Discrete Jacobi-Dunkl transform and approximation theorems. (English) Zbl 1524.41070
Summary: This paper uses some basic notions and results from the discrete harmonic analysis associated with the Jacobi-Dunkl operator to study some problems in the theory of approximation of functions in the space \(\mathbb{L}_2^{(\alpha, \beta)}\). Analogs of the direct Jackson theorems of approximations for the modulus of smoothness (of arbitrary order) constructed using the translation operators which was defined by Vinogradov are proved. In conclusion of this work, we show that the modulus of smoothness and the \(K\)-functionals constructed from the Sobolev-type space corresponding to the Jacobi-Dunkl Laplacian operator are equivalent.
MSC:
| 41A36 | Approximation by positive operators |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 44A20 | Integral transforms of special functions |
Keywords:
Jacobi-Dunkl operator; Jacobi polynomials; discrete Jacobi-Dunkl transform; Jacobi-Dunkl translation operator; Jacobi-Dunkl Laplacian operator; \(K\)-functionals; modulus of smoothnessReferences:
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