The combined Shepard-Lidstone bivariate operator

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1 Trends and Applications in Constructive Approximation (Eds.) M.G. de Bruin, D.H. Mache & J. Szabados International Series of Numerical Mathematics Vol. 151 c 2005 Birkhäuser Verlag Basel (ISBN ) The combined Shepard-Lidstone bivariate operator Teodora Cătinaş Abstract We extend the Shepard operator by combining it with the Lidstone bivariate operator. We study this combined operator and give some error bounds. 1 Preliminaries 1.1 The Shepard bivariate operator Recall first some results regarding the Shepard operator for the bivariate case [7], [17]. Let f be a real-valued function defined on X R 2, (x i,y i ) X, i = 0,...,N some distinct points and r i (x,y), the distances between a given point (x,y) X and the points (x i,y i ), i = 0,1,...,N. The Shepard interpolation operator is defined by where with µ R +. It follows that (Sf) (x,y) = A i (x,y) = A i (x,y)f (x i,y i ), N r µ j (x,y) j=0 j i k=0j=0 j k, (1) N r µ j (x,y) A i (x,y) = 1. (2) 0 This work has been supported by CNCSIS under Grant 8/139/

2 78 Teodora Cătinaş Because of its small degree of exactness we are interested in extending the Shepard operator S by combining it with some other operators. Let Λ := {λ i : i = 0,...,N} be a set of functionals and P the corresponding interpolation operator. We consider the subsets Λ i Λ, i = 0,...,N such that N Λ i = Λ and Λ i Λj, excepting the case Λ i = {λ i }, i = 0,...,N, when Λ i Λj = for i j. We associate the interpolation operator P i to each subset Λ i, for i = 0,...,N. The combined operator of S and P, denoted by S P, is defined in [9] by (S P f) (x,y) = A i (x,y) (P i f) (x,y). Remark 1.1 [16] If P i, i = 0,...,N, are linear and positive operators then S P is a linear and positive operator. Remark 1.2 [16] Let P i, i = 0,...,N, be some arbitrary linear operators. If dex (P i ) = r i, i = 0,...,N, then dex (S P ) = min {r 0,...,r N }. 1.2 Two variable piecewise Lidstone interpolation We recall first some results from [1] and [2]. Consider a,b,c,d R, a < b, c < d and let : a = x 0 < x 1 <... < x N+1 = b and : c = y 0 < y 1 <... < y M+1 = d denote uniform partitions of the intervals [a,b] and [c,d] with stepsizes h = (b a)/(n +1) and l = (d c)/(m +1), respectively. Denote by ρ = the resulting rectangular partition of [a, b] [c, d]. For the univariate function f and the bivariate function g and each positive integer r we denote by D r f = d r f/dx r, D r xg = r g/ x r and D r yg = r g/ y r. Definition 1.3 [2] For each positive integer r and p, 1 p, let PC r,p [a,b] be the set of all real-valued functions f such that: (i) f is (r 1) times continuously differentiable on [a, b]; (ii) there exist s i, 0 i L+1 with a = s 0 <... < s L+1 = b, such that on each subinterval (s i,s i+1 ), 0 i L, D r 1 f is continuously differentiable; (iii) the L p -norm of D r f is finite, i.e., ( L D r f p = For the case p = it reduces to si+1 s i D r f = max 0 i L x (s i,s i+1) ) 1/p D r f(x) p dx <. sup D r f(x) <.

