Global smoothness and shape preserving interpolation by classical operators

Global Smoothness and Shape Preserving Interpolation by Classical Operators

Volumn , Issue , 2005, Pages 1-146

  Gal, Sorin G ^a   Anastassiou, George A ^b  

^a UNIVERSITY OF ORADEA   (Romania)

^b UNIVERSITY OF MEMPHIS   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84891991571     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/b137115     Document Type: Book

References (111)

1. Akima H (1978) A method of bivariate interpolation and smooth surface fitting for irregularly distributed data points. ACM Transactions on Mathematical Software, 4, 146-159
2. Akima H (1984) On estimating partial derivatives for bivariate interpolation of scattered data. Rocky Mountain J. Math., 14, No. 1, 41-52
3. Allasia G (1995) A class of interpolating positive linear operators: theoretical and computational aspects. In: Approximation Theory: Wavelets and Applications (Singh SP ed), Kluwer, Dordrecht, 1-36
4. Allasia G (2003) Simultaneous interpolation and approximation by a class of multivariate positive operators. Numerical Algorithms (in print)
5. Anastassiou GA, Cottin C, Gonska HH (1991) Global smoothness of approximating functions. Analysis, 11, 43-57
6. Anastassiou GA, Gal SG (2000) Moduli of Continuity and Global Smoothness Preservation. Birkhäuser Boston
7. Anastassiou GA, Gal SG (2001) Partial shape-preserving approximation by bivariate Hermite-Fejér polynomials. Computers and Mathematics with Applic., 42, 57-64 (Pubitemid 32541826)
8. Anastassiou GA, Gal SG (2001) Partial shape-preserving approximation by bivariate Shepard operators. Computers and Mathematics with Applic., 42, 47-56 (Pubitemid 32535437)
9. Bari NK, Stechkin SB (1956) Best approximation and differential properties of two conjugate functions. (Russian) Trudy Moskov. Mat. Obshch., 5, 483-522
10. Barnhill RE (1977) Representation and approximation of surfaces. In: Mathematical Software, III, (Rice JR ed), Academic Press, New York, 69-120
11. Barnhill RE (1983) Asurvey of the representation and designofsurfaces. IEEE Computer Graphics and Applications, 3, No. 7, 9-16
12. Barnhill RE (1985) Surfaces in Computer Aided Geometric Design: A survey with new results. Comput. Aided Geom. Design, 2, 1-17
13. Barnhill RE, Dube RP, Little FF (1983) Properties of Shepard's surfaces. Rocky Mountain J. Math., 11, 365-381
14. Barnhill RE, Little FF (1984) Three-and four-dimensional surfaces. Rocky Mountain J. Math., 14, 77-102
15. Barnhill RE, Piper BR, Rescorla KL (1987) Interpolation to arbitrary data on a surface. In: Geometric Modelling: Algorithms and New Trends (Farin GE ed), SIAM, 281-290
16. Barnhill RE, Mansfield L (1972) Sard kernel theorems on triangular and rectangular domains with extensions and applications to finite-element error. Technical Report, Nr. 11, Department of Mathematics, Brunel University
17. Barnhill RE, Stean SE (1984) Multistage trivariate surfaces. Rocky Mountain J. Math., 14, No. 1, 103-118
18. Berman DL (1950) On some linear operators. (in Russian), Dokl. Akad. Nauk SSSR, 73, 249-252
19. Berman DL (1952) On a class of linear operators. (in Russian) Dokl. Akad. Nauk SSSR, 85, 13-16
20. Berman DL (1958) Divergence of the Hermite-Fejér interpolation processes. (in Russian) Uspekhi Mat. Nauk, 13(80), 143-148
21. Cavaretta AS, Sharma, JA, Varga RS (1980) Hermite-Birkhoff interpolation in the nth roots of unity. Trans. Amer. Math. Soc., 259, 621-628
22. Cheney EW (1966) Introduction to Approximation Theory. McGraw-Hill, New York
23. Coman Gh (1987) The remainder of certain Shepard-type interpolation formulas. Studia Univ. "Babeş-Bolyai, " 32, No. 4, 24-32
24. Coman Gh (1988) Shepard-Taylor interpolation. In: Itinerant Seminar on Functional Equations, Approximation and Convexity, "Babeş- Bolyai" University, Cluj-Napoca, 5-14
25. Coman Gh, Tambulea L (1988) A Shepard-Taylor approximation formula. Studia Univ. "Babeş-Bolyai, " 33, No. 3, 65-73
