Investigations in the harmonic and classical analysis II.  Page description

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Details of project

  
Identifier
46192
Type K
Principal investigator Móricz, Ferenc
Title in Hungarian Vizsgálatok a harmonikus és klasszikus analízisben II.
Title in English Investigations in the harmonic and classical analysis II.
Panel Mathematics and Computing Science
Department or equivalent Bolyai Institute (University of Szeged)
Participants Bagota, Mónika
Fekete, Árpád
Fülöp, Vanda
Németh, József
Németh, Zoltán
Starting date 2004-01-01
Closing date 2007-12-31
Funding (in million HUF) 5.509
FTE (full time equivalent) 0.00
state closed project





 

Final report

  
Results in Hungarian
FOURIER-ANALÍZIS: Lebesgue integrálható függvény Fourier integrálja majdnem mindenütt statisztikusan konvergens az adott függvényhez. Tetszõleges Lebesgue integrálható függvényre alkalmazott Fejér operátor akkor és csak akkor $L^1$-beli, ha az adott függvény a valós $H^1$ Hardy térbeli. A maximál kongugált és Hilbert operátorok nem korlátosak a valós $H^1$ Hardy térbõl az $L^1$ térbe. R.P. Boas több tételének, amelyek nemnegatív együtthatójú, abszolút konvergens szinusz és koszinusz soroknak és klasszikus függvényosztályoknak a kapcsolatára vonatkoznak, általánosítása tetszõleges abszolút konvergens Fourier sorokra. FUNKCIONÁLANALÍZIS: Beppo Levi klasszikus monoton konvergencia tétele nem érvényes nemkommutatív $L^2$-terekben. APPROXIMÁCIÓELMÉLET: L. Leindler tételének, amely az általánosított $\Omega_\alpha$ Zygmund függvényosztályoknak és a Fourier sorokkal történõ erõs approximációnak a kapcsolatára vonatkozik, kiterjesztése $\alpha=1$-rõl $0<\alpha<2$-re. SZUMMÁLHATÓSÁG: J. Karamata elegendõ Tauber feltételének élesítése szükséges és elegendõ Tauber feltétellé. I. J. Schoenberg Tauber tételének kiterjesztése számsorozatokról mérhetõ függvények sorozataira.
Results in English
FOURIER ANALYSIS: The Fourier integral of a Lebesgue integrable function is statistically convergent almost everywhere to the given function. The maximal Fejér operator applied to any integrable function $f$ is in $L^1$ if and only if $f$ is in the real Hardy space $H^1$. The maximal conjugate and Hilbert operators are not bounded from the real Hardy space $H^1$ to the space $L^1$. A number of theorems of R.P. Boas on the interrelation of absolutely convergent sine and cosine series with nonnegative coefficients and classical function classes were generalized for arbitrary absolutely convergent Fourier series. FUNCTIONAL ANALYSIS: The classical monotone convergence theorem of Beppo Levi fails in noncommutative $L_2$-spaces. APPROXIMATION THEORY: A theorem of L. Leindler on the interrelation between the generalized Zygmund classes $\Omega_\alpha$ of functions and strong approximation by Fourier series was extended from $\alpha=1$ to $0<\alpha<2$. SUMMABILITY: A sufficient Tauberian condition of J. Karamata was sharpened into a necessary and sufficient Tauberian condition. A Tauberian theorem of I. J. Schoenberg was extended from sequences of numbers to sequences of measurable functions.
Full text http://real.mtak.hu/1329/
Decision
Yes





 