3 The combined Shepard-Lidstone bivariate operator 79 Definition 1.4 [2] For each positive integer r and p, 1 p, let PC r,p ([a,b] [c,d]) be the set of all real-valued functions f such that: (i) f is (r 1) times continuously differentiable on [a,b] [c,d], i.e., D µ xd ν yf, 0 µ + ν r 1, exist and are continuous on [a,b] [c,d]; (ii) there exist s i, 0 i L + 1 and v j, 0 j R + 1 with a = s 0 <... < s L+1 = b and c = v 0 < v 1 <... < v R+1 = d, such that on each open subrectangle (s i,s i+1 ) (v j,v j+1 ), 0 i L, 0 j R and for all 0 µ r 1, 0 ν r 1 with µ + ν = r 1, D µ xd ν yf are continuously differentiable; (iii) for all 0 µ r, 0 ν r such that µ + ν = r, the L p -norm of D µ xd ν yf is finite, i.e., D µ x Dyf ( L ν p = R si+1 vj+1 D µ x Dyf(x,y) ) 1/p ν p dxdy <. j=0 s i v j For the particular case p = it reduces to D µ xd ν yf = max 0 i L 0 j R sup D xd µ yf(x,y) ν <. (x,y) (s i,s i+1) (v j,v j+1) Definition 1.5 [2] Let PC r1,r2,p ([a,b] [c,d]) be the set of all real-valued functions f such that: (i) D µ xd ν yf, 0 µ r 1 1,0 ν r 2 1 exist and are continuous on [a,b] [c,d]; (ii) on each open subrectangle (s i,s i+1 ) (v j,v j+1 ), 0 i L, 0 j R and for all 0 µ r 1,0 ν r 2, D µ xd ν yf exist and are continuous; (iii) for all 0 µ r 1,0 ν r 2 the L p -norm of D µ xd ν yf is finite. The Lidstone polynomial is the unique polynomial Λ n of degree 2n+1, n N on the interval [0,1], defined by (see, e.g., [1], [2]) Λ 0 (x) = x, Λ n(x) = Λ n 1 (x), Λ n (0) = Λ n (1) = 0, n 1. As in [1] and [2], for a fixed partition denote the set L m ( ) = { h C[a,b] : h is a polynomial of degree at most 2m 1 in each subinterval [x i,x i+1 ], 0 i N }. Its dimension is 2m(N + 1) N.

4 80 Teodora Cătinaş Definition 1.6 [2] For a given function f C 2m 2 [a,b] we say that L mf is the Lidstone interpolant of f if L mf L m ( ) with D 2k (L mf)(x i ) = f (2k) (x i ), 0 k m 1, 0 i N + 1. According to [2], for f C 2m 2 [a,b] the Lidstone interpolant L mf uniquely exists and on the subinterval [x i,x i+1 ], 0 i N can be explicitly expressed as (L mf) [xi,x i+1](x) = m 1 k=0 [ Λ k ( xi+1 x h ) f (2k) ( (x i ) + Λ x xi ) k h f (2k) (x i+1 )] h 2k. (3) It follows that (L mf)(x) = N+1 m 1 r m,i,j (x)f (2j) (x i ), j=0 where r m,i,j, 0 i N + 1, 0 j m 1 are the basic elements of L m ( ) satisfying D 2ν r m,i,j (x µ ) = δ iµ δ 2ν,j, 0 µ N + 1,0 ν m 1 (4) and ( Λ xi+1 x) j h h 2j, x i x x i+1, 0 i N ( r m,i,j (x) = Λ x xi 1 ) j h h 2j, x i 1 x x i, 1 i N + 1 0, otherwise. Proposition 1.7 [1], [11] The Lidstone operator L m is exact for the polynomials of degree not greater than 2m 1. We have the interpolation formula f = L mf + R mf, where R mf denotes the remainder. Taking into account Theorem from [2] we have for f PC 2m 2, [a,b] the following estimation of the remainder: Rmf (5) d 2m 2,0 h 2m 2 max sup f (2m 2) (x) xi+1 x 0 i N h f (2m 2) (x i ) x (x i,x i+1) x xi h f(2m 2) (x i+1 ) 2d 2m 2,0 h 2m 2 f (2m 2),

