Fractal Fract, Vol. 7, Pages 100: 2D Linear Canonical Transforms on Lp and Applications

Fractal Fract, Vol. 7, Pages 100: 2D Linear Canonical Transforms on Lp and Applications

Fractal and Fractional doi: 10.3390/fractalfract7020100

Authors: Yinuo Yang Qingyan Wu Seong-Tae Jhang

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.