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  • 1 Tbilisi State University Department of Mathematics, Faculty of Exact and Natural Sciences Chavchavadze str. 1 Tbilisi 0128 Georgia
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In this paper we study the exponential uniform strong approximation of two-dimensional Walsh-Fourier series. In particular, it is proved that the two-dimensional Walsh-Fourier series of the continuous function f is uniformly strong summable to the function f exponentially in the power 1/2. Moreover, it is proved that this result is best possible.

  • Alexits, G. and Králik, D., Über den Annäherungsgrad der Approximation im starken Sinne von stetigen Funktionen (German. Russian summary), Magyar Tud. Akad. Mat. Kutató Int. Közl., 8 (1964), 317–327 (1964). MR 0185339 (32#2807)

    Králik D. , 'Über den Annäherungsgrad der Approximation im starken Sinne von stetigen Funktionen (German. Russian summary) ' (1964 ) 8 Magyar Tud. Akad. Mat. Kutató Int. Közl. : 317 -327.

    • Search Google Scholar
  • Fejér, Leopold, Untersuchungen über Fouriersche Reihen (German), Math. Ann., 58 (1903), no. 1–2, 51–69. MR 1511228

    Fejér L. , 'Untersuchungen über Fouriersche Reihen (German) ' (1903 ) 58 Math. Ann. : 51 -69.

    • Search Google Scholar
  • Fridli, S. and Schipp, F., Strong summability and Sidon type inequalities (English summary), Acta Sci. Math. (Szeged), 60 (1995), no. 1–2, 277–289. MR 1348694 (98f:42002)

    Schipp F. , 'Strong summability and Sidon type inequalities (English summary) ' (1995 ) 60 Acta Sci. Math. (Szeged) : 277 -289.

    • Search Google Scholar
  • Fridli, S. and Schipp, F., Strong approximation via Sidon type inequalities (English summary), J. Approx. Theory, 94 (1998), no. 2, 263–284. MR 1637418 (99e:41011)

    Schipp F. , 'Strong approximation via Sidon type inequalities (English summary) ' (1998 ) 94 J. Approx. Theory : 263 -284.

    • Search Google Scholar
  • Gogoladze, L., On the exponential uniform strong summability of multiple trigonometric Fourier series, Georgian Math. J., 16 (2009), 517–532. MR 2572672 (2010k:42016)

    Gogoladze L. , 'On the exponential uniform strong summability of multiple trigonometric Fourier series ' (2009 ) 16 Georgian Math. J. : 517 -532.

    • Search Google Scholar
  • Goginava, U. and Gogoladze, L., Strong Approximation By Marcinkiewicz Means of two-dimensionalWalsh-Fourier Series, Constructive Approximation, 35,1 (2012), 1–19.

    Gogoladze L. , 'Strong Approximation By Marcinkiewicz Means of two-dimensionalWalsh-Fourier Series ' (2012 ) 351 Constructive Approximation : 1 -19.

    • Search Google Scholar
  • Hardy, G. H. and Littlewood, J. E., Sur la serie de Fourier d’une fonction a carre sommable, C.R. Acad. Sci. Paris, 156 (1913), 1307–1309.

    Littlewood J. E. , 'Sur la serie de Fourier d’une fonction a carre sommable ' (1913 ) 156 C.R. Acad. Sci. Paris : 1307 -1309.

    • Search Google Scholar
  • Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities, 2nd ed., Cambridge University Press (Cambridge, 1952). MR 0046395 (13,727e)

    Polya G. , '', in Inequalities , (1952 ) -.

  • Leindler, L., Über die Approximation im starken Sinne, Acta Math. Acad. Hungar., 16 (1965), 255–262. MR 0174921 (30#5112)

    Leindler L. , 'Über die Approximation im starken Sinne ' (1965 ) 16 Acta Math. Acad. Hungar. : 255 -262.

    • Search Google Scholar
  • Leindler, L., On the strong approximation of Fourier series, Acta Sci. Math. (Szeged), 38 (1976), 317–324. MR 0433116 (55#6095)

    Leindler L. , 'On the strong approximation of Fourier series ' (1976 ) 38 Acta Sci. Math. (Szeged) : 317 -324.

