pres(P12 ) = {1, x, y, x2 , xy, y 2 , x2 y, xy 2 }. For e remainder of e corresponding interpolation formula, F = HP + R12 F, we have: TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ eorem 11 ([14]). If F B12 (0, 0) en e following inequality holds HP (R12 F )(x, y) (20) x[ - x]2 (3,0) F (·, 0) 6 xy[ - x]2 (2,1) F (·, 0) + 2 - x y[ - x]2 (h - y) (0,3) + F (0, ·) 6g 2 (y) xy[ - x][3 - 2x] (1,2) F (·, ·) + g 2 (y) H Proof. By (19) it follows at dex(P12 ) = 2 applying Peano's eorem we get HP (R12 F )(x, y) = h h 0 ~ wi (x - s)2 x[2 - x] ( - s)2 x[x - ] + + - - ( - s)+ , 2 (y) 2 g 2 xy[2 - x] xy[x - ] K21 (x, y, s) =y(x - s)+ - ( - s)+ - , 2 (y) g (y - t)2 x[2 - x] (y - t)2 + + K03 (x, y, t) = - 2 g 2 (y) 2 [x - ]2 y(2h - y) (h - t)2 y(y - h)(h - t) - [ + ] g 2 (y) h2 2 h x[2 - x] K12 (x, y, s, t) =(y - t)+ [(x - s)0 - ]. + g 2 (y) K30 (x, y, s)= We have 2 2 s [ - x] 0, 2g 2 (y) K30 (x, y, s) = - x[ - s] [(x - s)+(x - )s]0, 2g 2 (y) 0, s [0, x) s [x, ) s [, h], INTERPOLATION OPERATORS We obtain at sy s [0, x) - 2 [ - x]2 0, g (y) xy g 2 (y) - [( - s) + s(x - )] 0, s [x, 2-x ) g 2 (y) K21 (x, y, s)= g 2 (y) - xy [(-s)+s(x-)]0, s [ 2-x , ) 2 g (y) 0, s [, h], 2 [ - x] 2 t (h - y)2 0, t [0, y) 2g 2 (y) K03 (x, y, t) = [x - ]2 (h - t)y - [h(y - t) + t(y - h)] 0, t [y, h] 2h2 g 2 (y) 2 (y - t) [ - x] 0, (s, t) [0, x) × [0, y) g 2 (y) K12 (x, y, s, t) = (y - t) x[x - 2] 0, (s, t) [x, ) × [0, y) g 2 (y) ~ 0, (s, t) ([0, h] × [y, h]) . HP (R12 F )(x, y) F (3,0) (·, 0) h K30 (x, y, s)ds 0 h + F (2,1) (·, 0) + F (1,2) (·, ·) h 0 |K21 (x, y, s)| ds + F (0,3) (0, ·) |K12 (x, y, s, t)| dsdt, K03 (x, y, t)dt ~ whence, after some computation, we get (20). e Boolean sum of e operators H1 H2 is given by H (S12 F )(x, y) = [x - ]2 [y - f (x)]2 F (0, y) + F (x, 0) g 2 (y) f 2 (x) y[2f (x) - y] y[y - f (x)] (0,1) + F (x, f (x)) + F (x, f (x)) 2 (x) f f (x) [x - ]2 (y - h)2 y(2h - y) y(y - h) (0,1) - F (0, 0) + F (0, h) + F (0, h) . 2 (y) 2 2 g h h h 1) e interpolation properties: H ~ S12 F = F, on H F )(1,0) = F (1,0) , H (S12 (S12 F )(0,1) = F (0,1) , on 3 . TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ xn y, n H H 2) e orders of accuracy: dex(S12 ) = 2, pres(S12 ) = {1, x, y, x2 , y 2 , }. N H HS For e remainder of e interpolation formula, F = S12 F + R12 F, we have: eorem 12 ([14]). If F B12 (0, 0) en e following inequality holds HS (R12 F )(x, y) F (0,3) (0, ·) h 0 |K03 (x, y, t)|dt ~ + F (1,2) (·, ·) |K12 (x, y, s, t)| dsdt. H Proof. By (15) it follows at dex(S12 ) = 2 erefore by Peano's eorem we get h 0 h 0 HS (R12 F )(x, y) = h ~ wi K30 (x, y, s) =K21 (x, y, s) = 0, (y - t)2 g 2 (y) - (x - )2 + 2 g 2 (y) y(2f (x) - y) (f (x) - t)2 y(y - f (x))(f (x) - t)+ + - - f 2 (x) 