Boundedness of STFrFT on Both Unweighted and Weighted Hardy and BMO Spaces

Abstract

The Fourier transform of a signal lacks local information in that it does not reflect the change of frequency over time. The Fourier transform method allows for the investigation of problems in either the time domain or the frequency domain, but not in both simultaneously. The use of short-time Fourier transforms enables the incorporation of both time and frequency localisation properties within a single transform function, through the utilisation of a window function. In cases where the fractional Fourier transform, which is more general than the Fourier transform, is unable to identify the frequency contents within the fractional Fourier domain, the short-time fractional Fourier transform is proposed as a potential solution. The present paper addresses the issue of boundedness of the short-time fractional Fourier transform. It is shown that the short-time fractional Fourier transform provides boundedness results in Hardy, BMO, weighted Hardy and weighted BMO spaces. This study also examines the Hardy and BMO-distance and weighted Hardy and weighted BMO-distance between two short-time fractional Fourier transforms associated with different windows and different signals. © 2024, MTJPAM Turkey. All rights reserved.