It is well-known that the ordinary convolution is closely related with the Fourier transform. It is therefore natural to ask: for other important integral transforms, can we define generalized convolution operators having analogous properties? Actually, the answer depends on the existence of a product formula for the kernel of the integral transform. In this talk, I will explain the general connection between product formulas, generalized convolutions and integral transforms. I will report on recent progress in constructing the product formula and convolution associated with the index Whittaker transform. Some applications will be presented, and the probabilistic motivation behind this work will be discussed. (*) Rúben Sousa is a PhD student of the Joint PhD Program UC|UP, working at the University of Porto, in Analysis, under the supervision of Professor Semyon Yakubovich (Univ. Porto) and Professor Manuel Guerra (Univ. Lisboa).
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