2 f (x) 2 y(2h - y)(h - t)2 (x - ) y(y - h)(h - t) + + , 2 (y) 2 g 2h h y[2f (x) - y] K12 (x, y, s, t) =(x - s)0 [(y - t)+ - (f (x) - t)+ + f 2 (x) y(y - f (x)) - (f (x) - t)0 ]. + f (x) K03 (x, y, t) = 17 We have INTERPOLATION OPERATORS y(2f (x) - y) (f (x) - t)2 (y - t)2 g 2 (y) - (x - )2 - 2 g 2 (y) f 2 (x) 2 (x - )2 y(2h - y)(h - t)2 y(y - h)(h - t) + + 0, g 2 (y) 2h2 h t [0, y), y(2f (x) - y) (f (x) - t)2 - f 2 (x) 2 K03 (x, y, t) = (x - )2 y(2h - y)(h - t)2 y(y - h)(h - t) + + , g 2 (y) 2h2 h t [y, f (x)) (x - )2 y(2h - y)(h - t)2 y(y - h)(h - t) + 0, g 2 (y) 2h2 h t [f (x), h], K12 (x, y, s, t) (y - t) - y[2f (x) - y] (f (x) - t) - y(y - f (x)) 0, f 2 (x) f (x) (s, t) [0, x) × [0, y), y[2f (x) - y] y(y - f (x)) - (f (x) - t) - 0, f 2 (x) f (x) f 2 (x) (s, t) [0, x) × [y, ), 2f (x) - y = (y - t)+ · y[2f (x) - y] y(y - f (x)) - (f (x) - t) - 0, 2 (x) f f (x) f 2 (x) (s, t) [0, x) × [ , f (x)) 2f (x) - y 0, ~ (s, t) ([x, h] × [0, f (x)] [0, x] × [f (x), h]) . HS (R12 F )(x, y) F (0,3) (0, ·) h 0 We obtain at |K03 (x, y, t)|dt ~ + F (1,2) (·, ·) |K12 (x, y, s, t)| dsdt. TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ 2.3. Birkhoff-type operators In is section we give some examples of operators which interpolate e ~ given function F : R on a side of e triangle its first partial ~ derivatives on anoer side of , respectively. ~ First, we suppose at e function F : R has e partial deriva(1,0) F (0,1) on e side . We consider e Birkhoff-type operatives F 3 tors B1 B2 defined respectively by (B1 F ) (x, y) = F (0, y) + xF (1,0) (g (y) , y) , (B2 F ) (x, y) = F (x, 0) + yF (0,1) (x, f (x)) . 1) e interpolation properties: B1 F = F on 1 (B1 F )(1,0) = F (1,0) on 3 , B2 F = F on 2 (B2 F )(0,1) = F (0,1) on 3 . ese interpolation properties are illustrated in Figure 8. V2 V2 B1 B2 V1 V3 V1 Figure 8. e interpolation domains for B1 B2 . 2) e orders of accuracy: (21) dex (B1 ) = dex (B2 ) = 1, pres (B2 ) = 1, y, xi , i N . pres (B1 ) = 1, x, y j , j N , B 3) For e remainders of e interpolation formulas F = B1 F + R1 F B F we have: F = B2 F + R2 eorem 13 ([14]). If F B11 (0, 0) en x[x - 2] (2,0) F (, 0) + xyF (1,1) (1 , ), 2 ~ wi [0, h], (1 , ) , respectively B (R1 F )(x, y) = (22) B (R1 F )(x, y) h2 (2,0) F (·, 0) 2 h2 (1,1) F (·, ·) 4 INTERPOLATION OPERATORS Proof. By (21) it follows at dex(B1 ) = 1, applying Peano's eorem we obtain B R1 F (x, y) = h 0 ~ K20 (x, y, s) F (2,0) (s, 0) ds K11 (x, y, s, t) F (1,1) (s, t) dsdt + wi K11 (x, y, s, t) = (x - s)0 (y - t)0 . + + We have K20 (x, y, s) = (x - s)+ - x [ - s]0 , + K20 (x, y, s) 0, s [0, h] K11 (x, y, s, t) > 0, (s, t) [0, x] × [0, y] ~ K11 (x, y, s, t) = 0, (s, t) (([x, h] × [0, y]) ([0, x] × [y, h])) , by e Mean Value