Smoothness of a function and the growth of its Fourier transform or its Fourier coefficients

  • Authors:

  • Z. Ditzian

  • Affiliations:

  • Department of Mathematical, and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2010

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Abstract

In a recent paper Bray and Pinsky [1] estimated the growth of f@^(@x), the Fourier transform of f(x) where x@?R^d, by some moduli of smoothness. We show here that noticeably better results can be derived as an immediate corollary of previous theorems in [2]. The improvements include dealing with higher levels of smoothness and using the fact that for higher dimensions (when d=2) the description of smoothness requires less information. Using a similar technique, we also deduce relations between the smoothness of f(x) for x@?S^d^-^1 or x@?T^d and the growth of the coefficients of the expansion by spherical harmonic polynomials or trigonometric polynomials.