Recent zbMATH articles in MSC 46F05https://zbmath.org/atom/cc/46F052024-03-13T18:33:02.981707ZWerkzeugApproximation of functions on rays in \(\mathbb{R}^n\) by solutions to convolution equationshttps://zbmath.org/1528.300132024-03-13T18:33:02.981707Z"Volchkov, V. V."https://zbmath.org/authors/?q=ai:volchkov.valerii-vladimirovich"Volchkov, Vit. V."https://zbmath.org/authors/?q=ai:volchkov.vitalii-vladimirovichSummary: This is a first study of approximation of continuous functions on rays in \(\mathbb{R}^n\) by smooth solutions to a multidimensional convolution equation with a radial convolutor. We obtain an analog of the well-known Carleman's Theorem on tangent approximation by entire functions. As consequences, we give some new results of interest for the theory of convolution equations. These results concern the density in \(\mathbb{C}\) of the range of some solutions to the convolution equation as well as the possible growth of solutions on rays in \(\mathbb{R}^n \).Kernel theorems for Beurling-Björck type spaceshttps://zbmath.org/1528.460022024-03-13T18:33:02.981707Z"Neyt, Lenny"https://zbmath.org/authors/?q=ai:neyt.lenny"Vindas, Jasson"https://zbmath.org/authors/?q=ai:vindas.jassonThe authors consider a general class of Beurling-Björck spaces (of Beurling and Roumieu type) defined in terms of two weight function systems \({\mathcal V}\) and \({\mathcal W}\). In the case that \({\mathcal V} = \{e^{\frac{1}{\lambda}v}: \lambda > 0\}\) and \({\mathcal W} = \{e^{\frac{1}{\lambda}w}: \lambda > 0\}\) for appropriate non-negative continuous functions \(v, w\) on \({\mathbb R}^d\), the classical Beurling-Björck spaces are recovered. Section~3 (along with Appendix~A) characterizes when the general Beurling-Björck spaces are nuclear in terms of the weight systems, while Section~4 is devoted to the new kernel theorems. An important tool in the proof of the kernel theorem for Roumieu-type spaces is the projective description of said spaces using maximal Nachbin families associated with weight systems.
Reviewer: Antonio Galbis (València)