Periodic Functions Of Several Variables Research Articles

Approximation of classes of periodic functions of several variables

We study classes of periodic functions of several variables with bounded generalized derivative in the metric of the space Lp. We obtain order estimates of deviations of Fourier sums, which are constructed depending on the behavior of functions that define the operator of generalized differentiation. We find estimates of the Kolmogorov widths, which are realized by the Fourier sums that are constructed.

On a class of holomorphic almost periodic functions of several variables

On a class of holomorphic almost periodic functions of several variables

ORDER ESTIMATES OF SMALLEST NORMS, WITH RESPECT TO THE CHOICE OF $ N$ HARMONICS, OF DERIVATIVES OF THE DIRICHLET AND FAVARD KERNELS

The Dirichlet kernel is defined for periodic functions of several variables; it consists of harmonics and has minimal order of the norm with respect to the choice of harmonics of the mixed Weyl derivative in the space . A similar problem on the minimal order of the norm is solved for the Favard kernel. Both problems generalize to the case of several derivatives.

Approximation of the Besov classes of periodic functions of several variables in a space Lq

Order estimates are obtained for best approximations by polynomials constructed according to hyperbolic crosses on the classes B p r , θ of periodic functions of several variables. The order is found of the Kolmogorov width on these classes in the spaces Lq for 1 <p ⩽q ⩽2.

KOLMOGOROV WIDTHS OF CLASSES OF PERIODIC FUNCTIONS OF ONE AND SEVERAL VARIABLES

The order of Kolmogorov widths are determined for the class that is the intersection of classes of periodic functions of one variable of higher smoothness, in the space for , and estimates from above for low smoothness, and also the order of Kolmogorov widths is calculated for periodic functions of several variables in the space for . The estimate from below for reduces to the estimate from below of the width of a finite-dimensional set whose width is determined. Bibliography: 28 titles.

Approximation of classes of periodic functions of several variables by nuclear operators

Approximation of classes of periodic functions of several variables by nuclear operators

Sets of convergence to zero of lacunary trigonometric series

Nauka, Moscow (1969). 6. L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York (1974). 7. Hua Loo Keng and Wang Yuan, Applications of Number Theory to Numerical Analysis, Springer-Verlag, Berlin (1981). 8. N. S. Bakhvalov, "On approximate computation of multiple integrals," Vestn. Mosk. Gos. Univ., Ser. Mat., Fiz., Astron., Khim., No. 4, 3-18 (1959). 9. I. F. Sharygin, "Lower bounds for the errors of quadrature formulas on function classes," Zh. Vychisl. Mat. Mat. Fiz., ~, No. 2, 370-376 (1963). i0. N. S. Bakhvalov, "Lower estimates for asymptotic characteristics of classes of functions with a dominating mixed derivative," Mat. Zametki, 12, No. 6, 655-664 (1972). ii. N. S. Bakhvalov, "On the optimal bounds of convergence of quadrature processes and of integration methods of Monte Carlo type on function classes," Zh. Vychisl. Mat. Mat. Fiz., ~, No. 4, Supplement, 5-63 (1964). 12. V. A. Bykovskii, "On the correct order of the error of optimal cubature formulas in spaces with dominating derivative and on square deviations of nets," Preprint, Akad. Nauk SSSR, Far Eastern Science Center (1985), pp. 3-31. 13. K. K. Frolov, "Upper bounds for the errors of quadrature formulas on function classes," Dokl. Akad. Nauk SSSR, 231, No. 4, 818-821 (1976). 14. V. N. Temlyakov, "On the reconstruction of periodic functions of several variables from the values at the nodes of number-theoretic nets," Analysis Mathematica, Akad. Kiado, Budapest, 12, No. 4, 287-305 (1986). 15. V. N. Temlyakov, "On the approximation by elements of a finite-dimensional subspace of functions from various classes of Sobolev or Nikol'skii," Mat. Zametki, 43, No. 6, 770-785 (1988).

