Weighted integrability of double trigonometric series

We study the double trigonometric series whose coefficients are such that Then its rectangular partial sums converge uniformly to some . We give sufficient conditions for the Lebesgue integrability of , where , or . For certain cases, they are also necessary conditions. Our results extend those of Boas and Móricz from the one-dimensional to the two-dimensional series.

     

On examples concerning integrability theorems for trigonometric series

Examples of trigonometric series are constructed that are not Fourier series, with coefficients whose differences are majorized by an arbitrary sequence of numbers given in advance, which satisfies certain natural conditions.     

On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series

For a trigonometric series

Martingales and some integrability criterions for multiple trigonometric series

Applying some martingale properties of the sequences of cubic partial sums of series in the Haar multiple system, some integrability criterions for multiple trigonometric series are established. Some of the proved theorems improve and extend to the multiple trigonometric series the classical theorems on integrability of one-dimensional trigonometric series, the proofs of which are based on Helly??s principal of choice.     

On the uniqueness and integrability of multiple trigonometric series

Under minimal constraints on the coefficients, we prove uniqueness theorems for multiple trigonometric series in which, instead of pointwise convergence, we consider the convergence of integral means of spherical, cubic, and other partial sums. We also obtain sufficient conditions for the integrability of multiple trigonometric series, i.e., conditions under which these series are Fourier series.     

A new condition for certain trigonometric series

The present paper proposes a new natural weak condition to replace the quasimonotone and O-regularly varying quasimonotone conditions, and shows the conclusion of the main result in [10] still holds. Moreover, we prove that any O-regularly varying quasimonotone condition implies this condition, thus tne vague implication of quasimonotonicity can be avoided in practice.     

On integrability and L-convergence differentiated trigonometric series

New integrability and L¹-convergence classes for the r-times differentiated trigonometric series are derived using known results on integrability and L¹-convergence of trigonometric series. These classes, Theorems 1 and 2, subsume all known integrability and L¹-convergence classes for differentiated series. In particular, the extensions of the Fomin type theorems to r-times differentiated series, due to S.S. Bhatia and B. Ram (Proc. Amer. Math. Soc. 124 (1996) 1821-1829) and S.Y. Sheng (Proc. Amer. Math. Soc. 110 (1990) 895-904), are deduced from our Theorem 2 and its corollary Theorem 4. Another extension of the known results for differentiated series is given in Theorem 6. This result is proved using Fomin's theorem on cosine and sine series (Mat. Zametki 23 (1978) 213-222), and as a corollary of Theorem 2. The first proof yields another interesting conclusion, namely that the Fomin type theorems for differentiated series can be deduced as consequences of the original Fomin's results.     

On weighted L p integrability and approximation of trigonometric series

We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first results on the relations among the weighted best approximation and the coefficients of trigonometric series. Supported in part by Hangzhou Normal University.     

Concerning the multiplicators of Fourier series in the trigonometric system

In this paper we prove theorems on multiplicators of Fourier series inL _p, where the conditions depend on a parameterp. An example illustrating the importance of these conditions is constructed. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 235–247, February, 1998.     

A new condition for the uniform convergence of certain trigonometric series

Summary The present paper proposes a new condition to replace both the (O-regularly varying) quasimonotone condition and a certain type of bounded variation condition, and shows the same conclusion for the uniform convergence of certain trigonometric series still holds.     

A sufficient condition for mean convergence of orthogonal series for Freud weights

Let S_n[f](x) be the n-th partial sum of the orthonormal polynomial series expansion for f corresponding to a Freud weight W(x)=e^−Q(x). We give a sufficient condition for the inequality      

A sufficient condition for mean convergence of orthogonal series for Freud weights

Let S_n[f](x) be the n-th partial sum of the orthonormal polynomial series expansion for f corresponding to a Freud weight W(x)=e^−Q(x). We give a sufficient condition for the inequality to hold. This sufficient condition is much easier to apply than that of Jha and Lubinsky [2].     

Mean bounded variation condition and applications in double trigonometric series

We introduce a new kind of double sequences named MVBVDS and some new classes of weight functions to study the weighted integrability of the double trigonometric series. Several results of Chen, Marzuq, Móricz, Ram and Singh Bhatia (see [2]?C[10]) are generalized and some new results are established.