Factorizing matrices by Dirichlet multiplication

Sowa, Artur

doi:10.1016/j.laa.2012.09.021

Cited by 11 publications

(10 citation statements)

References 3 publications

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“…However, when it is known and in addition its Fourier coefficients diminish relatively rapidly, then decryption is a simple matter, see [14]. Finally, we point out that when analogous operations are considered for digital signals the encryption as well as the decryption can be computed with high numerical efficiency, [15], although the problem of numerical stability of these processes would warrant a separate discussion in the case of slowly diminishing Fourier coefficients (small σ).…”

Section: Broadband Encryptions Of Periodic Functionsmentioning

confidence: 99%

Riemann’s zeta function and the broadband structure of pure harmonics

Sowa

2017

IMA Journal of Applied Mathematics

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Let a ∈ (0, 1) and let F s (a) be the periodized zeta function that is defined as F s (a) = n −s exp(2πina) for s > 1, and extended to the complex plane via analytic continuation. Let s n = σ n + it n , t n > 0, denote the sequence of nontrivial zeros of the Riemann zeta function in the upper halfplane ordered according to nondecreasing ordinates. We demonstrate that, assuming the Riemann Hypothesis, the Cesàro means of the sequence F sn (a) converge to the first harmonic exp(2πia) in the sense of periodic distributions. This reveals a natural broadband structure of the pure tone. The proof involves Fujii's refinement of the classical Landau theorem related to the uniform distribution modulo one of the nontrivial zeros of ζ.

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Section: Broadband Encryptions Of Periodic Functionsmentioning

confidence: 99%

Riemann’s zeta function and the broadband structure of pure harmonics

Sowa

2017

IMA Journal of Applied Mathematics

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“…It was demonstrated in [25] that an evaluation of an N-by-N matrix of type D({α}) -that is the upper left corner of the infinite matrix defined above -on a vector of length N can be performed via a lifting schema in O(N log N) arithmetical operations 5 . Since the well known FFT algorithm performs the discrete Fourier transform in O(N log N) arithmetical operations, the transform defined in (4) is a fast transform.…”

Section: Representation Of Hysteretic Signals In Special Nonorthogonamentioning

confidence: 99%

“…It has been established that such bases have a very unusual property -namely, they furnish fast, O(N log N), change-of basis transforms, [24]. In fact, the change of basis can be carried out numerically with extreme efficiency via a lifting schema, [25]. These facts form the precepts of a new method of signal analysis which is numerically efficient and customizable to specific nano-circuits.…”

Section: Introductionmentioning

confidence: 99%

“…This becomes a hindrance when one wishes to carry out, say, signal denoising or compression in a way that does not spoil the hysteretic features. Section 3 outlines an approach to this problem, which utilizes a novel fast transform, [24][25][26]. The discussion culminates in Section 4, where we argue that hysteretic distortion may also be simulated by certain fast transforms.…”

Section: Introductionmentioning

confidence: 99%

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Signals generated in memristive circuits

Sowa

2012

Nanoscale Systems: Mathematical Modeling, Theory and Applications

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Signals generated in circuits that include nano-structured elements typically have strongly distinct characteristics, particularly the hysteretic distortion. This is due to memristance, which is one of the key electronic properties of nanostructured materials. In this article, we consider signals generated from a memrsitive circuit model. We demonstrate numerically that such signals can be efficiently represented in certain custom-designed nonorthogonal bases. The proposed method ensures that the actual numerical representation can be implemented as a fast, O (N log N), algorithm. In addition, we discuss the possibility of modelling the hysteretic distortion via fast numerical transforms.

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“…Second, they are the basic tool in applications of the newly discovered phenomenon of broadband redundancy [19], in particular in its quantummechanical applications [20]. Third, their finite-dimensional reductions turn out to furnish the universal building blocks of generic matrices [17]-an observation that one may hope to generalize to infinite dimensions once the topological properties of infinite D-matrices are well understood.…”

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confidence: 99%

On the Dirichlet matrix operators in sequence spaces

Sowa¹

2017

Appl. Math. (Warsaw)

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We study the analytic properties of certain special operators, referred to as D-matrix operators, which arise naturally from classical Dirichlet series. There are a number of incentives for this work, including the applicability of D-matrix operators to signal processing via fast computational algorithms. It was observed in a prior publication that certain types of D-matrix operators are continuous in 2 (Sowa 2013). In this work the focus is on a complementary case that arises in relation to the special D-matrix associated with Riemann's zeta function, and on its continuity properties in suitable Hilbert and Banach sequence spaces. 2 are given in [18]. However, the approach taken therein does not pro

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Factorizing matrices by Dirichlet multiplication

Cited by 11 publications

References 3 publications

Riemann’s zeta function and the broadband structure of pure harmonics

Riemann’s zeta function and the broadband structure of pure harmonics

Signals generated in memristive circuits

On the Dirichlet matrix operators in sequence spaces

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