eorem we obtain B (R1 F )(x, y) = 20 (x, y)F (2,0) (, 0) + 11 (x, y)F (1,1) (1 , ), ~ wi [0, h], (1 , ) , 20 (x, y) = 0 y x 0 x[x - 2] , 2 11 (x, y) = As, |20 (x, y)| B R2 F. h2 2 , K11 (x, y, s, t)dsdt = xy. |11 (x, y)| e relation (22) follows. Remark 14. Analogous formula can be obtained for e remainder ~ Next, we suppose at e function F : R admits e partial (1,0) on F (0,1) on . derivatives F 1 2 We consider e Birkhoff-type operators B3 B4 defined by (B3 F ) (x, y) = F (, y) + [x - ]F (1,0) (0, y) , (B4 F ) (x, y) = F (x, f (x)) + [y - f (x)]F (0,1) (x, 0) . TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ 1) e interpolation properties: B3 F = F on 3 (B3 F )(1,0) = F (1,0) on 1 , B4 F = F on 3 (B2 F )(0,1) = F (0,1) on 2 . ese properties are illustrated in Figure 9. V2 V2 B3 B4 V3 V1 V3 V1 Figure 9. e interpolation domains for B3 B4 . 2) e orders of accuracy: dex (B3 ) = dex (B4 ) = 1, pres (B3 ) = 1, x, y j , j N , pres (B4 ) = 1, y, xi , i N . B 3) For e remainders of e interpolation formulas F = B3 F + R3 F B F, we have: F = B4 F + R4 eorem 15 ([14]). If F B11 (0, 0) en (23) B (R3 F )(x, y) = x2 - g 2 (y) (2,0) F (, 0) + y[x - ]F (1,1) (1 , ), 2 ~ wi [0, ], (1 , ) , respectively (24) B (R3 F )(x, y) h2 (2,0) F (·, 0) 2 h2 (1,1) F (·, ·) 4 Proof. As dex(B1 ) = 1, by Peano's eorem, we obtain B R3 F (x, y) = 0 h 0 K20 (x, y, s) F (2,0) (s, 0) ds K11 (x, y, s, t) F (1,1) (s, t) dsdt 21 where INTERPOLATION OPERATORS K20 (x, y, s) = (x - s)+ -[ - s] , K11 (x, y, s, t) = [(x - s)0 -1] (y - t)0 . + + Taking into account at K20 (x, y, s) 0, s [0, ] K11 (x, y, s, t) = -1, (s, t) D = [0, - x] × [0, y] ~ K11 (x, y, s, t) = 0, (s, t) D, 0 0 0 h x2 - g 2 (y) , 2 K11 (x, y, s, t)dsdt = y[x - ], 2 -g 2 (y) e relation (23) follows. Furer, as | x e inequality (24) follows. h2 2 , |y[x - ]| B Remark 16. Analogous formula can be obtained for e remainder R4 F. Example 17. We consider e following test function (see, e.g., [30]): (25) F1 (x, y) = exp[- 81 ((x - 0.5)2 + (y - 0.5)2 )]/3. 16 (Gentle) ~ We take triangle wi one curved side T1 , (h = 1), wi f : [0, 1] [0, 1], e 2 . In Figure 10 we plot e graphs of L F H F . f (x) = 1 - x 1 1 1 1 L1 F1 H1 F1 Figure 10. Graphs of L1 F1 H1 F1 . TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ Example 18. Table 1 contains some maximum approximation errors ~ for F1 , defined on T1 . Maximum error x Bm F1 0.0525 y Bn F1 0.0452 Pmn F1 0.0858 Qnm F1 0.0857 Smn F1 0.0095 Tnm F1 0.0095 Table 1. Maximum approximation errors for F1 . L1 F1 P13 F1 S12 F1 H1 F1 B1 F1 3. Bernstein-type operators on a triangle wi one curved side Since e Bernstein-type operators interpolate a given function at e endpoints of e interval, ese operators can also