×

zbMATH — the first resource for mathematics

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.)
PDF BibTeX XML Cite
Full Text: DOI Link
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.