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

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