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Discrete spectra for some complex infinite band matrices

Malejki Maria

Opuscula Mathematica

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2021

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Vol. 41, no. 6

861--879

EN

Under suitable assumptions the eigenvalues for an unbounded discrete operator A in l2 , given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let [formula] where [formula] is the set of all limit points of the sequence (λn) and An is a finite dimensional orthogonal truncation of A. The aim of this article is to provide the conditions that are sufficient for the relations σ (A) ⊂ Λ (A) or Λ(A) ) ⊂ σ (A) to be satisfied for the band operator A.

2

Spectra of some selfadjoint Jacobi operators in the double root case

Motyka W.

Opuscula Mathematica

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2015

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Vol. 35, no. 3

353--370

EN

In this paper we prove a mixed spectrum of Jacobi operators defined by λn = s(n)(1 + x(n)) and qn = — 2s(n)(l+y/(n)), where (s(n)) is a real unbounded sequence, (x(n)) and (y(n)) are some perturbations.

3

Asymptotics of the discrete spectrum for complex Jacobi matrices

Malejki M.

Opuscula Mathematica

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2014

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Vol. 34, no. 1

139--160

EN

The spectral properties and the asymptotic behaviour of the discrete spectrum for a special class of infinite tridiagonal matrices are given. We derive the asymptotic formulae for eigenvalues of unbounded complex Jacobi matrices acting in l2(N).

4

Statistical view on phase and magnitude information in signal processing

Popiński W.

Artificial Satellites : Journal of Planetary Geodesy

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2012

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Vol. 47, No. 3

127--136

EN

In this work the problem of reconstruction of an original complex-valued signal ot, t = 0, 1,...,n - 1, from its Discrete Fourier Transform (DFT) spectrum corrupted by random fluctuations of magnitude and/or phase is investigated. It is assumed that the magnitude and/or phase of discrete spectrum values are distorted by realizations of uncorrelated random variables. The obtained results of analysis of signal reconstruction from such distorted DFT spectra concern derivation of the expected values and bounds on variances of the reconstructed signal at the observation moments. It is shown that the considered random distortions in general entail change in magnitude and/or phase of the reconstructed signal expected values, which together with imposed random deviations with finite variances can blur the similarity to the original signal. The effect of analogous random amplitude and/or phase distortions of a complex valued time domain signal on band pass filtration of distorted signal is also investigated.

5

A first-order spectral phase transition in a class of periodically modulated Hermitian Jacobi matrices

Pchelintseva I.

Opuscula Mathematica

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2008

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Vol. 28, no. 2

137-150

EN

We consider self-adjoint unbounded Jacobi matrices with diagonal q(n) = b(n)n and off-diagonal entries λ(n) = n, where b(n) is a 2-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum of the operator is either purely absolutely continuous or discrete. We study the situation where the spectral phase transition occurs, namely the case of b(1)b(2) = 4. The main motive of the paper is the investigation of asymptotics of generalized eigenvectors of the Jacobi matrix. The pure point part of the spectrum is analyzed in detail.