The Multidimensional Quadratic Phase Fourier Transform: Theoretical Analysis and Applications

Abstract: The quadratic phase Fourier transform (QPFT) is a generalization of several well-known integral transforms, including the linear canonical transform (LCT), fractional Fourier transform (FrFT), and Fourier transform (FT). This paper introduces the multidimensional QPFT and investigates its theoretical properties, including Parseval's identity and inversion theorems. Generalized convolutions and correlation for multiple variables, extending the conventional convolution for single-variable functions, are proposed within the QPFT setting. Additionally, a Boas-type theorem for the multidimensional QPFT is established. As applications, multiplicative filter design and the solution of integral equations using the proposed convolution operation are explored.