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The author finds a sharp lower bound for the maximal measure of sets on which a Fourier series expansion in a uniformly bounded complete orthonormal system diverges.

Let be a natural number, . Then we shall say that a function of period in each variable is monotonic if there exist an open rectangular parallelepiped and numbers , each of which is either 0 or 1 , such that for , and if , and for , then .The main result of this paper is that the multiple trigonometric Fourier series of an integrable monotonic function is Pringsheim convergent almost everywhere, in particular at each point of continuity of in the interior of .

For the operator , , , defined on Fourier transforms of Schwartz functions by the relation the question of extension to a bounded linear operator is considered, where and are Lebesgue spaces of Bessel potentials, , and . Sharp conditions are obtained under which such an extension is possible. An explicit representation of is given for and , , , in the form of a difference hypersingular integral converging in the -norm and almost everywhere. For the operator generated by the Fourier multiplier an assertion is obtained regarding the convergence of , , as in the -norm and almost everywhere which generalizes a familiar result of Stein corresponding to the case . The results are applied to the investigation of the Cauchy problem for the wave equation in the scale of spaces .Figures: 4. Bibliography: 43 titles.

It is proved that if is a complete orthonormal system of bounded functions and , then there exists a measurable set with such that1) for any function there exists a function with on and such that the Fourier series of in the system converges in the -metric; and2) there exists a subsequence of natural numbers such that for any function there exists a function such that for , almost everywhere on and for all , where , .

It is shown that every system of collectively bounded orthonormal functions admits an integrable function whose Fourier series diverges on a set of positive measure.Bibliography: 6 titles.

This article gives the solution of a conjecture of V. Totik on strong summation of trigonometric Fourier series.Bibliography: 5 titles.

It is proved that there exists a continuous function defined on [0,1]2 whose double Fourier-Walsh-Paley series diverges almost everywhere in the sense of Pringsheim.Bibliography: 9 titles.

A set E with 0 < meas E < +∞ is constructed for which the Fourier transform of its characteristic function vanishes on an interval. The set is the union of a sequence of intervals whose lengths can be estimated asymptotically above and below. The construction is based on an infinite-dimensional version of the implicit function theorem. Bibiography: 6 titles.

For -periodic functions that have, on , only the point 0 as a nonsummable singular point, we consider generalized Fourier series depending on an integer-valued function . It is shown that if , where is an even nonnegative function, nonincreasing on , and , , then under a certain condition on the generalized Fourier series is almost everywhere summable to by the Abel method. The estimate and the hypothesis on are, in a certain sense, definitive.Bibliography: 3 titles.

The precise geometry is found of measurable sets in -dimensional Euclidean space on which generalized localization almost everywhere holds for multiple Fourier series which are rectangularly summable.Bibliography: 14 titles.

Let be a measurable function on . Define an operator , where and is the Fourier transform. Let be its domain. The operator is called local if implies for with . An entire function of order zero is constructed for which the operator is not local. Let be the Wiener algebra of absolutely convergent trigonometric series. We prove a theorem on correction in the spirit of Luzin's theorem: a condition is exhibited on a set of integers under which each function of can be corrected on a set of arbitrarily small measure so that the spectrum of the corrected function (also in ) is contained in .Bibliography: 7 titles.

The author gives a descriptive definition of the -integral. The -integral extends the Lebesgue integral and coincides with it for nonnegative functions. For a function , -integrable on , the -Fourier series is defined and is almost everywhere summable to ; the conjugate series is summable to , which is also -integrable on , and is its -Fourier series. Bibliography: 12 titles.

Let be a continuous -periodic function, the th partial sum of its Fourier series, the modulus of continuity and the modulus of variation of . In this paper the following theorems are proved.Theorem 1. For the estimate holds, where is an absolute constant.From this theorem there follows an estimate of Lebesgue and an estimate of Oskolkov.Theorem 2. In order that all Fourier series of class converge uniformly it is necessary and sufficient that Bibliography: 20 titles.

Both necessary conditions and sufficient conditions on a sequence are found for the family of translates of an -function whose Fourier transform is almost everywhere nonzero and rapidly decreasing to be dense in .Bibliography: 9 titles.

On the basis of the stationary phase method for oscillatory integrals with complex phase function, the authors prove the coincidence of Fourier integral operators and Maslov's canonical operator. Bibliography: 17 titles.

An -dimensional analog is proved of a theorem of Plessner and Ul'janov on equivalent conditions for convergence of certain series and integrals. There is obtained from it a sufficient condition on the quadratic modulus of continuity of a periodic function of variables ensuring the a.e. convergence of the spherical sums of its Fourier series. A two-dimensional analog of a theorem of Luzin and Denjoy and an -dimensional analog of the Dini-Lipschitz criterion are proved. A necessary and sufficient condition on a function is derived ensuring the pointwise convergence of the Riesz spherical means of critical order of multiple Fourier series of functions of bounded -variation.Bibliography: 33 items.

The paper is devoted to the absolute convergence of Fourier series of absolutely continuous functions in arbitrary bounded complete orthonormal systems of functions. The results obtained are new even for the classical systems, namely the trigonometric system and the Walsh system. Bibliography: 8 items.

The article is devoted to the study of analytic representations of a class of analytic functional which is closely related to the Fourier transformation and includes the Fourier hyperfunctions. Bibliography: 19 titles.

In this work there is constructed a complete orthonormal system with respect to which the Fourier series of piecewise continuous non-identically-zero functions diverge a.e. There is also constructed an orthonormal system with respect to which the Fourier series of continuous functions converge a.e., while the Fourier series of bounded functions having at least one discontinuity of the first kind diverge a. e. Bibliography: 10 items.

Necessary and sufficient conditions are given for a sequence of numbers to belong to the set of sequences of Fourier coefficients with respect to the Haar system of functions in the space ().Bibliography: 12 titles.

The paper contains inequalities for the absolute value of the Fourier coefficients of functions almost periodic in the sense of Stepanov (S-a.p. functions) having sparse spectrum, in a sense which we define. In the particular case in which the spectrum has a single limit point at infinity, we obtain generalizations of Theorem 1 of Chao Jai-arng, Proc. Japan Acad. 42 (1966), 308-312 and Theorem 1 of Hsieh Ting-fan, Acta Math. Sinica 16 (1966), 513-516, proved for 2π-periodic functions. The case in which the spectrum has a single limit point is considered. The results are then extended to the case of S-a.p. functions whose spectrum has a finite or countable number of isolated limit points. It is indicated how the results may be used to give sufficient conditions for absolute convergence for the Fourier series of S-a.p. functions. Bibliography: 14 items.