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UDC 517.5 In this survey paper, exact rate of approximation of functions by linear means of Fourier series and Fourier integrals and corresponding $K$-functionals are expressed via special moduli of smoothness.

We establish an asymptotic formula for the rate of approximation of Fourier series of individual periodic functions by linear averages with an error , . This formula is applicable to the means of Riesz, Gauss–Weierstrass, Picard and others. The result is new even for the arithmetic means of partial Fourier sums. We use the formula to determine the asymptotic behaviour of functions in a certain class. Separately, we consider the case of positive integral convolution operators.

We present a brief survey of works in the approximation theory of functions known to the author and connected with V. K. Dzyadyk’s research works.

Let \(L_{0,1}(\mathbb {R}_+)\) be the Banach space endowed with the norm $$\displaystyle \|f\|{ }_{0,1}=\int _0^\infty f^*,\qquad f^*(x)=\underset {t\geq x}{\text{ess sup}}~|f(t)|, $$ while \(L_{0,\infty }(\mathbb {R}_+)\) with the norm \(\displaystyle \sup \limits _{x>0}\frac {1}{x}\int _0^x|f|\).

Generalized Chebyshev polynomials are introduced and studied in this paper. They are applied to obtain a lower bound for the sup-norm on the closed interval for nonzero polynomials with integer coefficients of arbitrary degree.

In this paper, the following problem is studied. For what multipliers {λ k,n } do the linear means of the Fourier series of functions f ∈ L 1[−π, π], $$\begin{array}{*{20}c} {\sum\limits_{k = - \infty }^\infty {\lambda _{k,n} \widehat{f_k }e^{ikx} ,} } & {where \widehat{f_k } is the kth Fourier coefficient, } \\ \end{array} $$ , converge as n→∞ at all points at which the derivative of the function ∫ 0 f exists? In the case λ k,n = (1 − |k|/(n + 1)), a criterion of the convergence of the (C, 1)-means and, in the general case λ k,n = ϕ(k/(n + 1)), a sufficient condition of the convergence at all such points (i.e., almost everywhere) are obtained. In the general case, the answer is given in terms of whether ϕ(x) and xϕ′(x) belong to the Wiener algebra of absolutely convergent Fourier integrals. New examples are given.

We establish a general sufficient condition for the possibility of representation of functions $$ f\left( \max \left\{\left|{x}_1\right|,\left|{x}_2\right|\right\}\right) $$ in the form of absolutely convergent double Fourier integrals and study the possibility of its application to various problems of summability of double Fourier series, in particular, by using the Marcinkiewicz–Riesz method.

We study the convergence of linear means of the Fourier series of a function to as at all points at which the derivative exists (i.e. at the -points). Sufficient conditions for the convergence are stated in terms of the factors and, in the case of , in terms of the condition that the functions and belong to the Wiener algebra . We also study a new problem concerning the convergence of means of the Abel-Poisson type, , as depending on the growth of the function on the semi-axis. It turns out that cannot differ substantially from a power-law function.

The paper concerns the approximation properties of the Bernstein-Stechkin summability method for trigonometric Fourier series. The Jackson-Stechkin theorem is refined. Moreover, for any continuous periodic function not only is the exact upper estimate for approximation found, a lower estimate of the same order is also put forward. To do this special moduli of smoothness and the K-functional are introduced. Bibliography: 16 titles.

By using an exact estimate for approximation by known trigonometric polynomials, we strengthen a Jackson-type theorem. Moreover, we determine the exact order of approximation of some periodic functions by these polynomials. For this purpose, we introduce a special modulus of smoothness.

We obtain sufficient conditions for the representability of a function in the form of an absolutely convergent Fourier integral. These conditions are given in terms of the joint behavior of the function and its derivatives at infinity, and their efficiency and exactness are verified with the use of a known example. We also consider radial functions of an arbitrary number of variables.

We obtain new sufficient conditions for the representability of a function by an absolutely convergent Fourier integral in ℝ d . These conditions are given in terms of the simultaneous behavior of a function and its derivatives at ∞. We test the sharpness of the conditions using well-known examples.

The paper studies the approximation order of periodic functions by trigonometric polynomials with interpolation in arbitrary set of nodes. A method of construction of Hermite interpolation polynomials is pointed out.

We prove a general direct theorem on the simultaneous pointwise approximation of smooth periodic functions and their derivatives by trigonometric polynomials and their derivatives with Hermitian interpolation. We study the order of approximation by polynomials whose graphs lie above or below the graph of the function on certain intervals. We prove several inequalities for Hermitian interpolation with absolute constants (for any system of nodes). For the first time we get a theorem on the best-order approximation of functions by polynomials with interpolation at a given system of nodes. We also provide a construction of Hermitian interpolating trigonometric polynomials for periodic functions (in the case of one node, these are trigonometric Taylor polynomials).

