A generalization of the Paley–Wiener theorem for Mellin transforms and metric characterization of function spaces

Abstract

We characterize the function space whose elements have a Mellin transform with exponential decay at infinity. This result can be seen as a generalization of the Paley–Wiener theorem for Mellin transforms. As a byproduct in a similar spirit, we also characterize spaces of functions whose distances from Mellin–Paley–Wiener spaces have a prescribed asymptotic behavior. This leads to Mellin–Sobolev type spaces of fractional order.

Quadrature formulae for the positive real axis in the setting of Mellin analysis: sharp error estimates in terms of the Mellin distance

“…The final results give some interesting equivalence theorems which characterize function spaces in terms of the speed of convergence to zero of the remainders in (1). This generalizes and completes the results given in [39] by employing a modified notion of analyticity introduced in [5], called "polar-analyticity", and a generalization of the Paley-Wiener theorem in the Mellin frame, also established in [5] (also see [2] and [4]). 1 The characterizations are obtained by two different approaches (see Section 4): The first one applies to the so-called Mellin-even part of f and makes use of the Möbius inversion formula-a tool of number theory first used in numerical analysis by Lyness [28], Brass [9] and Loxton-Sanders [29], [30] for deducing properties of a function f from its remainders in a quadrature formula.…”

“…The next result can be deduced from Theorem C; for details see [5,Theorem 4.2]. We reproduce it in a concise form.…”

“…We note that W r+α,1 c (R + ) ⊂ W r+α, * c (R + ). Now the following proposition, stated in [5,Theorem 4.3], is a simple consequence of [3,Theorem 5].…”

“…In [3] we proved that functions from a Mellin-Sobolev space of order r have distances from B 1 c,σ that behave like O(σ −r ). However, as remarked in [5], the familiar Mellin-Sobolev spaces are not appropriate for characterizing this rate of convergence. The following modified space will accomplish the desired equivalence trivially.…”

Polar-Analytic Functions: Old and New Results, Applications

“…The next theorem characterizes the space of functions whose Mellin transform has an exponential decay at infinity. In order to do that, we have to introduce a new function space that is related to Mellin-Hardy type spaces (for details see [4]). For any function φ : R + → C that is integrable on every compact subinterval of R + , we define…”