Research in dyadic harmonic analysis  Page description

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Details of project

  
Identifier
48780
Type K
Principal investigator Gát, György
Title in Hungarian Kutatások a didaktikus harmonikus analízis körében
Title in English Research in dyadic harmonic analysis
Panel Mathematics and Computing Science
Department or equivalent Institute of Mathematics and Informatics (University of Nyíregyháza)
Starting date 2005-01-01
Closing date 2008-12-31
Funding (in million HUF) 1.872
FTE (full time equivalent) 3.20
state closed project





 

Final report

  
Results in Hungarian
A pályázat keretében írott cikkek között számosban foglalkoztam egy és kétváltozós integrálható függvények logaritmikus közepeinek konvergenciájával. Többek között vizsgáltuk, hogy mi a legbővebb norma konvergencia tér. A kutatási időszak fő 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 helyettesítjük 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 bővül”, akkor a tétel már nem fog teljesülni.
Results in English
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.
Full text http://real.mtak.hu/1866/
Decision
Yes





 

List of publications

  
Gát, G., Goginava, U., Tkebuchava, G.: Convergence of logarithmic means of multiple Walsh-Fourier series, Analysis in Theory and Applications 21 (4) (2005), 326-338., 2005
Gát, G.: On the divergence of the Fejer means of integrable functions on two- dimensional Vilenkin groups 107 (1-2) (2005), 17-33., Acta Math. Hungar., 2005
Gát, G.: Divergence of Fejér means of Lipschitz functions on noncommutative Vilenkin groups with respect to the character system 71 (2005), 181-193., Acta Sci. Math. (Szeged), 2005
Gát, G.: Properties of Fejér means of Fourier series with respect to unbounded Vilenkin systems, Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl. 76 (2005), 355-373., 2005
Gát, G., Goginava, U., Tkebuchava, G.: Convergence in measure of logarithmic means of double Walsh-Fourier series, Georgian Math. Journal 12 (4) (2005), 607-618., 2005
Gát, G., Goginava, U.: Uniform and L-convergence of logarithmic means of double Walsh-Fourier series, Georgian Math. Journal 12 (1) (2005), 075-088., 2005
Gát, G.: Fejér means of functions on noncommutative Vilenkin groups with respect to the character system, Analysis Math. 32 (2006), 25-48., 2006
Gát, G.: On convergence properties of logarithmic means of Walsh-Fourier series, Constructive Function Theory, Varna, july 1-7, 2005 (B. Bojanov ed.), Marin Drinov Academic Publishing House, Sofia, 2006, 113-120., 2006
Gát, G., Goginava, U.: Almost Everywhere Convergence of (C, a)-Means of Quadratical Partial sums of double Vilenkin-Fourier Series, Georgian Math. J., 13 (3) (2006), 447-462., 2006
Blahota, I., Gát, G., Goginava, G.: Maximal operators of Fej\'er means of Vilenkin-Fourier series, J. Inequal. Pure Appl. Math. 7 (4) (2007), 1-7., 2007
Blahota, I., Gát, G.: Norm summability of Nörlund logarithmic means on unbounded Vilenkin groups, Analysis in Theory and Applications, 24 (1) (2008),, 2008
Gát, G.: Convergence and divergence of Fejér means of Fourier series on one and two-dimensional Walsh and Vilenkin groups, Facta Universitatis (Series: University of Nis, Electronics and Energetics) published by the University of Nis, YU 21(3) (2008), 291-307., 2008
Gát, G., Nagy, K.: Almost everywhere convergence of a subsequence of the logarithmic means of Vilenkin Fourier series, Facta Universitatis (Series: University of Nis, Electronics and Energetics) published by the University of Nis, YU 21(3) (2008), 275-289., 2008
Gát, G., Goginava, U., Nagy, K.: Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system, Stud. Sci. Math. Hungar. (to appear), 2009
Gát, G., Goginava: On the divergence of NÄorlund logarithmic means of Walsh-Fourier series, Acta Math. Sinica (English series) (to appear), 2009
Gát, G., Toledo, R.: On the converge in L1-norm of Cesaro means with respect to representative product systems, Acta Math. Hungar. (to appear)., 2009
Gát, G., Nagy, K.: On the (C; a)-means of quadratical partial sums of double Walsh-Kaczmarz-Fourier series, Georgian Math. Journal (to appear), 2009
Gát, G., Goginava, U.: The weak type inequality for the maximal operator of the (C; a) -means of the Fourier series with respect to the Walsh-Kaczmarz system, Acta Math. Hungar. (to appear), 2009
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., 2007
Gát, G.: Almost everywhere convergence of Fejér means of $L^1$ functions on rarely unbounded Vilenkin groups, Acta Mathematica Sinica (English series) 23 (12) (2007), 2269-2294., 2007
Gát, G., Goginava, U.: Uniform and L-convergence of logarithmic means of Walsh-Fourier series, Acta Mathematica Sinica (English Series) 22 (2) (2006), 497-506., 2006
Gát, G., Goginava, U.: Maximal convergence space of a subsequence of the logarithmic means of rectangular partial sums of double Walsh-Fourier series, Real Analysis Exchange, 31 (2) (2006), 447-464., 2006
Gát, G., Goginava, U., Nagy, K.: On (Hpq,Lpq)-type inequality of maximal operator of Marcinkiewicz-Fejér means of double Fourier series with respect to the Kaczmarz system, Math. Ineq. and Applications 9 (3) (2006), 473-483., 2006
Gát, G., Goginava, U., Tkebuchava, G.: Convergence in measure of logarithmic means of quadratical partial sums of double Walsh-Fourier series, Journal of Math. Anal. and Appl. 323 (2006), 535-549., 2006
Blahota, I., Gát, G., Goginava, U.: Maximal operators of Fejér means of double Vilenkin-Fourier series, Colloquium Mathematicum 107 (2) (2007), 287-296., 2007
Gát, G.: Almost everywhere convergence of Cesaro means of Fourier series on the group of $2$-adic integers, Acta Math. Hungar. 116 (3) (2007), 209-221., 2007
Gát, G., Goginava, G.: Almost everywhere Convergence of a Subsequence of the Logarithmic means of quadratical partial sums of Double Walsh-Fourier Series, Publ. Math. Debrecen 71 (1-2) (2007), 173-184., 2007
Blahota, I., Gát, G.: Almost everywhere convergence of Marcinkiewicz means of Fourier series on the group of 2-adic integers, Studia Math. (to appear)., 2009



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