    • Search Google Scholar
  • Leindler, L., Strong approximation and classes of functions, Mitteilungen Math. Seminar Giessen, 132 (1978), 29–38. MR 0493404 (80d:42002)

    Leindler L. , 'Strong approximation and classes of functions ' (1978 ) 132 Mitteilungen Math. Seminar Giessen : 29 -38.

    • Search Google Scholar
  • Leindler, L., Strong approximation by Fourier series, Akadémiai Kiadó (Budapest, 1985). MR 0829707 (87g:42006)

    Leindler L. , '', in Strong approximation by Fourier series , (1985 ) -.

  • Rodin, V. A., BMO-strong means of Fourier series, Funct. anal. Appl., 23 (1989), 73–74, (Russian) translation in Funct. Anal. Appl., 23 (1989), no. 2, 145–147. MR 1011366 (90h:42033)

    Rodin V. A. , 'BMO-strong means of Fourier series ' (1989 ) 23 Funct. anal. Appl. : 73 -74.

  • Schipp, F., Über die starke Summation von Walsh-Fourier Reihen, Acta Sci. Math. (Szeged), 30 (1969), 77–87. MR 0241890 (39#3227)

    Schipp F. , 'Über die starke Summation von Walsh-Fourier Reihen ' (1969 ) 30 Acta Sci. Math. (Szeged) : 77 -87.

    • Search Google Scholar
  • Schipp, F., On strong approximation of Walsh-Fourier series, MTA III. Oszt. Közl., 19 (1969), 101–111 (Hungarian).

    Schipp F. , 'On strong approximation of Walsh-Fourier series ' (1969 ) 19 MTA III. Oszt. Közl. : 101 -111.

    • Search Google Scholar
  • Schipp, F. and Ky, N. X., On strong summability of polynomial expansions, Anal. Math., 12 (1986), 115–128. MR 0854534 (87i:40009)

    Ky N. X. , 'On strong summability of polynomial expansions ' (1986 ) 12 Anal. Math. : 115 -128.

    • Search Google Scholar
  • Schipp, F., Wade, W. R., Simon, P. and Pál, J., Walsh Series, an Introduction to Dyadic Harmonic Analysis, Adam Hilger (Bristol, New York, 1990).

  • Stećkin, S. B., The approximation of periodic functions by Fejér sums (Russian), Trudy Mat. Inst. Steklov., 62 (1961), 48–60; English transl.: Amer. Mat. Soc. Transl. (2), 28 (1963), 269–282. MR 0162085 (28#5287a)

    Stećkin S. B. , 'The approximation of periodic functions by Fejér sums (Russian) ' (1961 ) 62 Trudy Mat. Inst. Steklov. : 48 -60.

    • Search Google Scholar
  • Totik, V., On the strong approximation of Fourier series, Acta Math. Sci. Hungar., 35 (1980), no. 1–2, 151–172. MR 0588890 (82c:42004)

    Totik V. , 'On the strong approximation of Fourier series ' (1980 ) 35 Acta Math. Sci. Hungar. : 151 -172.

    • Search Google Scholar
  • Totik, V., On the generalization of Fejér’s summation theorem, in: Functions, Series, Operators; Coll. Math. Soc. J. Bolyai (Budapest) Hungary, 35, North Holland (Amsterdam-Oxford-New-Yourk, 1980), 1195–1199. MR 0751078 (85h:42008)

    Totik V. , '', in Functions, Series, Operators , (1980 ) -.

  • Totik, V., Notes on Fourier series: Strong approximation, J. Approx. Theory, 43 (1985), no. 2, 105–111. MR 0775778 (86g:42009)

    Totik V. , 'Notes on Fourier series: Strong approximation ' (1985 ) 43 J. Approx. Theory : 105 -111.

    • Search Google Scholar
  • Weisz, F., Strong summability of Ciesielski-Fourier series, Studia Math., 161 (2004), no. 3, 269–302. MR 2033018 (2004k:42008)

    Weisz F. , 'Strong summability of Ciesielski-Fourier series ' (2004 ) 161 Studia Math. : 269 -302.

    • Search Google Scholar
  • Weisz, F., Strong summability of more-dimensional Ciesielski-Fourier series, East J. Approx., 10 (2004), no. 3, 333–354. MR 2076892 (2005f:42072)

    Weisz F. , 'Strong summability of more-dimensional Ciesielski-Fourier series ' (2004 ) 10 East J. Approx. : 333 -354.

    • Search Google Scholar