The pointwise approximations of functions with a given modulus of continuity of their r-th derivatives on a closed interval of the real axis are done with various additional constraints. The paper attempts to review such constraints.

In this paper, exact rate of approximation of functions by linear means of Fourier series and Fourier integrals and corresponding $K$-functionals are expressed via special moduli of smoothness. . Introduction is given in $\S 1$. In $\S2$ functions on the line $\mathbb{R}$ are studied. A typical (well-known) result is as follows: for each $2\pi$-periodic function in $L_p$ on the period, for any $p\in[1,+\infty]$ ($L_\infty=C$) and $r\in\mathbb{N}$, there is a trigonometric polynomial $\tau_{r,n}(f)$ of degree not greater than $n$ such that \big\|f-\tau_{r,n}(f)\big\|_p\asymp\omega_r\Big(f;\frac{1}{n}\Big)_p\asymp \inf\limits_{g}\Big\{\|f-g\|_p+\frac{1}{n^r}\big\|g^{(r)}\big\|_p\Big\}, where the positive constants in these bilateral inequalities depend only on $r$. In $\S 3$ we deal with functions on $\mathbb{R}^d$ ($d\geq2$), while in $\S 4$ with functions on Banach spaces. The paper is partially of survey nature. The proofs are given only for Theorems 2.2, 3.9 and those in $\S 4$. Related open problems are formulated in $\S 5$. The list of references contains 52 items., Comment: 33 pages, the paper is in Russian, with the title, abstract and key words in English

In this paper, we study the question of the existence of the inequality $$\left\| {Q(D)f} \right\|_{L_q } \leqslant \gamma _0 \left\| {P(D)f} \right\|_{L_p } $$ , where P and Q are algebraic polynomials, D = d/dx, and γ 0 is independent of the function f. We obtain criteria (necessary and simultaneously sufficient conditions) for the existence of such inequalities for functions on the circle, on the whole line, and on the semiaxis. Besides, for the semiaxis, we obtain an inequality for q = ∞ and any p ≥ 1 with the smallest constant γ 0.

We determine the order of approximation of real functions of finite smoothness by polynomials with various restrictions: approximation of functions and their derivatives with Hermitian interpolation at the endpoints, one-sided approximation (for example, from above), and assumptions on the coefficients.

Even positive-definite splines with support in [–1,1] that are equal to real algebraic polynomials on [0,1] are investigated. Examples of such splines are presented. Under consideration are the -splines, which have several extremal properties, and the positive-definite -splines, which have the maximum possible smoothness on . An estimate of the approximation by a linear combination of shifts of an -spline is indicated. New relations for the hypergeometric function are found.

Direct theorems on the approximation of smooth functions by polynomials with integer coefficients are strengthened. First theorems on approximation by polynomials with natural coefficients are obtained.

For conditions on a?number sequence are indicated ensuring that the multiplier operator is continuous in the Hardy space (here can also be a?polydisc ). Some sufficient conditions are also established. These results are used to find out the precise order of approximation of multiple power series by Bochner-Riesz means and to evaluate the -functional for a?pair of spaces related to the polyharmonic operator.

New statements are proved regarding multipliers of trigonometric Fourier series in the space C of continuous periodic functions.

In this paper the connections among three algebras are discussed: the algebra of Fourier transforms of finite Borel measures on Rm, the algebra A of absolutely convergent Fourier integrals, and the algebra of functions which generate a bounded multiplier sequence. Necessary and sufficient conditions for membership in A are given, a Bernstein-Rogosinski type of summation method for multiple Fourier series is investigated, and a comparison principle is formulated for various methods of summation of Fourier series according to their approximation properties. In addition, in connection with the well-known theorem of Jackson and its converse, various moduli of smoothness are introduced and studied. Bibliography: 33 titles.

In this paper various transformations of trigonometrical polynomials are introduced; these are then used to deduce asymptotic formulas for the behavior of sequences of integrals of the moduli of the polynomials. As a consequence a definitive form is found for a relationship (in both directions) between the summability and absolute convergence of trigonometric Fourier series which has been noted earlier by the author.Bibliography: 26 titles.

We prove the following propositions. An even integrable function whose Fourier coefficients form a convex sequence is absolutely continuous if and only if its Fourier series converges absolutely. If the function f(t)is convex on [0, π],f(t)=f(π—t), then for odd n while for even n, b0=0.

We establish a connection between the summability and the absolute convergence of Fourier series and apply it in both directions.