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Á. P. Horváth

Publications and source records attributed to Á. P. Horváth.

17 recordsLinked to original sources
arXiv2026

Chromatic expansion with Bessel operator of fractional order

This paper develops a Bessel-chromatic expansion framework associated with fractional powers of the Bessel-Laplace operator. The construction combines methods of weighted polynomial approximation and of fractional differential operators. Using the spectral representation of $(-Δ_a)^{\frac{1}{2}}$, we define Bessel-chromatic derivatives and apply them to weighted spherical means both at a general point and at the origin. Different classes of weights on finite and infinite intervals are considered, with particular attention to cases where the inverse Hankel transform is explicit. The convergence of the expansions is studied through Cesàro and de la Vallée Poussin means. In the bandlimited case, the method gives reconstruction formulas for weighted spherical means and, under suitable assumptions, recovery formulas for the original function. Numerical examples illustrate the decay of the Bessel-chromatic coefficients and the accuracy of the corresponding reconstructions.

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arXiv2025

Fractional Bessel-Sobolev and Bessel B-potential spaces

After proving the equivalence of the Bessel $K$-functional and the corresponding spherical modulus of smoothness we define fractional Bessel-Sobolev spaces. As an analog of the classical one the imbedding relation of fractional Bessel-Sobolev and Bessel B-potential spaces is pointed out. Applying the defined notions a potential theoretic characterization of removable sets with respect to certain fractional partial differential operator is derived.

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arXiv2025

Removable Sets for Fractional Heat and Fractional Bessel-Heat Equations

We examine the fractional heat diffusion equations $L_{γ,a}:=(-Δ_a)^{\fracγ{2}}+\partial_t$, where $Δ_a$ is the Laplace- or the Bessel-Laplace operator. We give conditions for removability which are sufficient and which are necessary, by $L^p$-capacities. Introducing a spherical modulus of smoothness we can treat the Laplace and Bessel-Laplace cases together.

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arXiv2024

Trace inequality with Bessel convolution

Considering potentials defined by Bessel kernel with Bessel convolution a Kerman-Sawyer type characterization of trace inequality is given. As an application an estimate on the least eigenvalue of Schrödinger-Bessel operators is derived.

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arXiv2023

Translation beyond Delsarte

We introduce general translations as solutions to Cauchy or Dirichlet problems. This point of view allows us to handle the heat-diffusion semigroup as a translation. With the given examples Kolmogorov-Riesz characterization of compact sets in certain $L^p_μ$ spaces are given. Pego-type characterizations are also derived. Finally for some examples the equivalence of the corresponding modulus of smoothness and K-functional is pointed out.

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arXiv2023

$p$-capacity with Bessel convolution

We define and examine nonlinear potential by Bessel convolution with Bessel kernel. We investigate removable sets with respect to Laplace-Bessel inequality. By studying the maximal and fractional maximal measure, a Wolff type inequality is proved. Finally the relation of B-$p$ capacity and B-Lipschitz mapping, and the B-$p$ capacity and weighted Hausdorff measure and the B-$p$ capacity of Cantor sets are examined.

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arXiv2020

Multiplication operator and exceptional Jacobi polynomials

Below the normalized weighted reciprocal of the Christoffel function with respect to exceptional Jacobi polynomials is investigated. It is proved that it tends to the equilibrium measure of the interval of orthogonality in weak-star sense. The main tool of this study is the multiplication operator and examination of behavior of zeros of the corresponding average characteristic polynomial. Finally, as an application of multiplication operator, location of zeros of certain self-inversive polynomials are examined.

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arXiv2019

Asymptotics for Recurrence Coefficients of X1-Jacobi Polynomials and Christoffel Function

Computing asymptotics of the recurrence coefficients of X1-Jacobi polynomials we investigate the limit of Christoffel function. We also study the relation between the normalized counting measure based on the zeros of the modified average characteristic polynomial and the Christoffel function in limit. The proofs of corresponding theorems with respect to ordinary orthogonal polynomials are based on the three-term recurrence relation. The main point is that exceptional orthogonal polynomials possess at least five-term formulae and so the Christoffel-Darboux formula also fails. It seems that these difficulties can be handled in combinatorial way.

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arXiv2018

Translation operator with exceptional Laguerre polynomials

We extend the notion of general translation operator to exceptional Laguerre polynomials. To this we investigate the associated singular hyperbolic Cauchy problem. We derive a maximum principle with respect to this Cauchy problem and applying it we determine the norm of the translation operator. As an application we give Nikol'skii inequalities with respect to exceptional Laguerre polynomials.

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arXiv2016

Potential Theory and Quadratic Programming

We extend the notion of some energy-type expressions based on two sets, developed in the abstract potential theory. We also give the discretized version of the quantities defined, similar to Chebyshev constant. This extension allows to apply the potential-theoretic results to infinite quadratic programming problems. Together with a cutting plane algorithm, the Chebyshev-constant method ensures that under certain conditions, the infinite problem can be reduced to semi-infinite or to finite problems.

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arXiv2016

Chromatic Derivatives and Expansions with Weights

Chromatic derivatives and series expansions of bandlimited functions have recently been introduced in signal processing and they have been shown to be useful in practical applications. We extend the notion of chromatic derivative using varying weights. When the kernel function of the integral operator is positive, this extension ensures chromatic expansions around every points. Besides old examples, the modified method is demonstrated via some new ones as Walsh-Fourier transform, and Poisson-wavelet transform. Moreover the chromatic expansion of a function in some $L^p$-space is investigated.

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arXiv2016

The energy function with respect to the zeros of the exceptional Hermite polynomials

We examine the energy function with respect to the zeros of exceptional Hermite polynomials. The localization of the eigenvalues of the Hessian is given in the general case. In some special arrangements we have a more precise result on the behavior of the energy function. Finally we investigate the energy function with respect to the regular zeros of the exceptional Hermite polynomials.

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arXiv2014

The Electrostatic Properties of Zeros of Exceptional Laguerre and Jacobi Polynomials and stable interpolation

We will examine the electrostatic properties of exceptional and regular zeros of $X_m$-Laguerre and $X_m$-Jacobi polynomials. Since there is a close connection between the electrostatic properties of the zeros and the stability of interpolation on the system of zeros, we can deduce an Egerváry-Tur\' an type result as well. The limit of the energy on the regular zeros are also investigated.

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arXiv2013

Weighted Fejér Constants and Fekete Sets

We give the connections among the Fekete sets, the zeros of orthogonal polynomials, $1(w)$-normal point systems, and the nodes of a stable and most economical interpolatory process via the Fejér contants. Finally the convergence of a weighted Grünwald interpolation is proved.

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