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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 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]

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]

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 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]

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]