Some problems in the theory of multiple trigonometric series

Dyachenko, Mikhail Ivanovich

doi:10.1070/rm1992v047n05abeh000944

Cited by 24 publications

(9 citation statements)

References 102 publications

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“…In this section, we prove that these conditions are satisfied for a rather natural class of partial sums of Fourier series, where the domains Gn are convex with respect to each coordinate direction. The same result was established by D'yachenko [6] under the more restrictive assumptions that for each k E Gn all integer points of the rectangle [1, kl] x ..-x [1, kd] are also contained in G~.…”

Section: S(h G~ X) --+ H(x)supporting

confidence: 62%

“…The proof in [4] was based on Hardy's theorem and on Telyakovskii's result [5] on the uniform boundedness of some special trigonometric polynomials of several variables. Using the same scheme, D'yachenko [6,Theorem 7.12] extended Hardy's theorem to partial sums over domains that have the following property: if k belongs to a domain, then all points from the d-dimensional rectangle [1, kl] x ---x [1, kd] are also in the domain.…”

Section: Chandrasekharan and Minakshisundaram [3 Theorem 43] Provedmentioning

confidence: 98%

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Convergence of multiple fourier series for functions of bounded variation

Telyakovskiĭ

Temlyakov

1997

Math Notes

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ABSTRACT. For functions of bounded variation in the sense of Hardy, we consider, the pointwise convergence of the partial sums of Fourier series over a given sequence of bounded sets in the space of harmonics. We obtain sufficient conditions for convergence; necessary and sufficient conditions are obtained for the case in which these sets are convex with respect to each coordinate direction. The Pringsheim convergence of Fourier series in this problem was established by Hardy.KEY WORDS: multiple Fourier series, functions of bounded variation in the Sense of Hardy, generalized Pringsheim convergence.w IntroductionWe study the convergence of Fourier series for functions f(x) = f(xl,..., Xd) of bounded variation in the sense of Hardy (f E HV) and for the narrower class of functions whose mixed derivative Oaf/Oxl ... OXd is bounded in the L1 norm.The definitions of the function classes HV and W~ are given in w From now on we consider functions f whose Fourier coefficientsare zero whenever at least one component of the vector k is zero. This assumption is essentially unimportant but allows us to simplify the subsequent statements.Hardy Here we obtain a further generalization of Hardy's theorem in this direction (see Theorem 4 below). We study the convergence of partial sums of Fourier series of the form S(f, G,x) = ~ f^(k)e i(u'• (1.1) [klEG

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Section: S(h G~ X) --+ H(x)supporting

confidence: 62%

Section: Chandrasekharan and Minakshisundaram [3 Theorem 43] Provedmentioning

confidence: 98%

Convergence of multiple fourier series for functions of bounded variation

Telyakovskiĭ

Temlyakov

1997

Math Notes

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“…In this paper, we restrict ourselves to questions which were not covered by previous expository papers and which were actively developed over the last 25 years. For example, we do not discuss questions which are quite naturally linked to the mixed moduli of smoothness such as · Different types of convergence of multiple Fourier series (see Chapter I in the surveys [29,104] and the papers [17,28]); in particular, with characterization, representation, embeddings theorems, characterization of approximation spaces, m-term approximation, which are fast growing topics nowadays. Let us only mention a few basic older papers [42,43,44], the monograph [9] by Besov, Il'in and Nikol'skii, the monograph by Schmeisser and Triebel [71], the recent book by Triebel [93], and the 2006's survey [69] on this topic.…”

Section: What Is Not Included In This Surveymentioning

confidence: 99%

“…The Hardy-Littlewood-Paley theorem,[29]) Let the Fourier series of a function f ∈ L 0 1 (T 2 ) be given by (3.1). (A).…”

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confidence: 99%

Interrelations between mixed moduli of smoothness in metrics of L p and L ∞

Потапов

Симонов

2017

Moscow Univ. Math. Bull.

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In this paper we survey recent developments over the last 25 years on the mixed fractional moduli of smoothness of periodic functions from L p , 1 < p < ∞. In particular, the paper includes monotonicity properties, equivalence and realization results, sharp Jackson, Marchaud, and Ul'yanov inequalities, interrelations between the moduli of smoothness, the Fourier coefficients, and "angular" approximation. The sharpness of the results presented is discussed.MSC: 26A15, 26A33, 42A10, 41A25, 41A30, 42B05, 26B05

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“…Estimates of the Lebesgue constants play an important role in the summation of Fourier series, approximation and interpolation theory, and other branches of analysis. Different asymptotic formulas as well as upper and lower estimates of the Lebesgue constants on the d-dimensional torus T d have been known for years (see [9], [12], and [21,Ch. 9]).…”

Section: Introductionmentioning

confidence: 99%