A non-commutative Sobolev estimate and its application to spectral synthesis

TL;DR

This paper demonstrates that discrete sets serve as spectral synthesis sets for the Alpha transform, linking spectral synthesis problems to realizations of the canonical representation of the Heisenberg group.

Contribution

It establishes that discrete sets are spectral synthesis sets for the Alpha transform, advancing understanding of spectral synthesis in harmonic analysis.

Findings

Discrete sets are spectral synthesis sets for the Alpha transform

Connects spectral synthesis to canonical representation realizations

Provides new insights into spectral analysis of the Heisenberg group

Abstract

In [M. K. Vemuri, Realizations of the canonical representation], it was shown that the spectral synthesis problem for the Alpha transform is closely related to the problem of classifying realizations of the canonical representation (of the Heisenberg group). In this paper, we show that discrete sets are sets of spectral synthesis for the Alpha transform.