Marcinkiewicz-Fejer means of d-dimensional Walsh-Fourier series

The boundedness of Marcinkiewicz maximal operator for d-dimensional Walsh-Fourier series is studied from the martingale Hardy-Lorentz space Hp,q into the Lorentz space Lp,q.     

Convergence of logarithmic means of multiple Walsh-Fourier series

Norlund logarithmic means of multiple Walsh-Fourier series acting from space Llnd-1 L ([0, 1)d) ,d≥1 into space weak-L1([0,1)d) are studied. The maximal Orlicz space such that the Norlund logarithmic means of multiple Walsh-Fourier series for the functions from this space converge in d-dimensional measure is found.     

Divergence almost everywhere of rectangular partial sums of multiple fourier series of bounded functions

In this paper we establish the following results, which are the multidimensional generalizations of well-known theorems:

Almost everywhere summability of multiple Fourier integrals

We obtain sufficient conditions for the Riesz means of spectral expansions converge to the function to be expanded.     

Extension on absolute summability factors of infinite series

Extension and improvement has been made on a recent result proved by Mazhar [S.M. Mazhar, Absolute summability factors of infinite series, Kyungpook Math. J. 39 (1999) 67-73] concerning absolute summability factors of infinite series.     

Homogeneity equation almost everywhere

Summary In the paper the <InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"15"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"16"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"17"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"18"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"19"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"20"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"21"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"22"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"23"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"24"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"25"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\phi$-homogeneity equation almost everywhere is studied. Let $G$ and $H$ be groups with zero. Assume that $(X,G)$ is a $G$-space and $(Y,H)$ is an $H$-space. We prove, under some assumption on $(Y,H)$, that if the functions $\phi\: G\to H$ and $F\: X\to Y$ satisfy the equation of $\phi$-homogeneity $F(\alpha x)\eg \phi(\alpha)F(x)$ almost everywhere in $G\times X$ then either $F$ is a zero function or there exists a homomorphism $\widetilde{\phi}\: G\to H$ such that $\phi=\widetilde{\phi}$ almost everywhere in $G$ and there exists a function $\overline{F}\: X\to Y$ such that <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \overline{F}(\alpha x)=\widetilde{\phi}(\alpha)\overline{F}(x) \szo{for} \alpha\in G\setminus\{0\},\quad x\in X, $$ and $F=\overline{F}$ almost everywhere in $X$.     

The surprising almost everywhere convergence of Fourier-Neumann series

For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in L^p requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in L² is the celebrated Carleson theorem, proved in 1966 (and extended to L^p by Hunt in 1967).In this paper, we take the system

     

Convergence in measure of logarithmic means of quadratical partial sums of double Walsh-Fourier series

The main aim of this paper is to prove that the logarithmic means of quadratical partial sums of the double Walsh-Fourier series does not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of LlnL(I²), the set of the functions having logarithmic means of quadratical partial sums of the double Walsh-Fourier series convergent in measure is of first Baire category.     

The martingale hardy type inequality for Marcinkiewicz-Fejér means of two-dimensional conjugate Walsh-Fourier series

The main aim of this paper is to prove that for any 0 < p ≤ 2/3 there exists a martingale f ∈ H _p such that Marcinkiewicz-Fejér means of the two-dimensional conjugate Walsh-Fourier series of the martingale f is not uniformly bounded in the space L _p .     

Almost everywhere convergence of Banach space-valued Vilenkin-Fourier series

The duality between martingale Hardy and BMO spaces is generalized for Banach space valued martingales. It is proved that if X is a UMD Banach space and f ∈ L _p(X) for some 1 < p < ∞ then the Vilenkin-Fourier series of f converges to f almost everywhere in X norm, which is the extension of Carleson’s result. This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No. M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No. T043769, T047128, T047132.     

A note on strong summability of two-dimensional Walsh-Fourier series

The paper deals with the strong summability of Marcinkiewicz means with a variable power. Let $$H_n \left( {f,x,y,A_n } \right): = \tfrac{1} {n}\sum\nolimits_{l = 1}^n {\left( {e^{\left. {A_n } \right|\left. {S_{ll} \left( {f,x,y} \right) - f\left( {x,y} \right)} \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } - 1} \right)} .$$ It is shown that if A _n ↑ ∞ arbitrary slowly, there exists f ∈ C(I ²) such that lim _n→∞ H _n (f, 0, 0, A _n ) = +∞. At the same time, for every f ∈ C (I ²) there exists A _n (f) ↑ ∞ such that lim _n→∞ H _n (f, x, y, A _n ) = 0 uniformly on I ².     

On generalized absolute Cesàro summability

In this paper, a known result on ∣C, α; δ∣_k summability factors has been generalized for ∣C, α, γ; δ∣_k summability factors. Some new results have also been obtained.     

On (N, p, q) summability factors of infinite series

In this paper a necessary and sufficient condition has been obtained for Σa_n∈_n to be summable |N, q| whenever Σa_n is bounded (N, p, q).     

On general summability factor theorem of infinite series

A lower triangular matrix with nonzero principal diagonal entries is called a triangle. In this paper we obtain the sufficient conditions for ∑a_nλ_n to be summable ∣A∣_k whenever ∑a_n is summable ∣T∣_k for a triangle T.     

New Exceptional Sets and Convergence of the Square Partial Sums of Walsh–Fourier Series

For double Walsh–Fourier series and with f ∈ L~2([0, 1) × [0, 1)) we prove two almost orthogonality results relative to the linearized maximal square partial sums operator S_(N(x,y))f(x, y).Assumptions are N(x, y) non-decreasing as a function of x and of y and, roughly speaking, partial derivatives with approximately constant ratio ■≌2~(n_0) for all x and y, where n_0 is any fixed non-negative integer. Estimates, independent of N(x, y) and n_0, are then extended to L~r, 1 r 2.We give an application to the family N(x, y) = λxy on [0, 1) × [0, 1), any λ 10.     

Divergent Laguerre series

We prove failure of a.e. convergence of partial sums of Laguerre expansions of functions for 4$">. The idea which is used goes back to Stanton and Tomas. We follow Meaney's paper (1983), where divergence results were proved in the Jacobi polynomial case.

     

Conjectures and problems on Bochner-Riesz means

The aim of this paper is to state some conjectures and problems on Bochner-Riesz means in multiple Fourier series and integrals. The progress on these conjectures and problems are also mentioned.