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Pointwise convergence of cone-like restricted two-dimensional \((C,1)\) means of trigonometric Fourier series. (English) Zbl 1135.42007

Summary: The aim of this work is to generalize the more than 60 year old celebrated result of J. Marcinkiewicz and A. Zygmund [Fundam. Math. 32, 122–132 (1939; Zbl 0022.01804)] on the convergence of the two-dimensional restricted \((C,1)\) means of trigonometric Fourier series. They proved for any integrable function \(f\in L^1(T^2\) the a.e. convergence
\[ \sigma_{(n_1,n_2)} f\to f \]
provided \(n_1/\beta\leq n_2\leq\beta n_1\), where \(\beta>1\) is fixed constant. That is, the set of indices \((n_1,n_2)\) remains in some positive cone around the identical function. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets (of the set of indices) in order to preserve this convergence property.

MSC:

42B08 Summability in several variables

Citations:

Zbl 0022.01804
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References:

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