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It is known that the trigonometric Fourier series converges in L-norm if 1 < p <∞. This cla... more It is known that the trigonometric Fourier series converges in L-norm if 1 < p <∞. This claim is false for p = 1 and p =∞. Similar statements are true for the Walsh-system. The domain of parameter p for the Lnorm convergence in the case of orthogonal polynomials depends on the weight function. For example, for Legendre polynomials, the L-norm convergence holds only for 4/3 < p < 4 (see [5]). Bl. Sendov introduced in 1999 a generalization of the Walsh-system, the so-called “Walsh-similar” functions (see [10]). These are special cases of the Walsh system with respect the weight ρ (W ρ = Ψ = (ψ n, n ∈ N)), which was introduced by Schipp (see [7], [8]). We show that if the weight function ρ belongs to the class Lip (α,W ) (0 < α ≤ 1), and ρ ≥ ρ0 > 0, then the W -Walsh-Fourier-series is convergent in Lρ-norm if 1 < p < ∞. We study the behavior of such W -systems too, whose weight function has not positive lower bound. For example, for ρ(x) = x (α > −1) the W -s...
Publicationes Mathematicae Debrecen, 2022
International Journal of Wavelets, Multiresolution and Information Processing
We will prove that the analytic orthogonal wavelet-system, which was introduced by Feichtinger an... more We will prove that the analytic orthogonal wavelet-system, which was introduced by Feichtinger and Pap in [Hyperbolic wavelets and multiresolution in the Hardy space of the upper half plane, in Blaschke Products and Their Applications: Fields Institute Communications, Vol. 65 (Springer, New York, 2013), pp. 193–208] is discrete orthogonal too. We will discuss the discrete orthogonality and the properties of the reproducing kernel functions of the introduced wavelet-spaces.
We investigate the generalization of the Hardy operator, the so called dyadic Hausdorff operator.... more We investigate the generalization of the Hardy operator, the so called dyadic Hausdorff operator. We prove that the dyadic Hausdorff operator generated by an integrable function φ∈L 1 ([0,∞)) is bounded on the dyadic Hardy space H 1 . Under some weak condition on the integrable function φ, it is proved that the dyadic Hausdorff operator generated by this function is bounded on the spaces L p (ℝ + ) (1≦p<∞).
Journal of Fourier Analysis and Applications, 2013
A palyazat soran a kutatasi programnak megfelelő teruleteken osszesen 31 tudomanyos publikacio sz... more A palyazat soran a kutatasi programnak megfelelő teruleteken osszesen 31 tudomanyos publikacio szuletett. A palyazat egyik resztvevője 2005-ben vedte meg sikeresen Akademiai Doktori Ertekezeset a palyazati tervben szereplő kutatasi temaban. A program soran vegzett elmeleti kutatasok teremtettek meg az alapot ket alkalmazott, orvos biologiai projektbe valo bekapcsolodashoz is. Az ortogonalis sorok szummacioja teren szamos uj eredmenyt igazoltunk, peldaul Walsh, Walsh-Kaczmarcz, Vilenkin, Ciesielski rendszerekre vonatkozo Theta-szummacio konvergenciajara vonatozoan, valamint a Gabor analizisben tobb eddig nem vizsgalt ter eseten is. Multiplier operatorokkal kapcsolatban a Hormander-Mihlin-fele multiplier operatorok korlatossagat igazoltuk Hardy terekben a klasszikus es a diadikus esetre, utobbira tobb dimenzioban is. Az adott problemahoz illeszkedő ortogonalis rendszer konstrualasanak problemakore kapcsan a Malmquist-Takenaka, a gombfuggvenyek es a Zernike rendszer teruleten ertunk el...
It is known that the trigonometric Fourier series converges in L-norm if 1 < p <∞. This cla... more It is known that the trigonometric Fourier series converges in L-norm if 1 < p <∞. This claim is false for p = 1 and p =∞. Similar statements are true for the Walsh-system. The domain of parameter p for the Lnorm convergence in the case of orthogonal polynomials depends on the weight function. For example, for Legendre polynomials, the L-norm convergence holds only for 4/3 < p < 4 (see [5]). Bl. Sendov introduced in 1999 a generalization of the Walsh-system, the so-called “Walsh-similar” functions (see [10]). These are special cases of the Walsh system with respect the weight ρ (W ρ = Ψ = (ψ n, n ∈ N)), which was introduced by Schipp (see [7], [8]). We show that if the weight function ρ belongs to the class Lip (α,W ) (0 < α ≤ 1), and ρ ≥ ρ0 > 0, then the W -Walsh-Fourier-series is convergent in Lρ-norm if 1 < p < ∞. We study the behavior of such W -systems too, whose weight function has not positive lower bound. For example, for ρ(x) = x (α > −1) the W -s...
Publicationes Mathematicae Debrecen, 2022
International Journal of Wavelets, Multiresolution and Information Processing
We will prove that the analytic orthogonal wavelet-system, which was introduced by Feichtinger an... more We will prove that the analytic orthogonal wavelet-system, which was introduced by Feichtinger and Pap in [Hyperbolic wavelets and multiresolution in the Hardy space of the upper half plane, in Blaschke Products and Their Applications: Fields Institute Communications, Vol. 65 (Springer, New York, 2013), pp. 193–208] is discrete orthogonal too. We will discuss the discrete orthogonality and the properties of the reproducing kernel functions of the introduced wavelet-spaces.
We investigate the generalization of the Hardy operator, the so called dyadic Hausdorff operator.... more We investigate the generalization of the Hardy operator, the so called dyadic Hausdorff operator. We prove that the dyadic Hausdorff operator generated by an integrable function φ∈L 1 ([0,∞)) is bounded on the dyadic Hardy space H 1 . Under some weak condition on the integrable function φ, it is proved that the dyadic Hausdorff operator generated by this function is bounded on the spaces L p (ℝ + ) (1≦p<∞).
Journal of Fourier Analysis and Applications, 2013
A palyazat soran a kutatasi programnak megfelelő teruleteken osszesen 31 tudomanyos publikacio sz... more A palyazat soran a kutatasi programnak megfelelő teruleteken osszesen 31 tudomanyos publikacio szuletett. A palyazat egyik resztvevője 2005-ben vedte meg sikeresen Akademiai Doktori Ertekezeset a palyazati tervben szereplő kutatasi temaban. A program soran vegzett elmeleti kutatasok teremtettek meg az alapot ket alkalmazott, orvos biologiai projektbe valo bekapcsolodashoz is. Az ortogonalis sorok szummacioja teren szamos uj eredmenyt igazoltunk, peldaul Walsh, Walsh-Kaczmarcz, Vilenkin, Ciesielski rendszerekre vonatkozo Theta-szummacio konvergenciajara vonatozoan, valamint a Gabor analizisben tobb eddig nem vizsgalt ter eseten is. Multiplier operatorokkal kapcsolatban a Hormander-Mihlin-fele multiplier operatorok korlatossagat igazoltuk Hardy terekben a klasszikus es a diadikus esetre, utobbira tobb dimenzioban is. Az adott problemahoz illeszkedő ortogonalis rendszer konstrualasanak problemakore kapcsan a Malmquist-Takenaka, a gombfuggvenyek es a Zernike rendszer teruleten ertunk el...