2D Linear Canonical Transforms on Lp and Applications

Abstract

As Fourier transformations of Lp functions are the mathematical basis of various applications, it is necessary to develop Lp theory for 2D-LCT before any further rigorous mathematical investigation of such transformations. In this paper, we study this Lp theory for 1≤p<∞. By defining an appropriate convolution, we obtain a result about the inverse of 2D-LCT on L1(R2). Together with the Plancherel identity and Hausdorff–Young inequality, we establish Lp(R2) multiplier theory and Littlewood–Paley theorems associated with the 2D-LCT. As applications, we demonstrate the recovery of the L1(R2) signal function by simulation. Moreover, we present a real-life application of such a theory of 2D-LCT by encrypting and decrypting real images.

Discrete Octonion Linear Canonical Transform: Definition and Properties

“…The linear canonical transform (LCT) [1][2][3] is a generalized form of the fractional Fourier transform (FrFT). As a linear integral transform with three parameter class, the LCT is more flexible than the FrFT and is a widely used analytical and processing tool in applied mathematics and engineering [4][5][6][7][8].…”