Multiple fourier integral and multiple fourier series under square summation

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 1, pp. 39–52, January–February, 1990.     

Accuracy of some theorems on fourier series and fourier transforms

We consider necessary and sufficient condition of differentiability of the Fourier transform in the class of functions summable with a given weight. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 121–127, 1987.     

Multiple fourier series and integrals

One gives a brief survey of the investigations on the theory of multiple Fourier series and integrals, reviewed in Referativnyi Zhurnal Matematika in the period 1953–1980. Principal attention is given to the following questions: localization principles, uniform convergence and summability, convergence and summability at a point, in the L_p metric, and almost everywhere, absolute convergence, uniqueness theorems, conjugate Fourier series and integrals, equiconvergence and equisummability of Fourier series and integrals, properties of the kernel and of the Lebesgue constant of summation methods of Fourier series, Fourier coefficients and Fourier transforms.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 19, pp. 3–54, 1982.     

Winding numbers and fourier series

This is an expository paper on a topic of classical analysis arising from the BMO-theory of topological degree (Brezis-Nirenberg, 1995). We sketch the history of the subject and some of its recent developments.     

On some Diophantine fourier series

We continue our study on arithmetical Fourier series by considering two Fourier series which are related to Diophantine analysis. The first one was studied by Hardy and Littlewood in connection with the classification of numbers and the second one was studied by Hartman and Wintner by Lebesgue integration theory.     

Divergent fourier series: Numerical experiments

An open problem in the theory of Fourier series is whether there are functions f L ₁ such that the partial sums S _n(f, x) diverge faster than log log n, almost everywhere in x. For a class of particularly bad functions Kahane proved that the rate of divergence is faster than o(log log n). We give here a probabilistic interpretation of the Kahane result, which shows that the record values of the sums S _n(f, x) should behave essentially as the record values of a sequence of independent identically distributed random variables, for which we deduce the divergence rate log log n. Numerical computation is in good agreement with the prediction. One can argue that the Kahane examples are in some sense optimal, and conclude that, under this assumption, ...(log log n) is the highest possible rate for divergence almost everywhere of the Fourier partial sums for L ₁ functions.     

Geometry of the finite fourier transform

We provide a geometric interpretation for the multidimensional finite Fourier transform and give a reformulation of the Macwilliams identitity in terms of theta functions.     

Generalized fourier transform and its applications

We consider the generalized Fourier transform treated as an operator on the dual of an arbitrary locally convex space. We give a definition of this operator and establish its basic properties. Special attention is paid to cases in which the range of the generalized Fourier transform coincides with a weighted space of entire functions. The results are applied to finding the orders and types of operators in various spaces.     

Generalized fourier transform of slow-growth distributions

Krasnoyarsk City. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 2, pp. 94–103, March–April, 1990.