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.     

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 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.     

Integrability conditions for multiple trigonometric series

Sufficient conditions (the Boas-Telyakovskii conditions) on the coefficients of multiple trigonometric series are found that guarantee integrability of the sums of these series. Under these conditions, estimates are obtained for integrals of the moduli of the functions defined by multiple trigonometric series. In addition, the Boas-Telyakovskii conditions are compared with previously known conditions. It is shown that the Boas-Telyakovskii conditions are the most general ones.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 340–365, March, 1992.     

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 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.     

On sets of absolute convergence for multiple trigonometric series

We obtain a sufficient condition for a set of measure zero in N-dimensional space to be a set of absolute convergence (A.C. set) for an N-tuple trigonometric series. We also show that, in a certain subclass of sets of measure zero (namely in the subclass of monotonic curves), this condition cannot be sharpened.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 625–635, May, 1973.The author wishes to thank E. M. Nikishin for some valuable remarks.     

Concerning a sufficient condition of sidon for the integrability of trigonometric series

We consider problems concerning integrability and convergence in the metric L of trigonometric series, the coefficientsa _k of which satisfy the conditions:a _k 0, there exist numbers A_k such that A_k 0, < , and ¦¦ A_k. The integrability of a series in cosines under these conditions is equivalent to a theorem of Sidon.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 317–328, September, 1973.     

On some properties of multiple conjugate trigonometric series

We have obtained an estimate, in terms of partial and mixed moduli, of the continuity of deviation of the Cesáro (C, ) means ( = (₁,...,_n),_i , ₁ > –1, ) of the sequence of rectangular partial sums ofn-multiple (n>1) conjugate trigonometric series from then-multiple truncated conjugate function. This estimate implies the result on them -convergence (1) of (C, ) means (₁ > 0, ) provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown that them -convergence cannot be replaced by ordinary convergence.     

Uniqueness of multiple trigonometric series

Translated from Matematicheskie Zametki, Vol. 52, No. 2, pp. 148–150, August, 1992.     

Uniqueness of multiple trigonometric series

For multiple trigonometric series we state uniqueness theorems that are multidimensional analogs of the Steinhaus, Kahane, and Katznelson theorems.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 596–603, October, 1995.     

On convergence of multiple trigonometric series with monotone coefficients

We study the Pringsheim pointwise convergence of multiple trigonometric series. We obtain a condition on the coefficients of the series that guarantees its Pringsheim convergence and prove the unimprovability of this condition.     

On strong Carleman means of multiple trigonometric Fourier series

We prove the strong Carleman summability of the Fourier series of continuous functions on the m-dimensional torus, with partial sums constructed over polyhedra of a certain class.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 275–279, February, 1992.     

Uniqueness theorems for multiple lacunary trigonometric series

Translated from Matematicheskie Zametki, Vol. 51, No. 6, pp. 27–31, June, 1992.