A direct approach to the mellin transform

On Hardy spaces on worm domains

“…The argument that it is used to prove (14) cannot work in this context. In particular, to prove (14) it is used the mean value property of the holomorphic function K. ; w/; the Szegő projection of a function F 2 C 1 0 .…”

“…In particular, to prove (14) it is used the mean value property of the holomorphic function K. ; w/; the Szegő projection of a function F 2 C 1 0 . @Dˇ/ is a new function S DˇF defined on @Dˇgiven by the integration against the Szegő kernel on the distinguished boundary @Dˇ.…”

“…The fact that we are integrating on the boundary Dˇprevents us to exploit the holomorphicity of the function K Dˇ. ; w/ in Dˇ, thus to use the mean value property and obtain a conclusion similar to (14).…”

Convergence and rate of approximation in $BV^{\varphi}(\mathbb R^N_+)$ for a class of Mellin integral operators

“…which is of bounded ϕ−variation on R + , but not ϕ−absolutely continuous, and the Mellin Gauss-Weierstrass kernels (see, e.g., [21] and [10] for their multidimensional version) defined as G w (t) =…”

“…The importance of Mellin operators in approximation theory is well-known: they are widely studied (see, e.g., [32,21]) and they have important applications in several fields. For example, we recall that Mellin analysis has deep connections with Signal Processing, in particular with the so-called Exponential Sampling (see [22]).…”