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The Dirichlet ring and unconditional bases in L 2[0, 2π]

Abstract

It is observed that the Dirichlet ring admits a representation in an infinite-dimensional matrix algebra. The resulting matrices are subsequently used in the construction of nonorthogonal Riesz bases in a separable Hilbert space. This framework enables custom design of a plethora of bases with interesting features. Remarkably, the representation of signals in any one of these bases is numerically implementable via fast algorithms.

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Correspondence to Artur Sowa.

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__________

Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 47, No. 3, pp. 75–81, 2013

Original Russian Text Copyright © by Artur Sowa

The author gratefully acknowledges the support of the Canadian Foundation for Innovation, grant LOF no. 22117.

Since all bounded unconditional bases are Riesz, we will henceforth use the two terms interchangeably.

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Sowa, A. The Dirichlet ring and unconditional bases in L 2[0, 2π]. Funct Anal Its Appl 47, 227–232 (2013). https://doi.org/10.1007/s10688-013-0028-6

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  • DOI: https://doi.org/10.1007/s10688-013-0028-6

Key words

  • unconditional basis
  • Riesz basis
  • fast transform
  • Dirichlet series