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The Dirichlet ring and unconditional bases in \(L_2[0, 2\pi]\). (English. Russian original) Zbl 1315.46015
Funct. Anal. Appl. 47, No. 3, 227-232 (2013); translation from Funkts. Anal. Prilozh. 47, No. 3, 75-81 (2013).
Summary: 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.

MSC:
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
42C15 General harmonic expansions, frames
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