APPROXIMATION BY TRIGONOMETRIC POLYNOMIALS OF FUNCTIONS OF SEVERAL VARIABLES ON THE TORUS

The paper is devoted to the approximation of classes of periodic functions of several variables whose derivative is given with the aid of the absolute value of mixed moduli of continuity. The author studies best approximations by Fourier sums and by spaces of trigonometric polynomials, the Kolmogorov widths of these classes and other related questions. In the study of these questions, the problem arises in a natural way of estimating integrals and sums over convex sets depending on a parameter or over their complements. Asymptotic orders are computed for such integrals and sums connected with the corresponding questions of approximation.Bibliography: 46 titles.

Best linear methods for approximation of classes of periodic functions of several variables

Best linear methods for approximation of classes of periodic functions of several variables

Approximation of classes of smooth functions of several variables

The paper deals with imbedding theorems and harmonic approximation for classes of periodic functions of several variables. Necessary and sufficient conditions for imbedding are found for various classes of smooth functions. The paper contains asymptotic estimates for fractional order derivatives of multidimensional Dirichlet and Favard kernels, for Fourier best approximation and for the Kolmogorov diameter of smooth function classes.

KOLMOGOROV WIDTHS IN THE SPACE ${\tilde L}_q$ OF THE CLASSES ${\tilde W}_p^{\overline \alpha}$ AND ${\tilde H}_p^{\overline \alpha}$ OF PERIODIC FUNCTIONS OF SEVERAL VARIABLES

The author finds the order of the Kolmogorov widths for all , where is the class of periodic functions of several variables determined by a Weyl mixed fractional derivative, and for or , where is the class determined by a mixed difference. Bibliography: 28 titles.

APPROXIMATION OF FUNCTIONS WITH A BOUNDED MIXED DIFFERENCE BY TRIGONOMETRIC POLYNOMIALS, AND THE WIDTHS OF SOME CLASSES OF FUNCTIONS

This paper investigates the approximation of periodic functions of several variables by trigonometric polynomials whose harmonics lie in hyperbolic crosses. It is shown that in many cases the order of the widths, in the sense of Kolmogorov, can be found for classes of functions with a bounded mixed derivative or difference. The possibilities of linear methods of approximation are investigated. Bibliography: 16 titles.

Parameters for integrating periodic functions of several variables

A number-theoretical method for numerical integration of periodic functions of several variables was developed some years ago. This paper presents lists of numerical parameters to be used in implementing that method. The parameters define quadrature formulas for functions of 2, 3, ..., 8 variables; error bounds for those formulas are also tabulated. The derivation of the parameters and error bounds is described.

Estimates of errors in interpolating trigonometric polynomials with equidistant nodes on classes of continuous periodic functions of several variables

Estimates of errors in interpolating trigonometric polynomials with equidistant nodes on classes of continuous periodic functions of several variables

Asymptotically exact estimates of the errors of partial fourier sums on classes of continuous periodic functions of several variables

Asymptotically exact estimates of the errors of partial fourier sums on classes of continuous periodic functions of several variables

Approximation by Fourier sums of classes of periodic functions of several variables defined by polyharmonic operators

Approximation by Fourier sums of classes of periodic functions of several variables defined by polyharmonic operators

Approximation by Fourier sums of classes of functions with several bounded derivatives

An ordered estimate is obtained for the approximation by Fourier sums, in the metric ofd=(d1, ...,dn), 1<dj<∞,j=1, ...,n of classes of periodic functions of several variables with zero means with respect to all their arguments, having m mixed derivatives of order a1..., am., aie rn. which are bounded in the metrics ofpi=p1i, ..., pni, i<Pji<∞,i=i, ...,n, j=1, ...,n by the constants Β1, η., Βm, respectively.

Constructive characteristics of certain classes of periodic functions of several variables

Constructive characteristics of certain classes of periodic functions of several variables

A reduction formula for the numerical integration of periodic functions of several variables

A reduction formula for the numerical integration of periodic functions of several variables