Generalized Convolution Operator Associated with the (k, a)-Generalized Fourier Transform on the Real Line and Applications

Recently in Amri (Product formula for one-dimensional (k, a)-generalized Fourier kernel. arXiv:2301.06587), the author proved the product formula for the one dimensional (k, a)-generalized Fourier kernel. Profiting for this result, the primary aim of the present paper, is to develop the harmonic analysis associated with (k, a) the generalized Fourier transform. Firstly we study the (k, a)-generalized translation operator associated with the (k, a)-generalized Fourier transform. By means of the generalized translation operator, we define and we investigate the generalized convolution product in the setting of the (k, a)-generalized Fourier transform. Nevertheless, significant attention is also devoted to the time-frequency analysis by examining some applications on the wavelet transform in the (k, a)-generalized Fourier transform setting.