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If Sn,p f x is of the form 2. A quantitative version of Theorem 2. Reasoning exactly as in the proof of Lemma 3 in Popoviciu  and taking into account the proof of Theorem 2. We have two possibilities. We have two cases. We would like to point out here an error appeared in the proof of Theorem 3. The correct proof and estimate are given by Theorem 2. Taking into account the proof of Theorem 2.
Similar to the proof of Theorem 2. The Remarks 1 and 2 after the proof of Theorem 2. At the end, we present some negative remarks on the shape preservation properties of the so-called Shepard—Lagrange operator, whose global smoothness preservation properties were studied by Corollary 1. Reasoning exactly as for the relation 2.
Unfortunately, by Popoviciu  see Theorem 2. Let us take the simplest case, i. For the study of convexity, from the proof of Theorem 2. Open Problem 2. For the Kryloff—Stayermann polynomials, Kn f x , in Theorems 2. What happens if in the Theorems 2. What happens if in the statements of Theorems 2.
Also, another question for this operator is if there exist points such that in some neighborhoods of them, it preserves the strict-convexity of function. For the local variants of Shepard operators in Open Problem 1. In this sense, we will use the following kinds of bivariate moduli of continuity. The properties of these moduli of continuity useful in the next sections are given by the following Lemma 3.
In all subsequent results of this section, the constants involved in the signs O, will depend only on the functions considered. Theorem 3. Indeed, by hypothesis and Lemma 3. From Lemma 3. According to Theorem 3. Adding the last to relations we get the theorem. Lemma 3.
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These prove the lemma. Indeed, from the univariate case see Theorem 1. Then reasoning exactly as in the proof of Corollary 1. Taking now into account Theorem 3. We present: Theorem 3. Since by, e. F i 0 Reasoning exactly as in the previous section, we get the following. Corollary 3. As in the proofs of Lemma 3. Now, following the ideas in the proof of Corollary 3. This kind of Shepard operator is useful in computer-aided geometric design see, e. In what follows, we consider another variant of the bivariate Shepard operator which is not a tensor product of the univariate case, has good approximation properties and implicitly better global smoothness preservation properties than that introduced in Shepard .
Equations 3. Equation 3. Next, using 3.
We apply the standard technique. By the first estimate in Theorem 3. With respect to this modulus we obtain the following: Corollary 3. But by, e. It is immediate by Lemmas 3. By Lemma 3. By Corollary 3. Open Problem 3. For the bivariate tensor product Shepard operators on the semi-axis, generated by the univariate case in Della Vecchia—Mastroianni—Szabados , prove global smoothness preservation properties. For the kinds of univariate Shepard operators considered by Chapters 1 and 2, various bivariate combinations different from those considered by Chapter 3 can be considered as follows.
Type 1. Type 2. Type 3. Type 4.
Starting from the local variants of Shepard operators in the univariate case defined in Open Problem 1. The problem is to find global smoothness preservation properties with respect to various bivariate moduli of continuity of these local bivariate Shepard operators. A key result used in the univariate case for the proofs of qualitative-type results, is the following simple one Lemma 4. Popoviciu . New aspects appear because of various possible natural concepts for bivariate monotonicity and convexity.
Also, three different kinds of bivariate Shepard operators are studied. The key result of this section is Theorem 4. Corollary 4. Taking into account Theorem 4.
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Let us consider Fn1 ,n2 f x, y given by 4. An immediate consequence of Theorem 4. In this sense, we need the following. Definition 4. Theorem 4. Combining this estimate with 4. That would be useful for a better estimate. The global smoothness preservation properties and convergence properties of these operators were studied in Section 3 of Chapter 3. In this section we consider their properties of preservation of shape.
In this sense, a key result is the following. It is immediate by Theorem 4. Because by Remark 1 after Theorem 2. In what follows, we will extend the convexity problem from the univariate case. From the proof of Corollary 4. Convergence properties of this kind of operators can be found in, e. On the other hand, concerning the partial shape-preserving property we first can prove the following qualitative results.
By using Lemma 4. But according to Remark 1 after the proof of Theorem 2. Now, let us discuss some properties of qualitative kind of Sn,2p f x, y related to the convexity. We present the following: Theorem 4. Let Sn,2p f x, y be given by 4. According to the Remark after Definition 4. Lemma 4. Firstly, by 4. Similarly, by Lemma 4. By the same Lemma 4. Immediate by the Lemmas 4. By the proof of Theorem 4. Then, combining the ideas in the bivariate case in the proof of Theorem 4. Reasoning exactly as in the proof of Theorem 2. Because of the complicated proof of the qualitative result see the proofs of Lemmas 4.
