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General Rogosinsky–Bernstein linear polynomial means of Fourier series are introduced and three convergence criteria as are obtained: for convergence in the space of continuous periodic functions and for convergence almost everywhere with two guaranteed sets (Lebesgue points and -points). For smooth functions, the rate of convergence in norm of , as well as of their interpolation analogues, is also studied. For approximation of functions in , the asymptotics is found along with the rate of decrease of the remainder term. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the approximation properties of Fourier sums in terms of the modified Meixner polynomials mn,Nα(x), n = 0,1,..., which generate, for α > -1, an orthonormal system on the grid Ωδ = {0, δ, 2δ,...} with weight N (x) = e x (N x + + 1) (N x + 1) (1 e ) + 1 , where = 1 N , N 1. The main attention is paid to the derivation of a pointwise estimate for the Lebesgue function λn,Nα(x) of Fourier sums in terms of the modified Meixner polynomials for x ∈ [θn/2, ∞) and θn = 4n + 2α + 2. [ABSTRACT FROM AUTHOR]

We consider the problem of the mean-square approximation of complex functions regular in a domain D ⊂ C by Fourier serieswith respect to an orthogonal (inD) systemof functions {ϕk(z)}, k = 0, 1, 2,.... For the case inwhich D = {z ∈ C: |z| < 1}, we obtain sharp estimates for the rate of convergence of the Fourier series in the orthogonal system {zk}, k = 0, 1, 2,..., for classes of functions defined by a special mth-order modulus of continuity. Exact values of the series of n-widths for these classes of functions are calculated. [ABSTRACT FROM AUTHOR]

We present a necessary and sufficient condition for the series of absolute values of blocks of Fourier series elements and blocks of series of summands in Parseval’s identity to converge in the class of two-variable functions of bounded variation in the sense of Hardy. [ABSTRACT FROM AUTHOR]

We study Fourier series of Jacobi polynomials P ( x), k = r, r +1,..., orthogonal with respect to the Sobolev-type inner product of the following form: $$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{\left( v \right)}}} \left( { - 1} \right){g^{\left( v \right)}}\left( { - 1} \right) + \int_{ - 1}^1 {{f^{\left( r \right)}}} {g^{\left( r \right)}}\left( t \right){\left( {1 - t} \right)^\alpha }dt$$ . It is shown that such series are a particular case of mixed series of Jacobi polynomials P ( x), k = 0, 1,..., considered earlier by the author. We study the convergence of mixed series of general Jacobi polynomials and their approximation properties. The results obtained are applied to the study of the approximation properties of Fourier series of Sobolev orthogonal Jacobi polynomials P ( x). [ABSTRACT FROM AUTHOR]

Sufficient conditions for the best convergence in L of series in systems ϕ( nx), where ϕ( x) are polynomials or series in trigonometric andWalsh systems, are obtained. [ABSTRACT FROM AUTHOR]

It was a question of Taibleson, open for a long time that the almost everywhere convergence of Fejér (or ( C, 1)) means of Fourier series of integrable functions with respect the character system of the group of 2-adic integers. This question was answered by Gá t in 1997. The aim of this paper is to investigate the maximal operator of the sup |σ|. Among other things, we prove that this operator is bounded from the Hardy space H to the Lebesgue space L if and only if 1/2 < p < ∞. The two-dimensional maximal operator is also discussed. [ABSTRACT FROM AUTHOR]

The properties of mappings by the solutions of second-order elliptic partial differential equations in the plane are studied. We obtain conditions on a function, continuous on the unit circle, that are sufficient for the solution of the Dirichlet problem in the open unit disk for the given equation with the given boundary function to be a homeomorphism between the open unit disk and a Jordan simply connected domain. The properties of the zeros of the solutions of the given equations are also studied. In particular, an analog of the main theorem of algebra is proved for polynomial solutions. [ABSTRACT FROM AUTHOR]

