Remark on spline unconditional bases in
Leszek Skrzypczak (1989)
Banach Center Publications
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Leszek Skrzypczak (1989)
Banach Center Publications
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István Blahota, György Gát, Ushangi Goginava (2007)
Colloquium Mathematicae
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The main aim of this paper is to prove that the maximal operator of the Fejér means of the double Vilenkin-Fourier series is not bounded from the Hardy space to the space weak-.
Peter Oswald (2006)
Banach Center Publications
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We extend results on constructing semiorthogonal linear spline prewavelet systems in one and two dimensions to the case of irregular dyadic refinement. In the one-dimensional case, we obtain sharp two-sided inequalities for the -condition, 1 < p < ∞, of such systems.
Ushangi Goginava (2011)
Banach Center Publications
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The main aim of this paper is to prove that there exists a martingale such that the Fejér means of the two-dimensional Walsh-Fourier series of f is not uniformly bounded in the space weak-.
Yuichi Kanjin (2001)
Studia Mathematica
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We prove that the Hausdorff operator generated by a function ϕ is bounded on the real Hardy space , 0 < p ≤ 1, if the Fourier transform ϕ̂ of ϕ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on , 2/(2α+1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of .
Kobza, Jiří
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Markus Passenbrunner (2014)
Studia Mathematica
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We prove that given any natural number k and any dense point sequence (tₙ), the corresponding orthonormal spline system is an unconditional basis in reflexive .
J. J. Guadalupe, V. I. Kolyada (2001)
Studia Mathematica
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We investigate the behaviour of Fourier coefficients with respect to the system of ultraspherical polynomials. This leads us to the study of the “boundary” Lorentz space corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients of -functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReH¹. Namely, we prove that for any the series is the Fourier series of some function φ ∈ ReH¹ with . ...
Alberto Fiorenza, Babita Gupta, Pankaj Jain (2008)
Studia Mathematica
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We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality holds with some c independent of f iff w belongs to the well known Muckenhoupt class , and therefore iff for some c independent of f. Some results of similar type are discussed for the case of small...
Shuichi Sato (2019)
Czechoslovak Mathematical Journal
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We consider Littlewood-Paley functions associated with a non-isotropic dilation group on . We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted spaces, , with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).
Zhixin Liu, Shanzhen Lu (1993)
Studia Mathematica
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The aim of this paper is to establish transference and restriction theorems for maximal operators defined by multipliers on the Hardy spaces and , 0 < p ≤ 1, which generalize the results of Kenig-Tomas for the case p > 1. We prove that under a mild regulation condition, an function m is a maximal multiplier on if and only if it is a maximal multiplier on . As an application, the restriction of maximal multipliers to lower dimensional Hardy spaces is considered. ...
Ferenc Weisz (2009)
Studia Mathematica
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It is proved that the multi-dimensional maximal Fejér operator defined in a cone is bounded from the amalgam Hardy space to . This implies the almost everywhere convergence of the Fejér means in a cone for all , which is larger than .
Ushangi Goginava (2008)
Studia Mathematica
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The main aim of this paper is to prove that the maximal operator is bounded from the Hardy space to weak- and is not bounded from to .
István Blahota, Lars-Erik Persson, Giorgi Tephnadze (2015)
Czechoslovak Mathematical Journal
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We prove and discuss some new -type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients . These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of...
Ferenc Weisz (1999)
Colloquium Mathematicae
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The two-dimensional classical Hardy spaces are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from to (1/2 < p ≤ ∞) and is of weak type where the Hardy space is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ ⊃ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on whenever 1/2 < p < ∞. Thus, in case f ∈ , the...
Zygmunt Wronicz (2002)
Annales Polonici Mathematici
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Chebyshevian box splines were introduced in [5]. The purpose of this paper is to show some new properties of them in the case when the weight functions are of the form , where the functions are periodic functions of one variable. Then we consider the problem of approximation of continuous functions by Chebyshevian box splines.
Evgeny A. Poletsky, Khim R. Shrestha (2015)
Banach Center Publications
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In this paper we completely characterize those weighted Hardy spaces that are Poletsky-Stessin Hardy spaces . We also provide a reduction of problems to problems and demonstrate how such a reduction can be used to make shortcuts in the proofs of the interpolation theorem and corona problem.
Elijah Liflyand, Akihiko Miyachi (2009)
Studia Mathematica
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Sufficient conditions for the boundedness of the Hausdorff operators in the Hardy spaces , 0 < p < 1, on the real line are proved. Two related negative results are also given.
Loukas Grafakos, Liguang Liu, Dachun Yang (2009)
Bulletin de la Société Mathématique de France
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An RD-space is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type having “dimension” , there exists a such that for certain classes of distributions, the quasi-norms of their radial maximal functions and grand maximal functions are equivalent when . This result yields a radial maximal function characterization for Hardy spaces on . ...
Richard Oberlin, Andreas Seeger, Terence Tao, Christoph Thiele, James Wright (2012)
Journal of the European Mathematical Society
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We strengthen the Carleson-Hunt theorem by proving estimates for the -variation of the partial sum operators for Fourier series and integrals, for . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.
Gevorkyan Gegham, Kamont Anna
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AbstractWe study general Franklin systems, i.e. systems of orthonormal piecewise linear functions corresponding to quasi-dyadic sequences of partitions of [0,1]. The following problems are treated: unconditionality of the general Franklin basis in , 1 < p < ∞, and , 1/2 < p ≤ 1; equivalent conditions for the unconditional convergence of the Franklin series in for 0< p ≤ 1; relation between Haar and Franklin series with identical coefficients; characterization of the spaces...
Bhikha Lila Ghodadra, Vanda Fülöp (2020)
Mathematica Bohemica
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For a Lebesgue integrable complex-valued function defined on let be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that as . But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of there is a definite rate at which the Walsh-Fourier transform tends...
Paul Alton Hagelstein (2001)
Studia Mathematica
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Let denote the strong maximal operator. Let and denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that but . It is shown that if f is a function supported on Q such that but , then there exists a set A of finite measure in ℝ² such that .