In what follows we consider their properties of preservation of shape. Let us note that with respect to preserving monotonicity, it is unfortunate that this does not seem to be a useful method for dealing with this kind of Shepard operator in the univariate case — as it had been in dealing with the original Shepard operator see the proof of Theorem 4. That is why we consider here only some properties related to the usual bivariate convexity. By Lemma 4.
Immediate by Lemmas 4. For a quantitative estimate of the length of convexity neighborhood V 0, 0 in Corollary 4. The quantitative estimate of convexity in Theorem 4. From the proof of Theorem 4. We get 0 which proves the theorem. A version of Theorem 4. All the other results of this chapter except those where the authors are mentioned , are from Gal—Gal . Open Problem 4. Prove Theorems 4. For the local bivariate tensor products, Shepard operators mentioned in Open Problem 3. A Appendix: Graphs of Shepard Surfaces Due to the usefulness of their properties in approximation theory, data fitting, CAGD, fluid dynamics, curves and surfaces, the Shepard operators univariate and bivariate variants have been the object of much work.
Let us mention here the following papers: —, , , —, —, , —, —, , , , , —, , , , , , , —. In this appendix we present some pictures for various kinds of bivariate Shepard operators, which illustrate the shape-preserving property of them. Recall that by the previous sections, we have considered nine main types of Shepard operators, given by the following formulas. Type 6. Type 7. It is a bidimensional upper monotone function, i. Example 2. Example 3. It is an ordinary strictly convex bivariate function, i.
Example 4. Simple calculations show that f5 is simultaneously bidimensional upper monotone, strictly double convex and ordinary strictly convex. The graphs of the first four functions, f1 , f2 , f3 , f4 , are given in Figure A. Also, Figures 5. Figure A. In Figure A. Finally, in Figure A. The graph of this function is given in Figure A. Graphs for f5 and Shepard operators of types 2, 4 and 8 References 1. Akima H A method of bivariate interpolation and smooth surface fitting for irregularly distributed data points.
Akima H On estimating partial derivatives for bivariate interpolation of scattered data. Rocky Mountain J. Allasia G A class of interpolating positive linear operators: theoretical and computational aspects. Allasia G Simultaneous interpolation and approximation by a class of multivariate positive operators.
Numerical Algorithms in print 5. Analysis, 11, 43—57 6. Computers and Mathematics with Applic. Russian Trudy Moskov.
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Studia Univ. Coman Gh Shepard—Taylor interpolation. East J. Acta Math. Della Vecchia B Direct and converse results by rational operators. Della Vecchia B, Mastroianni G Pointwise estimates of rational operators based on general distribution of knots. Facta Universitatis Nis , 6, 63—72 Budapest Sect. Optimization, 17, No. Mathemathische Annalen, , — Fleming W Functions of Several Variables. Second edition, Springer-Verlag, New York Foley TA Interpolation of scattered data on a spherical domain. Foley TA Scattered data interpolation and approximation with error bounds.
Design, 3, — Foley TA Three-stage interpolation to scattered data. Franke R Scattered data interpolation: tests of some methods. Franke R Locally determined smooth interpolation at irregularly spaced points in several variables. JIMA, 19, — Methods Eng. Gal SG Remarks on the approximation of normed spaces valued functions by some linear operators. Studia Sci.
Analysis in Theory and Appl. In: Functions, Series, Operators, vol. I, II Proc.
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Gonska HH On approximation in spaces of continuous functions, Bull. Gonska HH, Jetter K Jackson—type theorems on approximation by trigonometric and algebraic pseudopolynomials. Theory, 48, — Haussmann W, Pottinger P On the construction and convergence of multivariate interpolation operators. Theory, 19, — Pure Appl.
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Katkauskayte AY The rate of convergence of a Shepard estimate in the presence of random interference. Soviet J. Sciences, 24, No. Theory, 54, — Theory, 99, — Classe Sci. Ukraine, 1, 13—16 Lawson CL C 1 surface interpolation for scattered data on a sphere. Little FF Convex combination surfaces. Marcus S Functions monotone de deux variables. Pures Appl. Palermo, ser. II, No. II, J. Theory, 90, 1—8 McLain DH Two-dimensional interpolation from random data.