be used as interpolation operators bo on triangles wi straight sides (see, e.g., [9], [33], [34]) wi curved sides. Let F be a real-valued function defined on (0, y), (, y), respectively, (x, 0), (x, f (x)) be e points where e parallel lines to e coordinate axes, passing rough e point (x, y) , intersect e sides i , i = 1, 2, 3, (see Figure 1). One considers e Bernstein-type operators y x Bm Bn defined by m x (Bm F ) (x, y) = i=0 n y (Bn F ) (x, y) = j=0 F qn,j (x, y) F ,y , m f (x) n x, j wi = qn,j (x, y) = m i n j y f (x) i j y f (x) n-j , , x+y , 0 x + y f (x), are uniform where x = i i = 0, m y = j f (x) j = 0, n n m m n partitions of e intervals [0, ] [0, f (x)]. INTERPOLATION OPERATORS eorem 19 ([11]). If F is a real-valued function defined on en: y x (i) Bm F = F on 2 3 , Bn F = F on 1 3 , x x (ii) (Bm eij ) (x, y) = xi y j , i = 0, 1; (Bm e2j ) (x, y) = x2 + j N, x(-x) m yj , y y (iii) (Bn eij ) (x, y) = xi y j , j = 0, 1; (Bn ei2 ) (x, y) = xi y 2 + i N. y(f (x)-y) n Proof. e interpolation properties (i) follow from e relations: pm,i (0, y) = respectively by qn,j (x, 0) = 1, 0, for j = 0, for j > 0, qn,j (x, f (x)) = 0, for j < n, 1, for j = n. 1, 0, for i = 0, for i > 0, pm,i (, y) = 0, for i < m, 1, for i = m, Regarding e properties (ii), we have x x (Bm eij ) (x, y) = y j (Bm ei0 )(x, y), x (Bm e00 ) (x, y) = m x Bm e10 (x, y) = m i i=0 m-1 i=0 m x Bm e20 (x, y) = m i i=0 x x + jN = 1, i m = x, =x m-1 i 2 m i=0 m i i2 m = = m i(i - 1) +x m m-1 2 x[ - x] x +x = x2 + . m m m TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ Properties (iii) are proved in e same way. x x Now, we consider e approximation formula F = Bm F + Rm F. eorem 20 ([11]). If F (·, y) C[0, ] en 1+ 2 m y [0, h], where (F (·, y); ) is e modulus of continuity of e function F wi regard to e variable x. Moreover, if = 1/ m en (26) x (Rm F ) (x, y) 1+ h 2 1 (F (·, y); ), m x Proof. From e property (Bm e00 )(x, y) = 1, it follows at m x (Rm F ) (x, y) i=0 F (x, y) - F (i , y) . m Using e inequality F (x, y) - F (i one obtains m x |(Rm F )(x, y)| , y) m 1 x-i + 1 (F (·, y); ) m i=0 m i=0 1 x-i + 1 (F (·, y); ) m m 2 1/2 1 1+ = 1+ 1 x - i (F (·, y); ) x( - x) (F (·, y); ). m (y) Since max0x x(- x) = g 4 max0yh g 2 (y) = h2 , it follows 2 at max x( - x) = h , hence 4 1+ 2 m (F (·, y); ). Now, for = 1/ m, one obtains (26). INTERPOLATION OPERATORS eorem 21 ([11]). If F (·, y) C 2 [0, h] en = x[x - ] (2,0) F (, y), 2m h2 M20 F, 8m for [0, ] (x, y) , where Mij F = max F (i,j) (x, y) . x Proof. Taking into account at dex(Bm ) = 1, by Peano's eorem, it follows = K20 (x, y; s)F (2,0) (s, y)ds, where K20 (x, y; s) = (x - s)+ - i i=0 kernel K20 (x, y; ·) to e interval ( - 1) , , i.e., m m m K20 (x, y; ) = (x - s)+ - For a given {1, ..., m} one denotes by K20 (x, y; ·) e restriction of e i i= -s , m whence, x - s - - m i= m (i i= K20 (x, y; s) - s), m s<x s x. (i - s), m It follows at K20 (x, y; s) 0, for s x. For s < x we have m K20 (x, y; s) = x-s- i=0 i -s + m -1 i i=0 -s . m As, i i=0 = x - s, TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ it follows at -1 K20 (x, y; s) i=0 i So, K20 (x, y; ·)0 for any {1, . . . , m}, i.e., K20 (x, y; s)0, for s[0, ]. By e Mean Value eorem, one obtains = F (2,0) (, y) 0 K20 (x, y; s)ds, 0 . x(x - ) 2m Since, K20 (x, y; s)ds = max0x |x(x-)| 2m g 2 (y) 8m h2 8m , y [0, h] e conclusion follows. Remark 22. Analogous results are obtained for e remainder of e y y formula F = Bn F + Rn F. y x x y Let Pmn = Bm Bn , respectively, Qnm = Bn Bm be e products of e y x operators Bm Bn , i.e., m n (Pmn F ) (x, y) = qn,j i j i i , y F , f m m n m (Qnm F ) (x, y) = j i j j pm,i x, f (x) qn,j (x, y) F g f (x) , f (x) . n m n n Remark 23. e nodes of e operator Pmn , respectively, Qnm are given in Figure 11. Figure 11. e nodes for Pmn Qnm , for m = n = 4. INTERPOLATION OPERATORS eorem 24 ([11]). If F is a real-valued function defined on en: (i) (Pmn F )(V3 ) = F (V3 ), Pmn F = F, on 3 (ii) (Qnm F )(V3 ) = F (V3 ), Qnm F = F, on 3 . Proof. e proof follows from e properties x (Pmn F )(x, 0) = (Bm F )(x, 0), y (Pmn F )(0, y) = (Bn F )(0, y), (Pmn F )(x, f (x)) = F (x, f (x)), x, y [0, h] x y (Qnm F )(x, 0) = (Bm F )(x, 0), (Qnm F )(0, y) = (Bn F )(0, y), (Qnm F )(, y) = F (, y), x, y [0, h], which can be verified by a straightforward computation. Remark 25. e product operators Pmn Qnm interpolate e function F at e vertex (0, 0) on e side y = f (x) (or x = ). P Let us consider now e approximation formula F = Pmn F + Rmn F, P where Rmn is e corresponding remainder operator. eorem 26 ([11]). If F C( ) en 1 1 P Rmn F (x, y) (1 + h) F ; , , m n Proof. We have (x, y) 1 1 + + i , y m i , y m x- i m j f n i m n y- i , y m (F ; 1 , 2 ). TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ Since m n i , y m i , y m i , y m x- y- = 1, i m j i f n m x( - x) , m y(f (x) - y) , n it follows at 1+ h2 4 x( - x) 1 + m 2 y(f (x) - y) n (F ; 1 , 2 ). But - x y f (x) - y 1+ whence, (F ; 1 , 2 ) 1 h 1 h + 1 2 m 2 2 n 1 1 (1 + h) F ; , m n y x Next we consider e Boolean sums of e operators Bm Bn , i.e., x y x y x y Smn := Bm Bn = Bm + Bn - Bm Bn , y x y x y x Tnm := Bn Bm = Bn + Bm - Bn Bm . eorem 27 ([11]). If F is a real-valued function defined on en Smn F Proof. As, x (Pmn F ) (x, 0) = (Bm F ) (x, 0) , y (Pmn F ) (0, y) = (Bn F ) (0, y) , x y (Bm F ) (x, h - x) = (Bn F ) (x, h - x) = (Pmn F )(x, h - x) = F (x, h - x) Tnm F e conclusion follows. For e remainder