26. Coman Gh, Trimbitas RT (2001) Combined Shepard univariate operators.East J.Approx., 7, No. 4, 471-483
27. Coman Gh, Trimbitas RT (1999) Bivariate Shepard interpolation. Research Seminar, Seminarof Numerical and Statistical Calculus, Babeş-Bolyai University, Facultyof Mathematics and Computer Sciences, 41-83
28. Coman Gh, Trimbitas RT (1997) Shepard operators of Lagrange-type. Studia Univ. "Babes-Bolyai, " 42, No. 1, 75-83
29. Coman Gh, Trimbitas RT (2000) Bivariate Shepard interpolation in MATLAB. In: Proceeding of "Tiberiu Popoviciu" Itinerant Seminar, Cluj-Napoca, 41-56
30. Coman Gh, Trimbitas RT (2001) Univariate Shepard-Birkhoff interpolation. Revue d'Analyse Numer. Théor. Approx., Cluj-Napoca, 30, No. 1, 15-24
31. Criscuolo G, Mastroianni G (1993) Estimates of the Shepard interpolatory procedure. Acta Math. Hung., 61, No.1-2, 79-91
32. Della Vecchia B (1996) Direct and converse results by rational operators. Constr. Approx., 12, 271-285 (Pubitemid 126331791)
33. DellaVecchiaB, Mastroianni G (1991) Pointwise simultaneousapproximationbyrational operators. J. Approx. Theor., 65, 140-150
34. Della Vecchia B, Mastroianni G (1991) Pointwise estimates of rational operators based on general distribution of knots. Facta Universitatis (Nis), 6, 63-72
35. Della Vecchia B, Mastroianni G (1995) On function approximation by Shepard-type operators-a survey. In: Approximation Theory, Wavelets and Applications (Maratea 1994), vol. 454, Kluwer, Dordrecht, 335-346
36. Della Vecchia B, Mastroianni G, Szabados J (1996) Balász-Shepard operators on infinite intervals. Ann. Univ. Sci. Budapest Sect. Comp., 16, 93-102
37. Della Vecchia B, Mastroianni G, Szabados J (1998) Uniform approximation on the semi-axis by rational operators. East J. Approx., 4, 435-458
38. Della Vecchia B, Mastroianni G, Szabados J (1999) Weighted uniform approximation on the semiaxis by rational operators. J. of Inequal. and Appl., 4, 241-264
39. Della Vecchia B, Mastroianni G, Totik V (1990) Saturation of the Shepard operators. Approx. Theor. Appl., 6, 76-84
40. Della Vecchia B, Mastroianni G, Vértesi P (1996) Direct and converse theorems for Shepard rational approximation. Numer. Funct. Anal. Optimization, 17, No. 5-6, 537-561 (Pubitemid 126566065)
41. DeVore RA, Lorentz GG (1993) Constructive Approximation, Springer-Verlag, Berlin
42. Farwig R (1986) Rate of convergence of Shepard's global interpolation formula. Math. Comput., 46, 577-590
43. Fejér L (1930) Über Weierstrassche Approximation besonders durch Hermitesche Interpolation. Mathemathische Annalen, 102, 707-725
44. FlemingW (1977) Functions of Several Variables. Second edition, Springer-Verlag, New York
45. Foley TA (1987) Interpolation and approximation of 3-D and 4-D scattered data. Comput. Math. Applic., 13, 711-740 (Pubitemid 17600680)
46. Foley TA (1990) Interpolation of scattered data on a spherical domain. In: Algorithms for Approximation, II, (Mason JC, Cox MG eds), Chapman Hall, 303-310
47. Foley TA (1986) Scattered data interpolation and approximation with error bounds. Comput.-Aided Geom. Design, 3, 163-177
48. Foley TA (1984) Three-stage interpolation to scattered data. Rocky Mountain J. Math., 14, No. 1, 141-149
49. Foley TA, Nielson GM (1980) Multivariate interpolation to scattered data using delta iteration. In: Approximation Theory, III, (Cheney EW ed), Academic Press, New York, 419-424
50. Franke R (1982) Scattered data interpolation: tests of some methods. Math. Comput., 38, 181-200
51. Franke R (1977) Locally determined smooth interpolation at irregularly spaced points in several variables. JIMA, 19, 471-482
52. Franke R, Nielson GM (1983) Surface approximation with imposed conditions. In: Surfaces in CAGD, (Barnhill R, Boehm W, eds), North-Holland, Amsterdam, 135-146
53. Franke R, Nielson GM (1980) Smooth interpolation of large sets of scattered data. Int. J. Numer. Methods Eng., 15, 1691-1704
54. Gal SG (2004) Remarks on the approximation of normed spaces valued functions by some linear operators. In: Proceeding of the 6th Romanian-German Research Seminar, RoGer (to appear).
