Multiple Fourier Series Of Functions Research Articles

The Convergence Problems of Eigenfunction Expansions of Elliptic Differential Operators

In the present research we investigate the problems concerning the almost everywhere convergence of multiple Fourier series summed over the elliptic levels in the classes of Liouville. The sufficient conditions for the almost everywhere convergence problems, which are most difficult problems in Harmonic analysis, are obtained. The methods of approximation by multiple Fourier series summed over elliptic curves are applied to obtain suitable estimations for the maximal operator of the spectral decompositions. Obtaining of such estimations involves very complicated calculations which depends on the functional structure of the classes of functions. The main idea on the proving the almost everywhere convergence of the eigenfunction expansions in the interpolation spaces is estimation of the maximal operator of the partial sums in the boundary classes and application of the interpolation Theorem of the family of linear operators. In the present work the maximal operator of the elliptic partial sums are estimated in the interpolation classes of Liouville and the almost everywhere convergence of the multiple Fourier series by elliptic summation methods are established. The considering multiple Fourier series as an eigenfunction expansions of the differential operators helps to translate the functional properties (for example smoothness) of the Liouville classes into Fourier coefficients of the functions which being expanded into such expansions. The sufficient conditions for convergence of the multiple Fourier series of functions from Liouville classes are obtained in terms of the smoothness and dimensions. Such results are highly effective in solving the boundary problems with periodic boundary conditions occurring in the spectral theory of differential operators. The investigations of multiple Fourier series in modern methods of harmonic analysis incorporates the wide use of methods from functional analysis, mathematical physics, modern operator theory and spectral decomposition. New method for the best approximation of the square-integrable function by multiple Fourier series summed over the elliptic levels are established. Using the best approximation, the Lebesgue constant corresponding to the elliptic partial sums is estimated. The latter is applied to obtain an estimation for the maximal operator in the classes of Liouville.

Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation

The paper introduces a new concept of Λ-variation of multivariable functions and studies its relationship with the convergence of multidimensional Fourier series.

Summability of multiple Fourier series of functions of bounded generalized variation

The paper is devoted to summability problems for multiple Fourier series of functions of bounded generalized variation. We find necessary and sufficient conditions on a sequence Λ = {λn} that guarantee the convergence of the Cesaro means of multiple Fourier series of functions of bounded Λ-variation.

On the convergence of multiple Fourier series of functions of bounded partial generalized variation

The convergence of multiple Fourier series of functions of bounded partial Λ-variation is investigated. The sufficient and necessary conditions on the sequence Λ = {λn} are found for the convergence of multiple Fourier series of functions of bounded partial Λ-variation.

Localization for multiple Fourier series of functions of bounded harmonic variation

An example of a function of four variables belonging to the class of bounded harmonic variation (in a weak sense) is constructed. Cubic sums of a trigonometric Fourier series for this function does not possess the localization property.

Gibbs’ phenomenon and surface area

If a functionffis of bounded variation onTN (N≥1)T^N\ (N\geq 1)and{φn}\{{\varphi }_n\}is a positive approximate identity, we prove that the area of the graph off∗φnf*{\varphi }_nconverges from below to the relaxed area of the graph offf. Moreover we give asymptotic estimates for the area of the graph of the square partial sums of multiple Fourier series of functions with suitable discontinuities.

Majorant estimates for partial sums of multiple fourier series of functions from Orlicz spaces vanishing on some set

In this paper majorants supn |Sn (x; ϕ)| of partial sums of double trigonometric Fourier series of functions fromL(log+L)3 vanishing on some subsets ofT2=[−π, π)2 are considered. In particular, the following theorem is proved.

ON CONVERGENCE OF RIESZ SPHERICAL MEANS OF MULTIPLE FOURIER SERIES

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.

Localization in bounded summation by Cesaro, Riesz, and Abel methods for multiple Fourier series

We establish the conditions under which the localization principle is valid in bounded summation by Cesaro, Riesz, and Abel methods for multiple Fourier series of functions of Liouville classes.