Abstract
In this article we discuss the Nörlund means of cubical partial sums of Walsh-Fourier series of a function in L p (1 ≤ p ≤ ∞). We investigate the rate of the approximation by this means, in particular, in Lip(α, p), where α > 0 and 1 ≤ p ≤ ∞. In case p = ∞ by L p we mean C W , the collection of the uniformly W-continuous functions. Our main theorems state that the approximation behavior of the two-dimensional Walsh- Nörlund means is so good as the approximation behavior of the one-dimensional Walsh- Nörlund means.
As special cases, we get the Nörlund logarithmic means of cubical partial sums of Walsh-Fourier series discussed recently by Gát and Goginava [5] in 2004 and the (C, β)-means of Marcinkiewicz type with respect to double Walsh-Fourier series discussed by Goginava [10].
Earlier results on one-dimensional Nörlund means of the Walsh-Fourier series was given by Móricz and Siddiqi [14].
Пезуме
Вработе рассматриваются средние Нëрлунда сумм Фурье-Уолща по квадратам для функций из L p (1 < p < ∞). Изучаются порядки приближений функций с помошью этих средних, в частности для функций из классов Lip(α, p), где α > 0 и 1 ≤ p ≤ ∞. В случае p = ∞ мы считаем, что L ∞ это C w , т.е. класс всех W-непрерывных функций. Нащи основные теоремы утверждают, что для двумерных рядов Фурье-Уолща качество приближения средними Уолща-Нëрлунда не хуже, чем для одномерных рядов.
Как частные случаи нащих реэультатов получаются оценки, недавно полученные в работе Гата и Гогинавы [5] для логарифмических средних сумм по кубам ряда Фурье-Уолща, а также (C, а)-qsредних типа Марцинкевича для двойного ряда, которые изучал Гогинава [10].
Более ранние результаты для одномерных средних Нëрлунда ряда Фурье-Уолща были получены Морицем и Сиддики в работе [14].
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Dedicated to Professor Ferenc Móricz on the occasion of his seventieth birthday
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Nagy, K. Approximation by Nörlund means of quadratical partial sums of double Walsh-Fourier series. Anal Math 36, 299–319 (2010). https://doi.org/10.1007/s10476-010-0404-x
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DOI: https://doi.org/10.1007/s10476-010-0404-x