On (p,q)-Analogs of the α-th Fractional Fourier Transform and Some (p,q)-Generalized Spaces

In this article, the (p,q)-analogs of the α-th fractional Fourier transform are provided, along with a discussion of their characteristics in specific classes of (p,q)-generalized functions. Two spaces of infinitely (p,q)-differentiable functions are defined by introducing two (p,q)-differential symmetric operators. The (p,q)-analogs of the α-th fractional Fourier transform are demonstrated to be continuous and linear between the spaces under discussion. Next, theorems pertaining to specific convolutions are established. This leads to the establishment of multiple symmetric identities, which in turn requires the construction of (p,q)-generalized spaces known as (p,q)-Boehmians. Finally, in addition to deriving the inversion formulas, the generalized (p,q)- analogs of the α-th fractional Fourier transform are introduced, and their general properties are discussed.