55. Gal SG, Gal CS, Partial shape preserving approximation by three kinds of bivariate Shepard operators, submitted.
56. Gal SG, Szabados J (1999) On the preservation of global smoothness by some interpolation operators. Studia Sci. Math. Hung., 35, 393-414
57. Gal SG, Szabados J (2002) Partial shape preserving approximation by interpolation operators. Functions, Series, Operators, Alexits Memorial Conference, 1999 (Leindler L, Schipp F, Szabados J eds), Budapest, 225-246
58. Gal SG, Szabados J (2003) Global smoothness preservation by bivariate interpolation operators. Analysis in Theory and Appl., 19, No. 3, 193-208 (Pubitemid 41626028)
59. Gonska HH (1983) A note on pointwise approximation by Hermite-Fejér type interpolation polynomials. In: Functions, Series, Operators, vol. I, II (Proc. Int. Conf. Budapest 1980; Nagy BSz, Szabados J eds), Amsterdam, New York, 525-537
60. Gonska HH (1983) On approximation in spaces of continuous functions, Bull. Austral. Mat. Soc., 28, 411-432
61. Gonska HH, Jetter K (1986) Jackson-type theorems on approximation by trigonometric and algebraic pseudopolynomials. J. Approx. Theory, 48, 396-406
62. Gordon WJ, Wixom JA (1978) Shepard's method of "metric interpolation" to bivariate and multivariate interpolation. Math. Comput., 32, 253-264
63. Haussmann W, Pottinger P (1977) On the construction and convergence of multivariate interpolation operators. J. Approx. Theory, 19, 205-221
64. Jiang G (1990) The degree of approximation of Grünwald interpolation polynomial operators. Pure Appl. Math., 6, No. 2, 82-84
65. Katkauskayte AY (1991) The rate of convergence of a Shepard estimate in the presence of random interference. Soviet J. Automat. and Inform. Sciences, 24, No. 5, 19-24 (Pubitemid 23570129)
66. Kratz W, Stadtmüller U (1988) On the uniform modulus of continuity of certain discrete approximation operators. J. Approx. Theory, 54, 326-337
67. Kroó A, Révész S (1999) On Bernstein and Markov-type inequalities for multivariate polynomials on convex bodies. J. Approx. Theory, 99, 139-152
68. Kryloff N, Stayermann E (1922) Sur quelques formules d'interpolation convergentes pour toute fonction continue. Bull. Classe Sci. Phys. Math. Acad. Sci. Ukraine, 1, 13-16
69. 1 surface interpolation for scattered data on a sphere. Rocky Mountain J. Math., 14, No. 1, 177-202
70. Little FF (1983) Convex combination surfaces. In: Surfaces and Computer-Aided Geometric Design, (Barnhill R, Boehm W eds), North-Holland, Amsterdam, 99-107
71. Marcus S (1956) Functions monotone de deux variables. Rev. Roum. Math. Pures Appl., 1(2), 17-36
72. Mastroianni G, Szabados J (1993) Jackson order of approximation by Lagrange interpolation. Suppl. Rend. Circ. Mat. Palermo, ser. II, No. 33, 375-386
73. Mastroianni G, Szabados J (1997) Balász-Shepard operators on inifnite intervals. II, J. Approx. Theory, 90, 1-8
74. McLain DH (1976) Two-dimensional interpolation from random data. Comput. J., 19, 178-181; Errata, Comput. J., 19, 384
75. Nguyen KA, Rossi I, Truhlar DG (1995) A dual-level Shepard interpolation method for generating potential energy surfaces for dynamics calculations. J. Chem. Phys., 103, 5522
76. Nicolescu M (1952) Contribution to a hyperbolic kind analysis of plane. (in Romanian) St. Cerc. Mat. (Bucharest), 3, No.1-2, 7-51
77. Nielson GM, Franke R (1984)A method for construction of surfaces under construction. Rocky Mountain J. Math., 14, No. 1, 203-221
78. p spaces. Approx. Theory Appl., 1, No. 1, 1-10
79. Popoviciu T (1960-1961) Remarques sur la conservation du signe et de la monotonie par certains polynomes d'interpolation d'une fonctions d'une variable. Ann. Univ. Sci. Budapestinensis, III-IV, 241-246
80. Popoviciu T (1961) Sur la conservation de l'allure de convexité d'une fonction par ses polynomes d'interpolation. Mathematica (Cluj), vol. 3, No. 2, 311-329
