Jasson Vindas

New Classes of Weighted Hölder-Zygmund Spaces and the Wavelet Transform

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces 𝒮0(ℝn) and 𝒮(ℍn

The Ridgelet transform of distributions

We define and study the Ridgelet transform of (Lizorkin) distributions. We establish connections with the Radon and wavelet transforms.

On quasianalytic classes of Gelfand-Shilov type. Parametrix and convolution

We develop a convolution theory for quasianalytic ultradistributions of Gelfand-Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in the study of new topological and structural properties of several quasianalytic spaces of functions and ultradistributions. In particular, our results apply to Fourier hyperfunctions and Fourier ultra-hyperfunctions.

Boundary values of holomorphic functions and heat kernel method in translation-invariant distribution spaces

We study boundary values of holomorphic functions in translation-invariant distribution spaces of type . New edge of the wedge theorems are obtained. The results are then applied to represent as a quotient space of holomorphic functions. We also give representations of elements of via the heat kernel method. Our results cover as particular instances the cases of boundary values, analytic representations and heat kernel representations in the context of the Schwartz spaces , , and their weighted versions.

On PNT equivalences for Beurling numbers

In classical prime number theory several asymptotic relations are considered to be “equivalent” to the prime number theorem. In the setting of Beurling generalized numbers, this may no longer be the case. Under additional hypotheses on the generalized integer counting function, one can however still deduce various equivalences between the Beurling analogues of the classical PNT relations. We establish some of the equivalences under weaker conditions than were known so far.

Eigenfunction expansions of ultradifferentiable functions and ultradistributions in \mathbb R^n

We obtain a characterization of

Optimal embeddings of ultradistributions into differential algebras

{\mathcal S}^{\{M_p\}}_{\{M_p\}}(\mathbb R^n)

Some examples in the theory of Beurling’s generalized prime numbers

S{Mp}{Mp}(Rn) and