50 research outputs found
Almost everywhere convergence of Fej\'er means of two-dimensional triangular Walsh-Fourier series
In 1987 Harris proved (Proc. Amer. Math. Soc., 101) - among others- that for
each there exists a two-dimensional function such that
its triangular Walsh-Fourier series diverges almost everywhere. In this paper
we investigate the Fej\'er (or ) means of the triangle two variable
Walsh-Fourier series of functions. Namely, we prove the a.e. convergence
() for each integrable two-variable function
Kutatások a diadikus harmonikus analÃzis körében = Research in dyadic harmonic analysis
A pályázat keretében Ãrott cikkek között számosban foglalkoztam egy és kétváltozós integrálható függvények logaritmikus közepeinek konvergenciájával. Többek között vizsgáltuk, hogy mi a legbÅ‘vebb norma konvergencia tér. A kutatási idÅ‘szak fÅ‘ eredménye: Gát, G.: Pointwise convergence of cone-like restricted two-dimensional (C,1) means of trigonometric Fourier series, Journal of Approximation Theory, 149 (1) (2007), 74-102. Marcinkiewicz és Zygmund 1939-ben igazolta kétváltozós trigonometrikus Fourier sorok Fejér közepeivel kapcsolatban, hogy a integrálható függvények kétdimenziós Fejér közepei majdnem mindenütt a függvényhez tartanak, hacsak az közepek indexei úgy tartanak végtelenbe, hogy a hányadosuk korlátos, azaz egy egyenes köré húzott kúpban maradnak. A nevezett cikkben igazoltam, hogy, ha az egyenest helyettesÃtjük egy függvény görbéjével, azaz egy ''görbe köré húzott kúpban maradnak az indexek'', akkor is igaz marad a majdnem mindenütti konvergencia. Továbbá, ha a "kúp jellegű" halmaz "végtelenül bÅ‘vül", akkor a tétel már nem fog teljesülni. | Among the papers written in the project I discussed the convergence of logarithmic means of one and two dimensional functions in several papers. Among others, we determinded the largest norm convergence space. The main result of the project is: Gát, G.: Pointwise convergence of cone-like restricted two-dimensional (C,1) means of trigonometric Fourier series, Journal of Approximation Theory, 149 (1) (2007), 74-102. In 1939 Marcinkiewicz and Zygmund proved with respect to the Fejér means of the trigonometric Fourier series of two variable integrable functions that if the ratio of the indices of the means remain bounded as they tend to infinity (in other words, they remain in some positive cone around of the identical function), then the Fejér means converge to the function almost everywhere. In my paper above I verified the same result for a more general case. That is, the identical function can be substituted by an "arbitrary" function. That is, the set of indices remain in a "cone-like" set ("a cone around a curve"). Moreover, if the "cone-like set" enlarges "infinitely", then the theorem fails to hold
Ces\`{a}ro means of subsequences of partial sums of trigonometric Fourier series
In 1936 Zygmunt Zalcwasser asked with respect to the trigonometric system
that how "rare" can a sequence of strictly monotone increasing integers
be such that the almost everywhere relation is fulfilled for each integrable function . In this paper, we give an
answer to this question. It follows from the main result that this a.e.
relation holds for every integrable function and lacunary sequence
of natural numbers
On almost everywhere convergence and divergence of Marcinkiewicz-like means of integrable functions with respect to the two-dimensional Walsh system
AbstractLet |n| be the lower integer part of the binary logarithm of the positive integer n and α:N2→N2. In this paper we generalize the notion of the two dimensional Marcinkiewicz means of Fourier series of two-variable integrable functions as tnαf≔1n∑k=0n−1Sα(|n|,k)f and give a kind of necessary and sufficient condition for functions in order to have the almost everywhere relation tnαf→f for all f∈L1([0,1)2) with respect to the Walsh–Paley system. The original version of the Marcinkiewicz means are defined by α(|n|,k)=(k,k) and discussed by a lot of authors. See for instance [13,8,6,3,11]