5 The combined Shepard-Lidstone bivariate operator 81 where d 2m,k, 0 k 2m 2 are the numbers given by ( 1) m i E 2m 2i 2 2m 2i (2m 2i)!, k = 2i, 0 i m d 2m,k = ( 1) m i+1 2(2 2m 2i 1) (2m 2i)! B 2m 2i, k = 2i + 1, 0 i m 1 2, k = 2m + 1, (6) E 2m and B 2m being the 2m-th Euler and Bernoulli numbers (see, e.g., [2]). After some computations we get 1, m = 0 d 2m,0 = 4 sin(2k+1)πt π 2m+1 (2k+1), 2m+1 m 1. (7) k=0 For a fixed rectangular partition ρ = of [a,b] [c,d] the set L m (ρ) is defined as follows (see, e.g., [1] and [2]): L m (ρ) =L m ( ) L m ( ) =Span { r m,i,µ (x)r m,j,ν (y) } N+1 m 1 M+1 m 1 µ=0 j=0 ν=0 { = h C([a,b] [c,d]) : h is a two-dimensional polynomial of degree at most 2m 1 in each variable and subrectangle } [x i,x i+1 ] [y j,y j+1 ]; 0 i N, 0 j M and its dimension is (2m(N + 1) N)(2m(M + 1) M). Definition 1.8 [2] For a given function f C 2m 2,2m 2 ([a,b] [c,d]) we say that L ρ mf is the two-dimensional Lidstone interpolant of f if L ρ mf L m (ρ) with D 2µ x D 2ν y (L ρ mf)(x i,y j ) = f (2µ,2ν) (x i,y j ), 0 i N + 1, 0 j M + 1, 0 µ,ν m 1. According to [2], for f C 2m 2,2m 2 ([a,b] [c,d]), the Lidstone interpolant L ρ mf uniquely exists and can be explicitly expressed as (L ρ mf)(x,y) = N+1 m 1 µ=0 M+1 j=0 m 1 r m,i,µ (x)r m,j,ν (y)f (2µ,2ν) (x i,y j ), (8) ν=0 where r m,i,j, 0 i N + 1, 0 j m 1 are the basic elements of L m (ρ) satisfying (4). Lemma 1.9 [2] If f C 2m 2,2m 2 ([a,b] [c,d]) then (L ρ mf)(x,y) = (L ml m f)(x,y) = (L m L mf)(x,y).

6 82 Teodora Cătinaş Corollary 1.10 [2] For a function f C 2m 2,2m 2 ([a,b] [c,d]), from Lemma 1.9 we have that f L ρ mf = (f L mf) + L m(f L m f) (9) = (f L mf) + [ L m(f L m f) (f L m f) ] + (f L m f). 1.3 Estimation of the error for the Shepard-Lidstone univariate interpolation We recall some results regarding error bounds for the Shepard-Lidstone univariate interpolation formula, obtained by us in [4]. With the previous assumptions we denote by L,i m f the restriction of the Lidstone interpolation polynomial to the subinterval [x i,x i+1 ], 0 i N, given by (3), and in analogous way we obtain the expression of the restriction L,i m f to the subinterval [y i,y i+1 ] [c,d], 0 i N. We denote by S L the univariate combined Shepard-Lidstone operator, given by (S L f)(x) = A i (x)(l,i m f)(x). We have obtained the following result regarding the estimation of the remainder R L f of the univariate Shepard-Lidstone interpolation formula Theorem 1.11 [4] If f PC 2m 2, [a,b] then f = S L f + R L f. (10) R L f (11) d 2m 2,0 h 2m 2 max sup f (2m 2) (x) xi+1 x 0 i N h f (2m 2) (x i ) x (x i,x i+1) x xi h f(2m 2) (x i+1 ) 2d 2m 2,0 h 2m 2 f (2m 2), with d 2m,0 given by (7). 2 The combined Shepard-Lidstone bivariate interpolation 2.1 The combined Shepard-Lidstone bivariate operator We consider f C 2m 2,2m 2 ([a,b] [c,d]) and the set of Lidstone functionals Λ i Li = {f(x i,y i ),f(x i+1,y i+1 ),...,f (2m 2,2m 2) (x i,y i ),f (2m 2,2m 2) (x i+1,y i+1 )}

