Sums Of Trigonometric Series Research Articles

Uniform Approximation of the Curvature of Smooth Plane Curves with the Use of Partial Fourier Sums

An error bound for the approximation of the curvature of graphs of periodic functions from the class Wr for r ≥ 3 in the uniform metric is obtained with the use of the simplest approximation technique for smooth periodic functions, which is approximation by partial sums of their trigonometric Fourier series. From the mathematical point of view, the interest in this problem is connected with the specific nonlinearity of the graph curvature operator on the class of smooth functions Wr on a period or a closed interval for r ≥ 2. There are several papers on curvature approximation for plane curves in the mean-square and Chebyshev norms. In previous works, the approximation was performed by partial sums of trigonometric series (in the L2 norm), interpolation splines with uniform knots, Fejer means of partial sums of trigonometric series, and orthogonal interpolating wavelets based on Meyer wavelets (in the C∞ norm). The technique of this paper, based on the lemma, can possibly be generalized to the Lp metric and other approximation methods.

A-integrability of sums of trigonometric series

We investigate sine and cosine series such that their coefficients tend to zero and some subsequences of the coefficients have bounded variation.

Polynomial Solutions of the Boundary Value Problems for the Poisson Equation in a Layer

It is well known that the Dirichlet problem for the Laplace equation in a ball has a unique polynomial solution (harmonic polynomial) in the case if the given boundary value is the trace of an arbitrary polynomial on the sphere. S.M.Nikol'skii generalized this result in the case of a boundary value problem of the first kind for a linear differential self-adjoint operator of the order 2l with constant coefficients (in particular polyharmonic) and for a domain that is an ellipsoid in R n . For a polyharmonic equation in a ball (homogeneous and inhomogeneous), V.V. Karachik proposed the Almansi formula-based algorithm to construct a polynomial solution of the Dirichlet problem. The paper considers the Poisson equation with the polynomial right-hand side in a multidimensional infinite layer bounded by two hyper-planes. Shows that the Dirichlet boundary value problem and the mixed Dirichlet-Neumann boundary value problem with polynomial boundary conditions have a unique solution in the class of functions of polynomial growth, and this solution is a polynomial. Gives an algorithm for constructing this polynomial solution and considers examples. In particular, presents formulas to give exact values of certain integrals (including multi-dimensional ones) and sums of trigonometric series.

A two-dimensional extension of Zygmund’s theorem on the smoothness of the sum of trigonometric series

A two-dimensional extension of Zygmund’s theorem on the smoothness of the sum of trigonometric series

Generalization of Zygmund’s theorem on the smoothness of the sum of trigonometric series

Generalization of Zygmund’s theorem on the smoothness of the sum of trigonometric series

On the first derivative of the sums of trigonometric series with quasi-convex coefficients of higher order

In this paper, for the sum of sine or cosine series with quasiconvex coefficients of higher order, the representation of their first derivatives are found in terms of the r-th differences of coefficients of the series obtained by formal differentiation. Also some estimates in terms of coefficients of the series are obtained for the integrals of the absolute values of those derivatives.

Interaction of a rigid punch with a base-secured elastic rectangle with stress-free sides

The plane contact problem of the indentation of a rigid punch into a base-sucured elastic rectangle with stress-free sides is considered. The problem is solved by a method tested earlier and reduces to a system of two integral equations in functions describing the displacement of the surface of the rectangle outside the punch and the normal or shear stress on its base. These functions are sought in the form of the sum of trigonometric series and an exponential function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result of this are regularized by introducing small positive parameters. Because the matrix elements of the systems, and also the contact stresses, are defined by poorly converging numerical and functional series, the previously developed method of summation of these series is used. The contact pressure distribution and the dimensionless indenting force are found. Examples of a plane punch calculation are given.

Smoothness of sums of trigonometric series with monotone coefficients

Smoothness of sums of trigonometric series with monotone coefficients

Remarks on Mean Convergence (Boundedness) of Partial Sums of Trigonometric Series

Fairly general conditions on the coefficients \(\left\{ {a_n } \right\}_{n = 1}^\infty \) of even and odd trigonometric Fourier series under which L-convergence (boundedness) of partial sums of the series is equivalent to the relation \(\sum\nolimits_{k = \left[ {{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \right]}^{2n} {{{\left| {a_k } \right|} \mathord{\left/ {\vphantom {{\left| {a_k } \right|} {\left( {\left| {n - k} \right| + 1} \right) = o\left( 1 \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\left| {n - k} \right| + 1} \right) = o\left( 1 \right)}}} \left( { = O\left( 1 \right),{\text{ respectively}}} \right)\) are given.

Modulation of the Solar Magnetic Cycle

The modulation model of the solar magnetic cycle for the time interval from 1650 to 1996 A.D. describes an harmonic oscillator with a basic (22.13 ± 0.05)-yr period, which is subjected to amplitude and phase variations that can be represented by a sum of trigonometric series. The simulated sunspot data explain 97.9% of cycle peak height variance and the residual standard deviation is 8.6 mean annual sunspots. A peak height of 139 for cycle 23 occurring in 2001 is predicted, whereas cycle 24 would have a maximum around 132 in 2014. Simulation of the sunspot numbers from 1000 until 2400 A.D. shows that the model recreates recurring minima (Maunder and Sporer Minimum). The prediction also expects a high level of amplitude modulation in the interval 1950–2010 with a rapid decrease afterwards. A greatly reduced cycle activity is reproduced by the simulation from about 2065 to 2100 A.D. No direct explanation of the long-term periodicities of the model can be advanced. The high-frequency contribution of the phase modulation, which accounts for the skewness of the solar cycle, shows coincidences with the orbital periods of Jupiter and Saturn, but no physical basis for the matching periodicities can be conceived.

On summability of subsequences of partial sums of trigonometric Fourier series

В работе в терминах ча стных и смешанных мод улей непрерывности дана о ценка отклонения кратных арифметичес ких средних последов ательности диадических прямоуг ольных частичных суммn-крат ного (n≧1) тригонометри ческого ряда Фурье функцииf∈Lp([−π, π]n),р=1 илир=∞ (L∞≡C) от этой функции по норме соответствующего пр остранства. Доказана неусиляемо сть, в некотором смысл е, полученного результ ата дляf∈С([−π, π]n).

Estimates of sums of trigonometric series and polynomials useful in connection with problems in the theory of approximation of functions

Estimates of sums of trigonometric series and polynomials useful in connection with problems in the theory of approximation of functions