81. Popoviciu T (1962) Sur la conservation par le polynome d'interpolation de L. Fejér, du signe ou de la monotonie de la fonction. An. Stiint. Univ. Jassy, VIII, 65-84
82. Popoviciu T (1934) Sur quelques propriétés des fonctions d'une de/deux variables reélles. Mathematica (Cluj), VIII, 1-85
83. Prestin J, Xu Y (1994) Convergence rate for trigonometric interpolation of nonsmooth functions. J. Approx. Theory, 77, 113-122
84. Rubel LA, Shields AI, Taylor BA (1975) Mergelyan sets and the modulus of continuity of analytic functions. J. Approx. Theory, 15, 23-40
85. Runck PO, Szabados J, Vértesi P (1989) On the convergence of the differentiated trigonometric projection operators. Acta Sci. Math. (Szeged), 53, 287-293
86. Schumaker LL (1976) Fitting surfaces to scattered data. In: Approximation Theory, II, (Lorentz GG, Chui CK, Schumaker LL eds), Academic Press, New York, 203-268
87. Sendov B (1968) Approximation with Respect to Hausdorff Metric. Thesis (in Russian), Moscow, 1968
88. Sendov B, Popov VA (1983) Averaged Moduli of Smoothness, Bulgarian Mathematical Monographs, Vol. 4, Publishing House of the Bulgarian Academy of Science, Sofia
89. Sharma A, Szabados J (1988) Convergence rates for some lacunary interpolators on the roots of unity. Approx. Theory and Its Applic., 4, No. 2, 41-48
90. Sheng B, Cheng G (1995) On the approximation by generalized Grünwald interpolation in Orlicz spaces. J. Baoji Coll. Arts Sci. Nat. Sci., 15, No. 4, 7-13
91. Shepard D (1968)A two-dimensional interpolation function for irregularly spaced points. Proceedings 1968 Assoc. Comput. Machinery National Conference, 517-524
92. Shisha O, Mond B (1965) The rapidity of convergence of the Hermite-Fejér approximation to functions of one or several variables. Proc. Amer. Math. Soc., 16, No.6, 1269-1276
93. Somorjai G (1976) On a saturation problem. Acta Math. Acad. Sci. Hungar., 32, 377-381
94. Stechkin SB (1951) On the order of best approximation of periodic functions. (in Russian) Izv. Akad. Nauk SSSR, 15, 219-242
95. Szabados J. (1973) On the convergence and saturation problem of the Jackson polynomials. Acta Math. Acad. Sci. Hungary, 24, 399-406
96. Szabados J (1976) On a problem of R. De Vore. Acta Math. Acad. Sci. Hungary, 27, 219-223
97. Szabados J (1987) On the convergence of the derivatives of projection operators.Analysis, 7, 349-357
98. Szabados J (1991) Direct and converse approximation theorems for the Shepard operator. Approx. Theory Appl. (N.S.), 7, 63-76
99. Szabados J, Vértesi P (1989) On simultaneous optimization of norms of derivatives of Lagrange interpolation polynomials. Bull. London Math. Soc., 21, 475-481
100. Szabados J, Vértesi P (1990) Interpolation of Functions. World Scientific, Singapore
101. Szabados J, Vértesi P (1992) A survey on mean convergence of interpolatory processes. J. Comp. Appl. Math., 43, 3-18
102. Szegö G (1939) Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ., vol. XXIII, NewYork
103. TimanAF (1994) Theory of Approximation of Functions of a Real Variable, Dover, New York
104. Trimbitas RT (2002) Univariate Shepard-Lagrange interpolation. Kragujevac J. Math., 24, 85-94
105. Trimbitas RT (2002) Local bivariate Shepard interpolation. Rend. Circ. Matem. Palermo, Serie II, Suppl. 68, 865-878
106. Trimbitas MG, Trimbitas RT (2002) Range searching and local Shepard interpolation. In: Mathematical Analysis and Approximation Theory, The 5th Romanian-German Seminar on Approximation Theory and Its Applications, RoGer, Burg Verlag, Sibiu, 279-292
107. Vértesi P (1996) Saturation of the Shepard operator. Acta Math. Hungar., 72, No. 4, 307-317 (Pubitemid 126314675)
108. Xie T, Zhang RJ, Zhou SP (1998) Three conjectures on Shepard interpolatory operators. J. Approx. Theory, 93, No. 3, 399-414 (Pubitemid 128440208)
109. Zhou X (1998) The saturation class of Shepard operators. Acta Math. Hungar., 80, No. 4, 293-310
110. Zygmund A (1955) Trigonometric Series. Dover, New York
111. Zygmund A (1959) Trigonometric Series. 2nd edition, Cambridge Univ. Press, London, New York, 1959, vol. II, p. 30