Converse theorems in the theory of approximate integration

Abstract

We characterize certain function classes in terms of the remainder in o~ co q_ the quadrature formula S_~of(x)dx=(~r/cr)Y~ .... f(ulr/cr) R~[f]. In the process, we prove a generalization of the famous theorem of Paley and Wiener about entire functions of exponential type belonging to L 2 on the real line.

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

“…In this section we aim at the generalization of the Paley-Wiener theorem for the Mellin transform which will later be applied for characterizing function spaces in terms of the distance from a Mellin-Bernstein space. The Fourier counterpart was proved in [17]. We need the following space H * c (H a ), which lies between two Mellin-Hardy spaces.…”

“…With the aim to characterize certain classes of functions in terms of the remainders of certain Gaussian type quadrature formulae, the authors in [17] proved an interesting generalization of the Paley-Wiener theorem, characterizing a space of functions for which the Fourier transform has an exponential decay at infinity. This has several interesting applications for quadrature formulae.…”

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

“…In fact, it is the compound trapezoidal rule on R. When appropriately scaled, it is exact for bandlimited functions for which it has features of a Gaussian quadrature formula (see [34], [36], [37], [38], [18, § 2.11.2]). Recently precise estimates of the remainder in terms of a new metric have been established (see [20] which continues earlier results in [24]).…”

“…Next, from Theorem 4, we deduce that (24) shows that the estimate (14) is best possible in the sense described in Theorem 4.…”

“…Both approaches will be considered in this section. We mention that some basic results of this kind were obtained in [24], [35] for quadrature formulae over the whole real axis and in [39] for the positive real axis.…”