of e Boolean sum approximation formula, F = S Smn F + Rmn F, we have e following result: INTERPOLATION OPERATORS eorem 28 ([11]). If F C( ) en S (Rmn F )(x, y) (1 + h 1 h 1 ) F (·, y); + (1 + ) F (x, ·); 2 2 m n 1 1 (x, y) . + (1 + h) F ; , , m n y x Proof. e identity F - Smn F = F - Bm F + F - Bn F - (F - Pmn F ) implies at S x y (Rm F ) (x, y) + (Rn F ( x, y) + (Rmn F )(x, y) e conclusion follows. Example 29. Consider e test function e triangle from Example x 17. In Figure 12 we plot e graphs of Bm F1 Pmn F1 , for m = 5, n = 6. x Bm F1 Pmn F1 x Figure 12. Graphs of Bm F1 Pmn F1 . Example 30. Table 2 contains some maximum approximation errors for F1 . y x Bm F1 Bn F1 Pmn F1 Qnm F1 Smn F1 Tnm F1 Max error 0.0525 0.0452 0.0858 0.0857 0.0095 0.0095 Table 2. e maximum approximation error for F1 . TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ 4. Bernstein-type operators on a triangle wi all curved sides ~ Let F be a real-valued function defined on (g2 (y), y), (g3 (y), y), respectively, (x, f1 (x)), (x, f3 (x)) be e points where e parallel lines to e coordinate axes, passing rough e point (x, y) , intersect e sides 1 , 2 3 . We consider e uniform partitions of e intervals [g2 (y), g3 (y)] [f1 (x), f3 (x)], x, y [0, h], x = {g2 (y) + i g3 (y)-g2 (y) i = m m y f3 (x)-f1 (x) j = 0, n} e Bernstein0, m}, respectively, n = {f1 (x) + j n y x type operators Bm Bn defined by (27) (28) wi x (Bm F ) (x, y) = i=0 n y (Bn F ) (x, y) = j=0 F qn,j (x, y) F g2 (y) + i g3 (y) - g2 (y) ,y , m f3 (x) - f1 (x) n , x, f1 (x) + j = qn,j (x, y) = m i n j y - f1 (x) f3 (x) - f1 (x) y - f1 (x) f3 (x) - f1 (x) n-j Remark 31. In Figures 13 14 we plot e points g2 (y) + i g3 (y) - g2 (y) , y , i = 0, m m respectively, x, f1 (x) + j f3 (x)-f1 (x) , j = 0, n, for x, n V1 V3 V2 V2 Figure 13. Points of x , for m = 4. m Figure 14. Points of n , for n = 4. y ~ eorem 32 ([12]). If F is a real-valued function defined on en: INTERPOLATION OPERATORS y x (i) Bm F = F on 2 3 , Bn F = F on 1 3 , x x (ii) (Bm ei0 ) (x, y) = xi , i = 0, 1; (Bm e20 ) (x, y) = x2 + x e ) (x, y) = y j (B x e )(x, y), i = 0, 1, 2; j N; (Bm ij m i0 y y (iii) (Bn e0j ) (x, y) = y j , j = 0, 1; (Bn e02 ) (x, y) = y 2 + y y (Bn eij ) (x, y) = xi (Bn e0j )(x, y), j = 0, 1, 2; i N. [x-g2 (y)][g3 (y)-x] ; m [y-f1 (x)][f3 (x)-y] ; n Proof. e interpolation properties (i) follow by e relations: pm,i (g2 (y), y) = respectively by qn,j (x, f1 (x)) = 1, for j = 0, 0, for j > 0, qn,j (x, f3 (x)) = 0, for j < n, 1, for j = n. 1, for i = 0, 0, for i > 0, pm,i (g3 (y), y) = 0, 1, for i < m, for i = m, Regarding e properties (ii), we have x x (Bm eij ) (x, y) = y j (Bm ei0 )(x, y), m x (Bm e00 ) (x, y) = i=0 m x Bm e10 (x, y) = jN = 1, g2 (y) + i i=0 g3 (y) - g2 (y) m m! x - g2 (y) i!