On Fourier Re-Expansions

“…In [Lif14], a similar problem of the integrability of the re-expansion for Fourier transforms of functions defined on R + = [0, ∞) has been studied. Surprisingly, necessary and sufficient conditions in terms of the membership of the sine or cosine Fourier transform in a certain Hardy space have been found.…”

Section: Introductionmentioning

confidence: 99%

Re-expansions on compact Lie groups

Akylzhanov¹,

Liflyand²,

Ruzhansky³

2019

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper we consider the re-expansion problems on compact Lie groups. First, we establish weighted versions of classical re-expansion results in the setting of multi-dimensional tori. A natural extension of the classical re-expansion problem to general compact Lie groups can be formulated as follows: given a function on the maximal torus of a compact Lie group, what conditions on its (toroidal) Fourier coefficients are sufficient in order to have that the group Fourier coefficients of its central extension are summable. We derive the necessary and sufficient conditions for the above property to hold in terms of the root system of the group. Consequently, we show how this problem leads to the re-expansions of even/odd functions on compact Lie groups, giving a necessary and sufficient condition in terms of the discrete Hilbert transform and the root system. In the model case of the group SU(2) a simple sufficient condition is given.

show abstract

Summability to the Hilbert Transform

Liflyand

2021

Operator Theory and Harmonic Analysis

View full text Add to dashboard Cite

On Fourier Re-Expansions

Cited by 6 publications

References 18 publications

Interaction between the Fourier transform and the Hilbert transform

Interaction between the Fourier transform and the Hilbert transform

Re-expansions on compact Lie groups

Summability to the Hilbert Transform

Contact Info

Product

Resources

About