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On weighted integrability of double cosine series. (English) Zbl 0879.42008
The authors define the class $$L(p,\alpha,\beta)$$ of even, periodic functions $$f(x,y)$$ by the requirement $|f|_{p,\alpha,\beta}:= \Biggl\{\int^\pi_0 \int^\pi_0|f(x,y)|^p(\sin x)^{\alpha p}(\sin y)^{\beta p}dx dy\Biggr\}^{1/p}< \infty.$ They prove lower or/and upper estimates for this norm in the special case when the Fourier series $$\sum^\infty_{m=1} \sum^\infty_{n=1} a_{mn}\cos mx\cos ny$$ of $$f$$ is such that some of the conditions $a_{mn}\geq 0,\quad\Delta_{10} a_{mn}\geq 0,\quad \Delta_{01}a_{mn}\geq 0,\quad\Delta_{11}a_{mn}\geq 0$ are satisfied. In this way, they extend two theorems of R. Askey and S. Wainger [Duke Math. J. 33, 223-228 (1966; Zbl 0136.36501)] and a theorem of the reviewer [Proc. Am. Math. Soc. 109, No. 2, 417-425 (1990; Zbl 0741.42010)].
Reviewer: F.Móricz (Szeged)

##### MSC:
 42B05 Fourier series and coefficients in several variables 42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
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##### References:
 [1] Askey, R.; Wainger, S., Integrability theorems for Fourier series, Duke math. J., 33, 223-228, (1966) · Zbl 0136.36501 [2] Chen, C.P., Weighted integrability andL1, Studia math., 108, 177-190, (1994) [3] Hardy, G.H.; Littlewood, J.E., Some new properties of Fourier coefficients, J. London math. soc., 6, 3-9, (1931) · Zbl 0001.13504 [4] Marzuq, Maher M.H., Integrability theorem of multiple trigonometric series, J. math. anal. appl., 157, 337-345, (1991) · Zbl 0729.42012 [5] Móricz, F., On double cosine, sine and Walsh series with monotone coefficients, Proc. amer. math. soc., 109, 417-425, (1990) · Zbl 0741.42010 [6] Móricz, F., On the integrability andL1, Studia math., 93, 203-225, (1991) · Zbl 0724.42015
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