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Localized Summability Kernels for Jacobi Expansions

In: Mathematical Analysis, Approximation Theory and Their Applications

Author

Listed:
  • H. N. Mhaskar

    (California Institute of Technology
    Claremont Graduate University)

Abstract

While the direct and converse theorems of approximation theory enable us to characterize the smoothness of a function f : [ − 1 , 1 ] → ℝ $$f: [-1,1] \rightarrow \mathbb{R}$$ in terms of its degree of polynomial approximation, they do not account for local smoothness. The use of localized summability kernels leads to a wavelet-like representation, using the Fourier–Jacobi coefficients of f, so as to characterize the smoothness of f in a neighborhood of each point in terms of the behavior of the terms of this representation. In this paper, we study the localization properties of a class of kernels, which have explicit forms in the “space domain,” and establish explicit bounds on the Lebesgue constants on the summability kernels corresponding to some of these.

Suggested Citation

  • H. N. Mhaskar, 2016. "Localized Summability Kernels for Jacobi Expansions," Springer Optimization and Its Applications, in: Themistocles M. Rassias & Vijay Gupta (ed.), Mathematical Analysis, Approximation Theory and Their Applications, pages 417-434, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-31281-1_18
    DOI: 10.1007/978-3-319-31281-1_18
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