Abstract
In this note we discuss notions of convolutions generated by biorthogonal systems of elements of a Hilbert space. We develop the associated biorthogonal Fourier analysis and the theory of distributions, discuss properties of convolutions and give a number of examples.
Highlights
In this work we introduce a notion of a convolution generated by systems of elements of a Hilbert space H forming a Riesz basis
Such collections often arise as systems of eigenfunctions of densely defined nonself-asjoint operators acting on H, and a suitable notion of convolution leads to the development of the associated Fourier analysis
In this note we aim at discussing an abstract point of view on convolutions when one is given only a Riesz basis in a Hilbert space, without making additional assumptions on an operator for which it may be a basis of eigenfunctions
Summary
In this work we introduce a notion of a convolution generated by systems of elements of a Hilbert space H forming a Riesz basis. Such collections often arise as systems of eigenfunctions of densely defined nonself-asjoint operators acting on H, and a suitable notion of convolution leads to the development of the associated Fourier analysis. In this note we aim at discussing an abstract point of view on convolutions when one is given only a Riesz basis in a Hilbert space, without making additional assumptions on an operator for which it may be a basis of eigenfunctions.
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