7 The combined Shepard-Lidstone bivariate operator 83 regarding each subrectangle [x i,x i+1 ] [y i,y i+1 ],0 i N, with Λ i Li = 4m, 0 i N. We denote by L ρ,i m f the restriction of the polynomial given by (8) to the subrectangle [x i,x i+1 ] [y i,y i+1 ],0 i N. This 2m 1 degree polynomial in each variable solves the interpolation problem corresponding to the set Λ i Li, 0 i N and it uniquely exists. We have (L ρ,i m f) (2ν,2ν) (x k,y k ) = f (2ν,2ν) (x k,y k ), 0 i N; 0 ν m 1; k = i,i + 1. We denote by S Li the Shepard operator of Lidstone type, given by (S Li f)(x,y) = A i (x,y)(l ρ,i m f)(x,y), (12) where A i, i = 0,...,N are given by (1). We call S Li the combined Shepard-Lidstone bivariate operator. Theorem 2.1 The operator S Li is linear. Proof. For arbitrary h 1, h 2 C 2m 2,2m 2 ([a,b] [c,d]) and α, β R one gets S Li (αh 1 + βh 2 )(x,y) = = α A i (x,y)l ρ,i m (αh 1 + βh 2 )(x,y) A i (x,y)(l ρ,i m h 1 )(x,y) + β = αs Li (h 1 )(x,y) + βs Li (h 2 )(x,y). A i (x,y)(l ρ,i m h 2 )(x,y) Theorem 2.2 The operator S Li has the interpolation property: (S Li f) (2ν,2ν) (x k,y k ) = f (2ν,2ν) (x k,y k ), 0 ν m 1, 0 k N + 1, (13) for µ > 4m 4. Proof. It is not difficult to show the following relations [8] A (p,q) k (x i,y i ) = 0, 0 i N; 0 p,q 2m 2, i k, A (p,q) k (x k,y k ) = 0, p + q 1, for all k = 0,...,N and µ > max{p + q 0 p,q m 1}.

8 84 Teodora Cătinaş From we obtain (S Li f) (2ν,2ν) (x k,y k ) = (S Li f) (2ν,2ν) (x k,y k ) = ( Ai (x,y)(l ρ,i m f) ) (2ν,2ν) (xk,y k ), A i (x k,y k )(L ρ,i m f) (2ν,2ν) (x k,y k ), and taking into account the cardinality property of A i s we get (13). Theorem 2.3 The degree of exactness of the combined operator S Li is dex(s Li ) = 2m 1. Proof. By Proposition 1.7 we have that dex(l ρ,i m ) = 2m 1. This implies L ρ,i m e pq = e pq, where e pq (x,y) = x p y q, for p,q N, with p+q 2m 1. Taking into account (2), we get (S Li e pq )(x,y) = = A i (x,y)(l ρ,i m e pq )(x,y) A i (x,y)e pq (x,y) Therefore the result is proved. = e pq (x,y) A i (x,y) = e pq (x,y), for p + q 2m Estimation of the error for the Shepard-Lidstone bivariate interpolation We obtain the bivariate Shepard-Lidstone interpolation formula, f = S Li f + R Li f, where S Li f is given by (12) and R Li f denotes the remainder term. Theorem 2.4 If f PC 2m 2,2m 2, ([a,b] [c,d]) then R Li f 4d 2m 2,0 h 2m 2 f (2m 2) + + 2d 2m 2,0 h 2m 2 max (f L,i m f) (2m 2) (14) 0 i N

