Survey of Quasiasymptotic Behavior of Distributions in Relation to the Properties of Their Fractional Transforms

Fractional transforms have emerged as powerful analytical tools that bridge the time, frequency, and scale domains by introducing a fractional-order parameter into the kernel of classical transforms. This survey provides an overview of the mathematical foundations and distributional frameworks of several key fractional transforms, with emphasis on their formulation within appropriate spaces of generalized functions. Particular attention is devoted to the quasiasymptotic behavior of distributions in relation to the asymptotic properties of their corresponding fractional transforms. We demonstrate how individual transforms map illustrative signals into their corresponding domains and identify the values of the parameter α for which they produce the best results.