(m - i)! g3 (y) - g2 (y) = g2 (y) + [g3 (y) - g2 (y)] x - g2 (y) · g3 (y) - g2 (y) = g2 (y) + [x - g2 (y)] · i=0 i m m-1 i m-1 i=0 -1 = x, TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ m x Bm e20 (x, y) i=0 g3 (y) - g2 (y) g2 (y) + i m 2 =g2 (y) + 2g2 (y)[x - g2 (y)] + + · (m - 1)[g3 (y) - g2 (y)]2 m i=2 x - g2 (y) x - g2 (y) g3 (y) - g2 (y) g3 (y) - g2 (y) m-1 (2m - 1)g2 (y) + g3 (y) [x - g2 (y)]2 + = m m [x - g2 (y)][g3 (y) - x] 2 2 · [x - g2 (y)] + g2 (y) = x + . m Properties (iii) are proved in e same way. [g3 (y) - g2 (y)][x - g2 (y)] m m (m - 2)! (i - 2)!(m - i)! y x Remark 33. e interpolation properties of Bm F Bn F are illustrated in Figures 15 16. V1 V1 V3 V2 Figure 15. Interpolation x set of Bm F Figure 16. Interpolation y set of Bn F. x x Now, we consider e approximation formula F = Bm F + Rm F. eorem 34 ([12]). If F (·, y) C[g2 (y), g3 (y)], y [0, h], en 1+ g3 (y) - g2 (y) 2 m y [0, h], if M = max0yh |g3 (y) - g2 (y)| en we have 1+ M 2 m INTERPOLATION OPERATORS Moreover, if = 1/ m en (29) x (Rm F ) (x, y) 1+ M 2 1 (F (·, y); ), m x Proof. As (Bm e00 )(x, y) = 1, it follows at m x (Rm F ) (x, y) m i=0 F (x, y) - F g2 (y) + i g3 (y) - g2 (y) ,y m i=0 g3 (y) - g2 (y) 1 ] + 1 (F (·, y); ) x - [g2 (y) + i m g3 (y) - g2 (y) ) m 2 1/2 1+ 1+ m i=0 x - (g2 (y) + i (F (·, y); ) [x - g2 (y)][g3 (y) - x] (F (·, y); ). m [g3 (y)-g2 (y)]2 , 4 Since maxg2 (y)xg3 (y) [x - g2 (y)][g3 (y) - x] = 1 + one obtains g3 (y) - g2 (y) (F (·, y); ) 2 m e proof follows. Remark 35. Analogous results can be obtained for e remainder of y y e formula F = Bn F + Rn F. y x x y Let Pmn = Bm Bn , respectively, Qnm = Bn Bm be e products of e y x operators Bm Bn , i.e., m n (Pmn F ) (x, y)= qn,j (xi , y) F xi , f1 (xi )+j pm,i (x, yj ) qn,j (x, y) F g2 (yj )+i f3 (xi ) - f1 (xi ) , n (Qnm F ) (x, y)= g3 (yj ) - g2 (yj ) , yj , m wi xi = g2 (y) + i g3 (y)-g2 (y) , yj = f1 (x) + j f3 (x)-f1 (x) . m n Remark 36. e nodes of e operators Pmn , respectively Qnm are given in Figures 17 18, where f1 g2 f3 are arcs of e circles (C1) , 15 2 15 1 2 (x - 2 ) + (y + 2 ) = 4, (C2) (x + 2 )2 + (y - 1 )2 = 4, respectively, (C3) 2 x2 + y 2 = 1. TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ V1 V3 V2 Figure 17. e nodes of Pmn , for m = n = 4. Figure 18. e nodes of Qnm , for m = n = 4. ~ eorem 37 ([12]). If F is a real-valued function defined on en: (i) (Pmn F )(V3 ) = F (V3 ), Pmn F = F, on 3 (ii) (Qnm F )(V3 ) = F (V3 ), Qnm F = F, on 3 . Proof. e proof follows from e properties: x (Pmn F )(x, f1 (x)) = (Bm F )(x, f1 (x)), y (Pmn F )(g2 (y), y) = (Bn F )(g2 (y), y), (Pmn F )(x, f3 (x)) = F (x, f3 (x)), x (Qnm F )(x, f1 (x)) = (Bm F )(x, f1 (x)), y (Qnm F )(g2 (y), y) = (Bn F )(g2 (y), y), x, y [0, h] (Qnm F )(g3 (y), y) = F (g3 (y), y), x, y [0, h], which can be verified by a straightforward computation. e interpolation properties of Pmn F Qnm F are illustrated in Figure 19. V3 V2 Figure 19. Interpolation set for Pmn F Qnm F INTERPOLATION OPERATORS For e remainder of e product approximation formula F = Pmn F + P Rmn F we have: ~ eorem 38 ([12]). If F C( ) en P Rmn F (x, y) 1+ 21 m 22 n (F ; 1 , 2 ) , ~ (x, y) , respectively P Rmn F (x, y) 1+ N M + 2 2 1 1 F; , m n ~ (x, y) , where M = max0yh |g3 (y) - g2 (y)| N = max0xh |f3 (x) - f2 (x)|. Proof. We have (xi , y) |x - xi | f3 (xi ) - f1 (xi ) ) n + + n (xi , y) y - (f1 (xi ) + j (xi , y) (F ; 1 , 2 ) n qn,j (xi , y) j=0 m [x - g2 (y)][g3 (y) - x] m [y - f1 (x)][f3 (x) - y] + 1 (F ; 1 , 2 ), n i=0 furer, [x - g2 (y)][g3 (y) - x] m [x - f1 (x)][f3 (x) - y] + 1 (F ; 1 , 2 ) n 1 g3 (y) - g2 (y) 1 f3 (x) - f1 (x) 1+ + + 1 (F ; 1 , 2 ) 1 2 m 2 2 n 1 M 1 N + + 1 (F ; 1 , 2 ), 1+ 1 2 m 2 2 n TEODORA CATINAS, PETRU BLAGA GHEORGHE COMAN ¸ e proof follows. Q For e remainder Rnm F = F - Qnm F we can also obtain Q Rnm F (x, y) 1+ N M + 2 2 1 1 F; , m n ~ (x, y) . y x We consider e Boolean sums of e operators Bm Bn , i.e., x y x y x y Smn := Bm Bn = Bm + Bn - Bm Bn , y x y x y x Tnm := Bn Bm = Bn + Bm - Bn Bm . ~ eorem 39 ([12]). If F is a real-valued function defined on en Smn F Proof. As x (Pmn F ) (x, f1 (x)) = (Bm F ) (x, f1 (x)) , y (Pmn F ) (g2 (y), y) = (Bn F ) (g2 (y), y) , Tnm F (Pmn F ) (x, f3 (x)) = F (x, f3 (x)) , e proof follows. For e remainder of e Boolean sum approximation formula, F = S Smn F + Rmn F, we have e following result: ~ eorem 40 ([12]). If F C( ) en S (Rmn F )(x, y) (1 + 1 N 1 M ) F (·, y); + (1 + ) F (x, ·); 2 m 2 n M N 1 1 + (1 + + ) F ; , , (x, y) . 2 2 m n y x Proof. e identity F -Smn F = F -Bm F +F -Bn F -(F -Pmn F ) imS F )(x, y) (Rx F ) (x, y) + (Ry F ( x, y) + (RP F )(x, y) plies at (Rmn n m mn e proof follows. T An analogous inequality can be obtained for e error Rnm F -Tnm F. Example 41. Consider e test function from Example 17. In Figure 20 x we plot e graphs of Bm F1 Smn F1 , wi h = 1, m = 5, n = 6, 15 15 f1 , g2 , f3 : [0, 1] [0, 1], f1 (x) = - 2 - 4 - (x - 0.5)2 , g2 (y) = - 2 - 4 - (y - 0.5)2 f3 (x) = 1 - x2 . INTERPOLATION OPERATORS x Figure 20. Graphs of Bm F1 Smn F1 . Example 42. Table 3 contains e maximum approximation errors. y x Bm F1 Bn F1 Pmn F1 Qnm F1 Smn F1 Tnm F1 Max error 0.0847 0.0684 0.1269 0.1266 0.0239 0.0240 Table 3. e maximum approximation error for F1 .

Journal

Annals of the Alexandru Ioan Cuza University - Mathematics – de Gruyter

Published: Nov 24, 2014

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