9 The combined Shepard-Lidstone bivariate operator 85 2d 2m 2,0 h 2m 2 [ 2 f (2m 2) + (f L m f) (2m 2) ], with d 2m 2,0 given by (7). Proof. Taking into account (12) and (2) we get (R Li f)(x,y) = f(x,y) (S Li f)(x,y) = f(x,y) A i (x,y)(l ρ,i m f)(x,y) = A i (x,y)f(x,y) A i (x,y)(l ρ,i m f)(x,y) = A i (x,y) [ f(x,y) (L ρ,i m f)(x,y) ]. Next, applying formulas (9) and (2) we get { (R Li f)(x,y) = A i (x,y) (f L,i m f)(x,y) + [ L,i m (f L,i m f)(x,y) (f L,i m f)(x,y) ] } + (f L,i m f)(x,y) = A i (x,y)(f L,i m f)(x,y) + + [ = f(x,y) A i (x,y) [ L,i m (f L,i m f)(x,y) (f L,i m f)(x,y) ] A i (x,y)(f L,i m A i (x,y) f)(x,y) A i (x,y)(l,i m f)(x,y) [ ] A i (x,y) (f L,i m f)(x,y) L,i m (f L,i f)(x,y) + [ f(x,y) A i (x,y) A i (x,y)(l,i m ] m ] f)(x,y)

10 86 Teodora Cătinaş [ = f(x,y) ] A i (x,y)(l,i m f)(x,y) [ ] A i (x,y) (f L,i m f)(x,y) L,i m (f L,i f)(x,y) + [ f(x,y) whence it follows that A i (x,y)(l,i m R Li f f A i L,i + max m f sup 0 i N x (x i,x i+1) ] f)(x,y), (f L,i + f A i (L,i m f). 0 i N x (x i,x i+1) m m f) L,i m (f L,i m f) We have f(,y) PC 2m 2, [a,b], (f L,i m f)(,y) PC 2m 2, [a,b], for all y [c,d] and f(x, ) PC 2m 2, [c,d], for all x [a,b]. From (11) we get that R Li f 4d 2m 2,0 h 2m 2 f (2m 2) + max sup (f L,i m f) L,i m (f L,i m f) (15) and from (5) we obtain max sup 0 i N x (x i,x i+1) (f L,i 2d 2m 2,0 h 2m 2 max m 0 i N f) L,i m (f L,i (f L,i m f) (2m 2) 2d 2m 2,0 h 2m 2 (f L m f) (2m 2). Finally, replacing (16) in (15) we are led to (14). m f) (16) Next, we give an estimation of the approximation error in terms of the mesh length and using the modulus of smoothness of order k. Recall that the k th modulus of smoothness of f L p [a,b], 0 < p <, or of f C[a,b], if p =, is defined by ( see, e.g., [13]): ω k (f;t) p = sup k h f(x) p, 0<h t

11 The combined Shepard-Lidstone bivariate operator 87 where k hf(x) = k ( 1) k+i( k i) f(x + ih). We will use some results for spline approximation given in [12]. Definition 2.5 [12, p.134] Let T := (t i ) s 1 or T := (t i ) + be a finite or infinite strictly increasing sequence of points of R; in the second case, we assume that t i for i ±. A function S on R is a spline of order r (r = 1,2,...), equivalently of degree m = r 1, with the breakpoints T if on each interval (t i,t i+1 ), and on the intervals (,t 1 ), (t s,+) if T is finite, it is a polynomial of degree m, and on one of them of degree exactly m. At the breakpoints t i, S and its derivatives (which are also splines) are defined by continuity. Definition 2.6 [12, p. 135] For given A = [a,b] and T = (t i ) s 1 (we assume that a < t i < b, i = 1,...,s) we form the Schoenberg space, denoted by S r (T,A), which is the space of all splines of order r on A whose breakpoints are contained in T, and of smoothness m i at t i (0 m i r), i = 1,...,s. A Schoenberg space S r (T,A) contains the space P r 1, of polynomials of degree r 1 [12, p. 135]. Definition 2.7 [12, p. 144] A projection operator Q from L 1 onto the Schoenberg space S r := S r (T,A), and thereby from each L p onto S r, for each 1 p, is called a quasi-interpolant of order r. Theorem 2.8 [12, Th. 7.3., p. 225] Given a quasi-interpolant Q of order r, for each f C[a,b], one has the following estimation: f Q(f) C r ω r (f;δ), where C r is a constant and δ is defined by: δ := max 0 j N (x i+1 x i ). By Definition 2.7, it follows that the operators L m and L m, given in Subsection 1.3 are quasi-interpolants of order 2m. Therefore, we can apply Theorem 2.8 for f C[a,b] and g C[c,d] and we obtain f L m(f) C 2m ω 2m (f;δ 1 ), (17) where g L m (g) C 2mω 2m (g;δ 2 ), δ 1 = max 0 j N (x i+1 x i ), (18) δ 2 = max 0 j N (y j+1 y j ),

