Unconditional Bases of Systems of Bessel Functions

Vynnyts’kyi, B. V.; Khats, R. V.; Sheparovych, Iryna

doi:10.32523/2077-9879-2020-11-4-76-86

Cited by 4 publications

(3 citation statements)

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“…x s k J ν (x s k ) in the space L 2 ((0; 1); x 2p dx), where ν ≥ 1/2, p ∈ R and ( s k ) k∈N is a sequence of distinct nonzero complex numbers, have been studied in [6,7,20,21,22,23,24]. [1,27]), and…”

Section: Operator B νmentioning

confidence: 99%

Generalized eigenvectors of linear operators and biorthogonal systems

Khats

2022

Constructive Mathematical Analysis

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The notions of a set of generalized eigenvalues and a set of generalized eigenvectors of a linear operator in Euclidean space are introduced. In addition, we provide a method to find a biorthogonal system of a subsystem of eigenvectors of some linear operators in a Hilbert space whose systems of canonical eigenvectors are over-complete. Related to our problem, we will show an example of a linear differential operator that is formally adjoint to Bessel-type differential operators. We also investigate basis properties (completeness, minimality, basicity) of the systems of generalized eigenvectors of this differential operator.

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Section: Operator B νmentioning

confidence: 99%

Generalized eigenvectors of linear operators and biorthogonal systems

Khats

2022

Constructive Mathematical Analysis

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“…Various approximation properties of the systems of Bessel functions has been studied in many papers (see, for example, [1][2][3][4][5][6][7][10][11][12][13][14][15][16][17][18][19][20][21][22]). In particular, it is well known that the system √ xJ ν (x ρ k ) : k ∈ N is an orthogonal basis for the space L 2 (0; 1) if ν > −1 and ( ρ k ) k∈N is a sequence of positive zeros of J ν (see [1,2,4,12,22]).…”

Section: Introductionmentioning

confidence: 99%

“…Basis properties (completeness, minimality, basicity) of the above systems of Bessel functions and more general systems x −p−1 x ρ k J ν (x ρ k ) : k ∈ N in the space L 2 (0; 1); x 2p dx , where ν ≥ 1/2, p ∈ R and ( ρ k ) k∈N is a sequence of distinct nonzero complex numbers, have been studied in [6,7,[15][16][17][18][19]. Those results are formulated in terms of sequences of zeros of functions from certain classes of entire functions.…”

Section: Introductionmentioning

confidence: 99%

Completeness of the systems of Bessel functions of index $-5/2$

Khats'

2024

Carpathian Math. Publ.

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Let $L^2((0;1);x^4 dx)$ be the weighted Lebesgue space of all measurable functions $f:(0;1)\rightarrow\mathbb C$, satisfying $\int_{0}^1 t^4 |f(t)|^2\, dt<+\infty$. Let $J_{-5/2}$ be the Bessel function of the first kind of index $-5/2$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system $\big\{\rho_k^2\sqrt{x\rho_k}J_{-5/2}(x\rho_k):k\in\mathbb N\big\}$ in the space $L^2((0;1);x^4 dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(\rho_k)_{k\in\mathbb N}$. In this case, we study an integral representation of some class $E_{4,+}$ of even entire functions of exponential type $\sigma\le 1$. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than $-1$, due to B. Vynnyts'kyi, V. Dilnyi, O. Shavala and the author.

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Completeness of the System of Generalized Eigenfunctions for a Bessel-Type Differential Operator

Khats’

2023

J Math Sci

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Unconditional Bases of Systems of Bessel Functions

Cited by 4 publications

References 0 publications

Generalized eigenvectors of linear operators and biorthogonal systems

Generalized eigenvectors of linear operators and biorthogonal systems

Completeness of the systems of Bessel functions of index $-5/2$

Completeness of the System of Generalized Eigenfunctions for a Bessel-Type Differential Operator

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