Binomial polynomials and their applications in Approximation Theory

2 years ago

(original), Approximation Theory, not-at-ICTP, paper

Abstract

In this paper we are concerned with the sequences of polynomials of binomial type. In particular we point out their remarkable algebraic-combinatorial properties related to the so called delta operators as used in a sseries of papers on the foundations of combinatorial theory, see [23], [27]. In order to detail theis field, the theoretical aspects are illustrated with several concrete examples. The paper is also a survey of the role of these polynomials in Approximation Theory and it includes the construction of general binomial type operators and their main approximation properties.

Authors

Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

Keywords

Paper coordinates

O. Agratini, Binomial polynomials and their applications in Approximation Theory, Conferenze del Seminario di Matematica dell Universita di Bari, 281 (2001), pp. 1-22.

PDF

https://ictp.acad.ro/wp-content/uploads/2023/03/2001a-Agratini-Binomial-polynomials-and-their-applications-in-approximation-theory.pdf

About this paper

Journal

Publisher Name

DOI

Print ISSN

Online ISSN

google scholar link

[1] M. Abramowits, I.A., Stegun (eds), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National of Standards Applied Mathematics Series 55, Issued June, 1964.
[2] O. Agratini, “On a certain class of approxcimation operators” Pure Mathematics and Applications, 11, 2000, in print.
[3] F. Altomare, M. Campiti, Korokvin-Type Approximation Theory and its Applications, de Guyter Series Studies in Mathematics, vol. 17, Walter de Gruyter, Berlin-New York, 1994.
[4] P. Appel, Sur une classe de polynomes, Ann. Sci. Ecole Norm. Sup (2), 9, 1980, 119-144.
[5] A. Di Bucchianico, D.E. Loeb, Natural exponential families and umbral calculus, in Mathematical Essays in Honor of Gian-Carlo Rota (Bruce E. Sagan, Richard P. Stanley, eds.), Progress in Mathematics, vol. 161, Birkhauser, 1998.
[6] B.C. Carlson, Polynomials satisfying a binomial theorem, J. Math. Anal. Appl., 32, 1970, 543-558.
[7] E.W. Cheney, A. Sharma, On generalization of Bernstein polynomials, Revista di Matematica Univ. Parma 5, 1964, 77-84.
[8] C. Cottin, H.H. Gonska, Simultaneous approximation and global smoothness preservation, Rend. Circ. Mat. Palermo, Serie 2, 33, 1993, 259-279.
[9] Z. Ditzian, V. Totik, Moduli of Smmothness, Springer Series in Computational Mathematics, vol. 9, Springer Verlag, Berlin/Heidelber/New York, 1987.
[10] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York, 1953.
[11] F.B. Hildbrand, Introduction to Numerical Analysis, Mc Graw-Hill, New York 1956.
[12] Mourad E.H. Ismail, Polynomials of binomial type and approximation theory, Journal of Approx. Theory 23, 1978, 177-186.
[13] Mourad E.H. Ismail, C.P. May, On a family of approximation operators, J. Math. Anal. Appl. 63, 1978, 446-462.
[14] C. Jordan, Calculus of Finite Differences, Chelsea Publishing Company, New York 1950.
[15] T. Lindvall, Bernstein polynomials and the law of large numbers, Math. Scientist, 7, 1982, 127-139.
[16] G.G. Lorentz, Bernstein Polynomials, Univ. of Toronto Press, Toronto 1953.
[17] A. Lupas, Approximation operators of binomial type, New developments in approximation theory (Dortmund, 1998), 175-198, International Series of Numerical Mathematics, vol. 132, Birkhauser Verlag Basel/Switerland 199
[18] L. Lupas, A. Lupas, Polynomials of binomial type and approximation operators, Studia Univ. Bab es-Bolyai, Mathematica, 32, 1987, 4, 61-69.
[19] C. Manole, Dezvoltări în serii de polinoame Appell genralizate cu aplicații la aproximarea funcțiilor, Ph.D Thesis, Cluj-Napoca 1984.
[20] C.P. May, Saturation and inverse theorems for combinations of a class of exponential-type operators, Canadian J. Math., 28, 1976, no.6, 1224-1250.
[21] V. Miheșan, Aproximarea funcțiilor continue prin operatori liniari și pozitivi, Ph. D. Thesis, Cluj-Napoca 1997.
[22] G.V. Milovanovic, D.S. Mitrinovic, Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific Publishing Co. Pte Ltd., Singapore 1994.
[23] R. Mullin, G.=C. Rota, On the Foundations of Combinatorial Theory. III. Theory of binomial enumeration, in Graph Theory and its Applications, (B. Harris, ed.), Academic Press, 1970, 167=213.
[24] T. Popoviciu, Remarques sur les polynomes binomiaux, Bul. Soc. Sci. Cluj (Roumanie), 6, 1931, 146-148 (also reproduced in Mathematica Cluj, 6, 1932, 8-10).
[25] T. Popoviciu, Les Fonctions Convexes, Actualites Scientifique et Industrielles, 992, (Publies sous la direction de Paul Montel), XVII, Herman & C^{ie}, Editeurs, Paris 1994.
[26] S. Roman, G.-C. Rota, The umbral calculus, Advances in Math., 27, 1978, 95-188.
[27] G.-C. Rota, D. Kahaner, A. Odlyzko, On the Foundations of Combinatorial Theory. VIII. Finite operator calculus, Journal of Mathematical Analysis and Applicaitons, 42, 1973, 685-760.
[28] P. Sablonniere, Positive Bernstein-Sheffer operators, Journal of Approx. Theory 83, 1995, 330-341.
[29] C. Scaravelli, Su i polinomi di Appell, Rivista di Matematica Univ. Parma, serie 2, 6, 1965, 95-108.
[30] C. Scaravelli, Polinomi di Appell nel senso del Calcolo delle differenze finite, Rivista di Matematica Univ. Parma, Serie 2, 8, 1967, 355-366.
[31] I.M. Sheffer, Some properties of polynomials sets of type zero, Duke Math.Journal 5, 1939, 590-622.
[32] D.D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roum. Pures et Appl. 13, 1968, no.8, 1173-1194.
[33] D.D. Stancu, M.R., Occorsio, On approximation by binomial operators of Tiberiu Popoviciu type, Revue d’Analyse Num.et.de Theorie de l’Approxi., Tome 27, 1998, no.1, 167=181.
[34] J.F., Steffensen, Interpolation, Williams and Wilkins Co., Baltimore 1927.
[35] X.-H. Sun, New charateristics of some polynomial sequences in combinatorial theory, J. Math. Anal. Appl. 175, 1993, 199-205.
[36] A.H. Zemanian, Generalized Integral Transformations, Interscience, New York 1965.
[37] Ding-Xuan Zhou, On a problem of Gonska, Results in Math. 28, 1995, 169-183.