Sharp Jackson-Stechkin type inequalities in which the modulus of continuity of mth order of functions is defined via the Steklov function are obtained. For the classes of functions defined by these moduli of continuity, exact values of various n-widths are derived. [ABSTRACT FROM AUTHOR]

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$, k = 0, 1, ..., is obtained by passing to the limit as α→−1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> −1, an orthonormal system with weight (1 − x) α on [−1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $. In particular, it is shown that the operator S( f) = S ( f, x) is the projection onto the subspace of algebraic polynomials p = p( x) of degree at most n, i.e., S( p) = p; in addition, S( f, x) coincides with f( x) at the endpoints ±1, i.e., S ( f,±1) = f(±1). It is proved that the Lebesgue function Λ( x) of the partial sums S( f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$. [ABSTRACT FROM AUTHOR]

For a mixed modulus of smoothness of positive order, its relationship with the norms of partial sums of a Fourier series in a mixed metric is obtained. The relation between mixed moduli of smoothness of positive order in different mixed metrics is established. [ABSTRACT FROM AUTHOR]

We introduce the notion of spherical jump of a function of several variables at a given point with respect to a homogeneous harmonic polynomial. Here, if the function is integrable over spheres of sufficiently small radius centered at the given point and is continuous at this point, then its spherical jump at this point with respect to any homogeneous harmonic polynomial, distinct from a constant, is zero. Under certain conditions on a function of n variables ( n ≥ 2) at a point where the spherical jump of this function with respect to a homogeneous harmonic polynomial P is distinct fromzero, we calculate the first termof the asymptotics of the spherical Bochner-Rieszmeans of the critical order ( n − 1)/2 of the series (integral) conjugate to the n-multiple Fourier series (integral) of this function with respect to the Riesz-type kernel generated by the polynomial P. This first term of the asymptotics contains the spherical jump of the function as a multiplicative constant. [ABSTRACT FROM AUTHOR]

We find sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity of mth order in the space L as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of n-widths in L. [ABSTRACT FROM AUTHOR]

We establish a lower bound for the uniform norm of the trigonometric polynomial of special form via the sum of the L-norms of its summands. This result generalizes a theorem due to Kashin and Temlyakov, which, in turn, generalizes the classical Sidon inequality. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of the roots of polynomials given by a linear summation method for partial sums of the Fourier series. [ABSTRACT FROM AUTHOR]

We obtain exact values of different n-widths for classes of differentiable periodic functions in the space L2[0, 2π] satisfying the constraint , where 0 < h < ∞, 1/ r < p ≤ 2, r ∈ ℕ, and ω m( f( r); t) is the modulus of continuity of mth order of the derivative f( r)( x) ∈ L2[0, 2 π]. [ABSTRACT FROM AUTHOR]

We obtain order-sharp estimates of best approximations for the classes B of periodic functions of several variables by trigonometric polynomials whose spectra are generated by the level surfaces of the function Ω( t). [ABSTRACT FROM AUTHOR]

In this paper, we study the question of the existence of the inequality , 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. [ABSTRACT FROM AUTHOR]

For a function ƒ ∈, L 1( $$\mathbb{T}$$ ), we investigate the sequence ( C, 1) of mean values Φ(¦ S k ( x, ƒ) − ƒ( x)¦), where Φ( t): [0, +∞) → [0,+∞), Φ(0) = 0, is a continuous increasing function. We prove that if Φ increases faster than exponentially, then these means can diverge everywhere. Divergence almost everywhere of such means was established earlier. [ABSTRACT FROM AUTHOR]

As A. S. Belov proved, the partial sums of an even 2π-periodic function f expanded in a Fourier series with convex coefficients {α n }, are uniformly bounded below if the conditions a n = O( n −1), n → ∞, are satisfied; moreover, this assertion is no longer valid if the exponent −1 in this condition is replaced by a greater one. In this paper, we obtain analogs of these results for trigonometric polynomials of best approximation to the function f in the metric of L . [ABSTRACT FROM AUTHOR]