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Integrability and $$L$$-convergence of multiple trigonometric series. (English) Zbl 0795.42007
Summary: ft is proved that under the following condition the sum $$f$$ of double trigonometric series with coefficients $$c_{jk}$$ is integrable and the rectangular partial sums $$s_{mn}(f,x,y)$$ converge to $$f$$ in $$L^ 1$$- norm: $\sum^ \infty_{j=-\infty} \sum^ \infty_{k=-\infty} (\ln| j|)^ \top(\ln| k|)^ \top\;|\Delta_{11} c_{jk}|< \infty,$ where $$\xi^ \top=\max(1,\xi)$$ and $$\Delta_{11} c_{jk}= c_{j,k}- c_{j,k+1}- c_{j+1,k}+ c_{j+1,k+1}$$. This generalizes the corresponding results of F. Móricz [Stud. Math. 98, No. 3, 203-225 (1991; Zbl 0724.42015)]. We also prove that the aforementioned condition is sharp. A more general version of this result is established for double series of orthonormal functions, which generalizes one of F. Móricz, F. Schipp and W. R. Wade [Mich. Math. J. 37, No. 2, 191-201 (1990; Zbl 0714.42017)]. An extension to higher dimensions is given.

##### MSC:
 42B05 Fourier series and coefficients in several variables 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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##### References:
 [1] Schipp, Walsh Series. An Introduction to Dyadic Harmonic Analysis (1990) [2] DOI: 10.1307/mmj/1029004125 · Zbl 0714.42017 [3] Zygmund, Trigonometric Series (1959) [4] DOI: 10.2307/2047238 · Zbl 0666.42004 [5] DOI: 10.2307/1990619 · Zbl 0036.03604 [6] Móricz, Studia. Math. 98 pp 203– (1991)
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