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

In a recent paper Bray and Pinsky [1] estimated the growth of View the MathML sourcef̂(ξ), the Fourier transform of View the MathML sourcef(x) where View the MathML sourcex∈Rd, 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≥2d≥2) the description of smoothness requires less information. Using a similar technique, we also deduce relations between the smoothness of View the MathML sourcef(x) for View the MathML sourcex∈Sd−1 or View the MathML sourcex∈Td and the growth of the coefficients of the expansion by spherical harmonic polynomials or trigonometric polynomials.