On the Dirichlet matrix operators in sequence spaces

Sowa, Artur

doi:10.4064/am2328-4-2017

Cited by 4 publications

(4 citation statements)

References 16 publications

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“…Matrices a (s) , 1 , denoted by A = [(a n )] , were introduced in [8], [9]. In [9], [10], they are considered as matrices of the operators in the Hilbert space and are called (in [10]) the Dirichlet matrices, or D-matrices.…”

Section: Riordan Matricesmentioning

confidence: 99%

The Riordan–Dirichlet Group

Burlachenko

2019

Math Notes

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Riordan matrices are infinite lower triangular matrices that correspond to certain operators in the space of formal power series. In this paper, we introduce similar matrices for the space of formal Dirichlet series. We show that these matrices form a group similar to the Riordan group, and we derive an analog of the Lagrange inversion formula for this group. As an example of the application of these matrices, we obtain an analog of the Abel identities.

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Section: Riordan Matricesmentioning

confidence: 99%

The Riordan–Dirichlet Group

Burlachenko

2019

Math Notes

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“…Remark 2. More properties of the action of matrices ( 26) in 2 and other select sequence spaces are described in [22] and [26].…”

Section: The Generalized Q-transform Related To the Dirichlet Ringmentioning

confidence: 99%

“…By assumption σ > 3/2 and hence D −1 [σ+iτn] are homeomorphisms. Recall the matrices are populated with coefficients b n = µ(n)n −(σ+iτn) and their complex conjugates according to the pattern (26). Thus, we obtain…”

Section: The Periodized Zeta Function and The Broadband Redundancymentioning

confidence: 99%

A nonlocal transform to map and track quantum dynamics

Sowa¹

2019

J. Phys. A: Math. Theor.

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The focus of this work is a correspondence between the Hilbert space operators on one hand, and doubly periodic generalized functions on the other. The linear map that implements it, referred to as the Q-transform, enables a direct application of the classical Harmonic analysis in a study of quantum systems. In particular, the Q-transform makes it possible to reinterpret the dynamic of a quantum observable as a (typically nonlocal) dynamic of a doubly periodic real function. From this point of view we carry out an analysis of an open quantum system whose dynamics are governed by an asymptotically harmonic Hamiltonian and compact type Lindblad operators. It is established that the initial value problem of the equivalent nonlocal but classical evolution is well posed in the appropriately chosen Sobolev spaces. The second set of results pertains to a generalization of the basic Q-transform and highlights a certain type of asymptotic redundancy. This phenomenon, referred to as the broadband redundancy, is a consequence of a well-known property of the zeros of the Riemann zeta function, namely, the uniform distribution modulo one of their ordinates. Its relevance to the analysis of quantum dynamics is only a special instance of its utility in harmonic analysis in general. It remains to be seen if the phenomenon is significant also in the physical sense, but it appears well-justified—in particular, by the results presented here—to pose such a question.

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“…More properties of the action of matrices ( 27) in 2 and other select sequence spaces are described in [22] and [26].…”

Section: The Generalized Q-transform Related To the Dirichlet Ringmentioning

confidence: 99%

Encoding spatial data into quantum observables

Sowa

2016

Preprint

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The focus of this work is a correspondence between the Hilbert space operators on one hand, and doubly periodic generalized functions on the other. The linear map that implements it, referred to as the Q-transform, enables a direct application of the classical Harmonic Analysis in a study of quantum systems. In particular, the Q-transform makes it possible to reinterpret the dynamic of a quantum observable as a (typically nonlocal) dynamic of a classical observable. From this point of view we carry out an analysis of an open quantum system whose dynamics are governed by an asymptotically harmonic Hamiltonian and compact type Lindblad operators. It is established that the initial value problem of the equivalent nonlocal but classical evolution is well posed in the appropriately chosen Sobolev spaces. The second set of results pertains to a generalization of the basic Q-transform and highlights a certain type of asymptotic redundancy. This phenomenon, referred to as the broadband redundancy, is a consequence of a well-known property of the zeros of the Riemann zeta function, namely, the uniform distribution modulo one of their ordinates. Its relevance to the analysis of quantum dynamics is only a special instance of its utility in harmonic analysis in general. It remains to be seen if the phenomenon is significant also in the physical sense, but it appears well-justified-in particular, by the results presented here-to pose such a question.

show abstract

On the Dirichlet matrix operators in sequence spaces

Cited by 4 publications

References 16 publications

The Riordan–Dirichlet Group

The Riordan–Dirichlet Group

A nonlocal transform to map and track quantum dynamics

Encoding spatial data into quantum observables

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