List of publications

  
V. Fülöp: Double cosine series with nonnegative coefficients, Acta Sci. Math. 70, 91-100, 2004
V. Fülöp: Double sine and cosine-sine series with nonnegative coefficients, Acta Sci Math. 70, 101-116, 2004
V. Fülöp - F. Móricz: Order of magnitude of multiple Fourier coefficients of functions of bounded variation, Acta Math. Hungar. 104, 95-104, 2004
G. Brown - Feng Dai - F. Móricz: Strong approximation by Fourier-Laplace series on the unit sphere S^{n-1}, Acta Math. Hungar. 102, 91-115, 2004
F. Móricz - B. E. Rhoades: Necessary and sufficient Tauberian conditions for certain weighted mean methods of summability. II, Acta Math. Hungar. 102, 279-285, 2004
F. Móricz: Necessary and sufficient Tauberian conditions in the case of weighted mean summable integrals over R_+, Math. Ineq. & Appl. 7, 87-93., 2004
F. Móricz: Fejér type theorems for Fourier - Stieltjes series, Analysis Math. 30, 123-136, 2004
F. Móricz: Theorems relating to statistical harmonic summability and ordinary convergence of slowly decreasing or oscillating sequences, Analysis 24, 127-145, 2004
F. Móricz: Ordinary convergence follows from statistical summability (C,1) in the case of slowly decreasing or oscillating sequences, Colloq.Math. 99, 207-219, 2004
B. Le Gac - F. Móricz: A new two-parameter SLLN in noncommutative L_2-spaces in terms of bundle convergence, Acta Sci. Math. 70, 213-228, 2004
G. Brown - Feng Dai - F. Móricz: The maximal Fejér operator on real Hardy spaces, Periodica Math. Hungar. 49, 15-25, 2004
F. Móricz - C. Orhan: Tauberian conditions under which statistical convergence follows from statistical summability by weighted means, Studia Sci. Math. Hungar. 41, 391-403, 2004
B. Le Gac - F. Móricz: Bundle convergence in a von Neumann algebra and in a von Neumann subalgebra, Bull. Polish Acad. Sci. Math. 52, 283-295, 2004
F. Móricz: Statistical convergence of Walsh--Fourier series, Acta Math. Acad. Paedagogicae Nyíregyháziensis elektronikus folyóirat 20:2, 7. cikk, 2004
F. Móricz - U. Stadtmüller: Summability of double sequences by weighted mean methods and Tauberian conditions for convergence in Pringsheim\'s sense, Internat. J. Math. Math. Sci. 65, 3499-3511, 2004
F. Móricz: Konvex és konkáv függvények, Polygon 13, 29-50, 2004
A. Fekete: Tauberian conditions for double sequences that are statistically summable by weighted means, Sarajevo J. Math. (1) 14 , 197-210., 2005
G. Brown - Feng Dai - F. Móricz: The maximal conjugate and Hilbert operators on real Hardy spaces, Acta Math. Hungar. 109 53-63, 2005
F. Móricz: Multivariate Hausdorff operators on the spaces $H^1 (R^n)$ and BMO $(R^n)$, Analysis Math. 31, 31--41, 2005
F. Móricz: Strong Ces\`aro summability and statistical limit of double Fourier integrals, Acta Sci. Math. Szeged, 71, 159--174, 2005
F. Móricz: Constructive Function Theory: I. Orthogonal Series, A Panorama of Hungarian Mathematics in the Twentieth Century, Bolyai Society Math. Studies 14, Springer, 29--54., 2005
F. Móricz: Regular statistical convergence of multiple sequences, Analysis 25, 171--182, 2005
F. Móricz: Valós és komplex számok végtelen szorzatai, Polygon 14:1, 19--43, 2005
F. Móricz: Valós vagy komplex változó hatványsorai, Polygon 14:2, 12--34, 2005
J. Németh: Embedding relations and generalized Lipschitz classes, Acta Sci. Math. Szeged 71, 175--180., 2005
J. Németh: A remark on the degree of approximation of continuous functions, Acta Math. Hungar. 106, 83--88., 2005
A. Fekete: Tauberian conditions under which the statistical limit of an integrable function follows from its summability, Studia Sci. Math. Hungar., 43, 131-145, 2006
Á. Jenei and F. Móricz: Extension of the Dini test to double Fourier series, Acta Sci. Math. Szeged, 72, 135-145, 2006
Á. Jenei and F. Móricz: Pointwise convergence of series conjugate to double Fourier series, Acta Sci. Math. Szeged 72, 555-567., 2006
F. Móricz: On the harmonic averages of numerical sequences, Arch. Math. Basel, 86, 375-384., 2006
F. Móricz and W.R. Wade: On the problem of uniqueness of the trigonometric moment constants, Acta Math. Hungar., 111, 313-324, 2006
F. Móricz: Pointwise behavior of double Fourier series of functions of bounded variation, Monatsh. Math., 148, 51-59, 2006
B. Le Gac and F. Móricz: Bundle convergence of sequences of pairwise uncorrelated operators in von Neumann algebras and vectors in their $L_2$-spaces, Indag. Mathem., N. S., 7, 221-230, 2006
G. Brown - Feng Dai - F. Móricz: Strong Cesaro summability of double Fourier integrals, Acta Math. Hungar., 115, 1-12., 2007
Á. Fekete - F. Móricz: A characterization of the existence of statistical limit of real-valued measurable functions, Acta Math. Hungar., 114, 235-246, 2007
F. Móricz: A Dini type test on the pointwise convergence of double Fourier integrals, Analysis Math., 33, 45-54., 2007
F. Móricz: Statistical extendsions of some classical Tauberian theorems in nondiscrete setting, Colloq. Math., 107, 45-56, 2007
F. Móricz: Pringsheim type test on the pointwise convergence of integrals conjugate to double Fourier integrals, Analysis Math., 33, 287-299., 2007
F. Móricz - A. Veres: On the absolute convergence of multiple Fourier series, Acta Math. Hungar., 117, 275-292., 2007
Z. Németh: Az arkhimédeszi tulajdonságról, Polygon, 15(2), 37-47, 2007
G. Brown - Feng Dai - F. Móricz: The maximal Riesz, Fejér, and Cesàro operators on real Hardy spaces, J. Fourier Anal. Appl. 10, 27-50, 2004
F. Móricz: Statistical limits of measurable functions, Analysis 24, 1-18, 2004
F. Móricz: Strong Cesaro summability and statistical limit of Fourier integrals, Analysis 25, 79--86, 2005
F. Móricz: Pointwise behavior of Fourier integrals of functions of bounded variation over R, J. Math. Anal. Appl. 297, 527-539, 2004
F. Móricz: Absolutely convergent Fourier series and function classes, J. Math. Anal. Appl., 324, 1168-1177, 2006
V. Fülöp: Double sine series with nonnegative coefficients and Lipschitz classes, Colloq. Math., 105, 25-34, 2006
B .Le Gac - F. Móricz: The classical monotone convergence theorem of Beppo Levi fails in noncommutative $L_2$-spaces, Proc. Amer. Math. Soc. 133 2559--2567, 2005
F. Móricz - J. Németh: Generalized Zygmund classes of functions and strong approximation by Fourier series, Acta Sci. Math. Szeged, 73, 637-647., 2007
A. Fekete - F. Móricz: Necessary and sufficient Tauberian conditions in the case of weighted mean summable integrals over $R_+$, II., Publ. Math. Debrecen 67 65--78, 2005
F. Móricz: Regular statistical convergence of double sequences, Colloq. Math. 102 217--227, 2005



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