Dilation‐and‐modulation frame sets on the half real line

Abstract

Wavelet and Gabor frame sets on the real line have been extensively studied. Due to it not being a group under addition, the half real line admits no traditional wavelet and Gabor frames as the real line. In this paper, we introduce a class of dilation‐and‐modulation frame sets ( ‐frame sets) on the half real line. We characterize ‐frame sets and ‐tight frame sets and obtain some sufficient conditions for ‐tight frame sets. In particular, we prove that ‐tight frame sets must be bounded, but ‐frame sets need not be bounded. Some examples are also provided.