12 88 Teodora Cătinaş and C 2m, C 2m are some constants. We obtain an estimation of R L f from (10), in terms of the modulus of smoothness of high order. Theorem 2.9 If f PC 2m 2, [a,b] then R L f C 2m ω 2m (f;δ 1 ), (19) with Proof. We have δ 1 = max 0 j N (x i+1 x i ). (R L f)(x) = f(x) A i (x)(l,i m f)(x) = A i (x)f(x) A i (x)(l,i m f)(x) = A i (x)[f(x) (L,i m f)(x)], and taking into account that N A i(x) = 1 and (17), the conclusion follows. We obtain an estimation of the remainder for the bivariate Shepard-Lidstone formula, in terms of the modulus of smoothness of high order. Theorem 2.10 If f PC 2m 2,2m 2, ([a,b] [c,d]) then R Li f C 2m max y [c,d] ω 2m (f(,y);δ 1 ) + C 2m max y [c,d] ω 2m ((f L m f)(,y);δ 1 ) + C 2m max x [a,b] ω 2m (f(x, );δ 2 ), where δ 1 and δ 2 are given in (18) and C 2m, C 2m are some constants. Proof. This result follows using the same procedure as in proof of Theorem 2.4 and applying formula (17) and two times Theorem 2.9.

13 The combined Shepard-Lidstone bivariate operator 89 Example 2.11 Let f : [ 2,2] [ 2,2] R, f(x,y) = xe (x2 +y 2 ) and consider the nodes z 1 = ( 1, 1), z 2 = ( 0.5, 0.5), z 3 = ( 0.3, 0.1), z 4 = (0,0), z 5 = (0.5,0.8), z 6 = (1,1). Below we plot the graphics of f and S Li f Fig. 1. The graphic of f(x,y) = xe (x2 +y 2) (left) and S Li f for µ = 1 (right) References [1] R. Agarwal, P. J. Y. Wong, Explicit error bounds for the derivatives of piecewise-lidstone interpolation, J. Comp. Appl. Math., 58 (1993), pp [2] R. Agarwal, P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and their Applications, Kluwer Academic Publishers, Dordrecht, [3] T. Cătinaş, The combined Shepard-Abel-Goncharov univariate operator, Rev. Anal. Numér. Théor. Approx., 32 (2003) no. 1, pp [4] T. Cătinaş, The combined Shepard-Lidstone univariate operator, Tiberiu Popoviciu Itinerant Seminar of Functional Equations, Approximation and Convexity, Cluj-Napoca, May 21 25, 2003, pp [5] W. Cheney and W. Light, A Course in Approximation Theory, Brooks/Cole Publishing Company, Pacific Grove, [6] Gh. Coman, The remainder of certain Shepard type interpolation formulas, Studia Univ. Babeş-Bolyai, Mathematica, 32 (1987) no. 4, pp [7] Gh. Coman, Shepard-Taylor interpolation, Itinerant Seminar on Functional Equations, Approximation and Convexity, Cluj-Napoca, (1988), pp [8] Gh. Coman, Shepard operators of Birkhoff type, Calcolo, 35 (1998), pp

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