A note on a family of two-variable polynomials

AbstractThe main object of this paper is to construct a two-variable analogue of Jacobi polynomials and to give some properties of these polynomials. We show that these polynomials are orthogonal, then we obtain various recurrence formulas for them. Furthermore, we give some integral representations for these polynomials

Fourier transform of the orthogonal polynomials on the unit ball

Fourier transform of multivariate orthogonal polynomials on the unit ball are obtained. By using Parseval's identity, a new family of multivariate orthogonal functions are introduced. The results are expressed in terms of the continuous Hahn polynomials.Comment: 15 page

On approximation properties of generalized Lupas type operators based on Polya distribution with Pochhammer k-symbol

The purpose of this paper is to introduce a Kantorovich variant of Lupas-Stancu operators based on Polya distribution with Pochhammer k-symbol. We obtain rates of convergence for these operators by means of the classical modulus of continuity. Also, we give a Voronovskaja type theorem for the pointwise approximation. Furthermore, we construct a bivariate generalization of these operators and we discuss some convergence properties of them. Finally, we present some figures to compare approximation properties of our new operators with those of other operators which are mentioned in this paper. We observe that the approximation of our operators to the function f is better than that of some other operators in a certain range of values of k

Three term relations for multivariate Uvarov orthogonal polynomials

Three term relations for orthogonal polynomials in several variables associated to a moment linear functional obtained by a Uvarov modification of a given moment functional are studied. Existence of Uvarov orthogonal polynomials is analyzed, stating conditions to ensure it. The matrices of the three term relations of the Uvarov orthogonal polynomials are explicitly given in terms of the matrices of the three term relations satisfied by the original family. Two examples are presented in order to show that the results are valid for positive definite linear functionals and also for some quasi definite linear functionals which are not positive definite.Agencia Estatal de Investigación | Ref. PID2020-113275GB-I00IMAG-María de Maeztu | Ref. CEX2020-001105-MAgencia Estatal de Investigación | Ref. PGC2018-094932-B-I00Junta de Andalucía | Ref. A-FQM-246-UGR20Universidade de Vigo/CISU