4 research outputs found
Growth and integrability of Fourier transforms on Euclidean space
A fundamental theme in classical Fourier analysis relates smoothness
properties of functions to the growth and/or integrability of their Fourier
transform. By using a suitable class of multipliers, a rather general
inequality controlling the size of Fourier transforms for large and small
argument is proved. As consequences, quantitative Riemann-Lebesgue estimates
are obtained and an integrability result for the Fourier transform is developed
extending ideas used by Titchmarsh in the one dimensional setting
Moduli of smoothness and growth properties of Fourier transforms: two-sided estimates
We prove two-sided inequalities between the integral moduli of smoothness of
a function on and the weighted tail-type integrals
of its Fourier transform/series.
Sharpness of obtained results in particular is given by the equivalence
results for functions satisfying certain regular conditions. Applications
include a quantitative form of the Riemann-Lebesgue lemma as well as several
other questions in approximation theory and the theory of function spaces.Comment: 22 page
Moduli of smoothness and growth properties of Fourier transforms : two-sided estimates
We prove two-sided inequalities between the integral moduli of smoothness of a function on R d[superscript] / T d[superscript] and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is given by the equivalence results for functions satisfying certain regular conditions. Applications include a quantitative form of the Riemann-Lebesgue lemma as well as several other questions in approximation theory and the theory of function spaces