Integrability of double lacunary sine series
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Abstract:
We consider mainly the series $\sum {\sum {{a_{jk}}\sin {2^j}x\sin {2^k}y} }$, which converges to a finite function $f(x,y)$ a.e. if $\sum {\sum {a_{jk}^2 < \infty } }$. We show that the (Lebesgue) integrability of ${x^{ - 1}}{y^{ - 1}}f(x,y)$ over the two-dimensional torus is essentially controlled by the quantity $\sum \nolimits _{m = 1}^\infty {\sum \nolimits _{n = 1}^\infty {{{\left ( {\sum \nolimits _{j = m}^\infty {\sum \nolimits _{k = n}^\infty {a_{jk}^2} } } \right )}^{1/2}}} }$. Our result is an extension of the corresponding one by M. C. Weiss [4] from one-dimensional to two-dimensional lacunary sine series.References
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R. P. Boas, Integrability theorems for trigonometric transforms, Springer-Verlag, Berlin, 1967.
- F. Móricz, On the $|C,\alpha >{1\over 2},\beta >{1\over 2}|$-summability of double orthogonal series, Acta Sci. Math. (Szeged) 48 (1985), no. 1-4, 325–338. MR 810889 W. Orlicz, Beiträge zur Theorie der Orthogonalentwicklungen, Studia Math. 6 (1936), 20-38. M. C. Weiss, The law of the iterated logarithm for lacunary series and its application to the Hardy-Littlewood series, dissertation, University of Chicago, 1957.
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 355-364
- MSC: Primary 42B05
- DOI: https://doi.org/10.1090/S0002-9939-1990-1021902-5
- MathSciNet review: 1021902