Approximation of functions from L p (ω) β by matrix means of their Fourier series

We formulate some special conditions for the integrable functions and moduli of continuity. We give the results on rate of approximation of such functions by matrix means of their Fourier series, where the entries of the rows of the matrix generate the sequences belonging to the classes MRBVS and MHBVS, We also present some results on norm approximation for functions from the generalized integral Lipschitz classes.     

Fourier coefficients of functions of classC Β ψ H X ω

Sharp upper bounds are obtained from the suprema of the Fourier coefficients of functionsC H _C andC H _L of several variables defined by multipliers (·). translations in the arguments and moduli of continuity in the spaces C and L.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1537–1545, November, 1990.     

Approximation by Riesz sums of periodic functions of Hölder classes

We have found asymptotic equalities for the least upper bounds of the deviations of Riesz sums on the Hölder classes W^rH, r is a nonnegative integer, (t) is an arbitrary convex modulus of continuity.Translated from Matematicheskie Zametki, Vol. 21, No. 3, pp. 341–354, March, 1977.     

Approximating periodic functions in Hölder type metrics by Fourier sums and Riesz means

Let M be a fixed space of 2π-periodic functions, L_p or C, let ω_r(f, h) be the continuity modulus of order r of the function f in the space M, and let ϕ(t) be a function such that ϕ(t) > 0 for t > 0. By S_n(f) we denote the Fourier sums and by R_n,r(f) we denote the Riesz sums (the Fejér sums for r = 1) of the function f. Set

Fourier coefficients of functions from many-dimensional classes of bounded Λ-variation

Estimates of trigonometric Fourier coefficients are obtained for functions from the classes of bounded many-dimensional Λ-variation and functions continuous in Λ-variation. In some cases it is proved that these estimates are unimprovable by the order.     

Uniform approximation of Poisson integrals of functions from the class H ω by de la Vallée Poussin sums

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallée Poussin sums on the sets C _?? ^q H _?? of Poisson integrals of functions from the class H _?? generated by convex upwards moduli of continuity ??(t) which satisfy the condition ??(t)/t ?? ?? as t ?? 0. As an implication, a solution of the Kolmogorov-Nikol??skii problem for de la Vallée Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H ^??, 0 < ?? < 1, is obtained.     

Two-side estimates for sums of absolute values of Fourier coefficients of functions from Hω(Tm)

The class of functions H ^ω is considered, where ω(t) is a continuity modulus monotone in the sense of Hardy and satisfying some condition C. The behavior of the value is obtained, where is the sum of absolute values of Fourier coefficients of a function f ∈ L(T ^m ) in pth power. Original Russian Text ? D.M. D’yachenko, 2008, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2008, Vol. 63, No. 3, pp. 19–26.     

On some estimates for best approximations of bivariate functions by Fourier–Jacobi sums in the mean

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions, i.e., to approximate representations of functions (solutions) by partial sums of corresponding expansions. However, the errors of these approximations are rarely estimated or minimized in certain classes of functions. In this paper, the convergence rate (of best approximations) of a Fourier series in terms of Jacobi polynomials is estimated in classes of bivariate functions characterized by a generalized modulus of continuity. An approximation method based on “spherical” partial sums of series is substantiated, and the introduction of a corresponding class of functions is justified. A two-sided estimate of the Kolmogorov N-width for bivariate functions is given.     

Multiplicative convolutions of functions from Lorentz spaces and convergence of series of Fourier–Vilenkin coefficients

Let f and g be functions from different Lorentz spaces L ^{p, q} [0, 1), h be theirmultiplicative convolution and xxxx be Fourier coefficients of h with respect to a multiplicative system with bounded generating sequence. We estimate the remainder of the series of xxxx with multiplicators of type k ^b in terms of the best approximations of f and g in the corresponding Lorentz spaces. We establish sharpness of this result and of its corollaries for the Lebesgue spaces.     

Two-side estimates for sums of absolute values of Fourier coefficients of functions from Hω(Tm) with the use of partial moduli of smoothness

Two-side estimates of sums of absolute values of Fourier coefficients for functions of many variables from the classes H ^ω (T ^m ) are established in the paper with the help of partial moduli of smoothness.     

A method for summing Fourier integrals of functions from H p (E 2n + ), 0 < p < ∞

Suppose that H ^p (E _2n ⁺ ) is the Hardy space for the first octant $$E_{2n}^ + = \{ z \in \mathbb{C}^n :\operatorname{Im} z_j > 0, j = 1, \ldots ,n\} $$ and P _? ^l (f, x), l > 0, is the generalized Abel-Poisson means of a function f ? H ^p (E _2n ⁺ ). In this paper, we prove the inequalities $$C_1 (l,p)\widetilde\omega _l (\varepsilon ,f)_p \leqslant \left\| {f(x) - P_\varepsilon ^l (f,x)} \right\|_p \leqslant C_2 (l,p)\omega _l (\varepsilon ,f)_p ,$$ where $\widetilde\omega _l (\varepsilon ,f)_p $ and ω _l (?, f) _p are the integral moduli of continuity of lth order. For n = 1 and an integer l, this result was obtained by Soljanik.     

On the order of approximation to functions ofH p (R) (0<p≦1) by certain means of Fourier integrals

H ^P (R ₊ ² ) — R ₊ ² ={zC: Imz>0} ^p (R) — H ^p (R ₊ ² ). P ^k (f,x) — ë- — ,W ^k (f,x) — — R ^k, (f,x) — f H (R) (. §1,1)–3)); _k (, f) _p — - . , fH ^p (R) 0<p1,kN; (1+)^–1<p1, 0<<,kN.     

Multiple Fourier sums and ϕ-strong means of their deviations on the classes of $$\bar \psi $$ -differentiable functions of many variables

We present the results concerning the approximation of -differentiable functions of many variables by rectangular Fourier sums in uniform and integral metrics and establish estimates for the ϕ-strong means of their deviations in terms of the best approximations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1075–1093, August, 2007.     

Traces of functions from H∞ (\mathbb{B}^n ) on certain sets of hyperplanes

One considers necessary and sufficient conditions in order that a collection of sections of a ball in ⁿ by hyperplanes be the set of zeros or the interpolation set for functions of class H in the ball.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 183–187, 1985.