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September 2018 $\mathcal{E}'$ as an algebra by multiplicative convolution
Dietmar Vogt
Funct. Approx. Comment. Math. 59(1): 117-128 (September 2018). DOI: 10.7169/facm/1719


We study the algebra $\mathcal{E}'(\mathbb{R}^d)$ of distributions with compact support equipped with the multiplication $(T\star S)(f)=T_x(S_y(f(xy))$ where $xy=(x_1 y_1,\dots,x_d y_d)$. This allows us a very elegant access to the theory of Hadamard type operators on $C^\infty(\Omega)$, $\Omega$ open in $\mathbb{R}^d$, that is, of operators which admit all monomials as eigenvectors. We obtain a representation of the algebra of such operators as an algebra of holomorphic functions with classical Hadamard multiplication. Finally we study global solvability for such operators, in particular of Euler differential operators, on open subsets of $\mathbb{R}_+^d$.


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Dietmar Vogt. "$\mathcal{E}'$ as an algebra by multiplicative convolution." Funct. Approx. Comment. Math. 59 (1) 117 - 128, September 2018.


Published: September 2018
First available in Project Euclid: 28 March 2018

zbMATH: 06979912
MathSciNet: MR3858282
Digital Object Identifier: 10.7169/facm/1719

Primary: 35A01 , 44A35 , 47L80
Secondary: 46F05 , 46F20 , 46H35 , 47B38

Keywords: algebra of operators , distributions with compact support , Euler operators , Hadamard operators , monomials as eigenvectors , operators on smooth functions

Rights: Copyright © 2018 Adam Mickiewicz University

Vol.59 • No. 1 • September 2018
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