Sharp inequality for deviation of Rogozinski sums and the second continuity modulus in the space of periodic continuous functions

The sharp constant (uniformly in n) is found in a Jackson-type inequality involving the Rogozinski sums of order n and the second modulus of continuity with step π/(n+1). Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 26–45. Translated by S. Yu. Pilyugin.     

Approximation estimates for convolution classes in terms of the second modulus of continuity

We establish the best approximation estimates for certain convolutions whose kernels are entire functions of exponential type in terms of the second modulus of continuity. The Jackson-type inequalities for even-order derivatives are particular cases of our estimates.     

Sharp estimates of integrals in terms of the second modulus of continuity

For a certain class of kernels, the exact constant in the estimate of the integral of the product of two functions in terms of the second modulus of continuity of one of them is obtained. Estimates of best approximations by entire functions of exponential type and by splines in terms of the second modulus of continuity of the second derivative of the approximated function are derived from the results obtained. The constants in these estimates are smaller than the previously known ones.     

Optimal embedding and sharp estimates of the continuity envelope for generalized Bessel potentials

In the theory of function spaces it is an important problem to describe the differential properties for the convolution u = G * f in terms of the behavior of kernel near the origin, and at the infinity. In our paper the differential properties of convolution are characterized by their modulus of continuity of order k ∈ N in the uniform norm. The kernels of convolution generalize the classical kernels determining the Bessel and Riesz potential. They admit non-power behavior near the origin. The order-sharp estimates are obtained for moduli of continuity of the convolution in the uniform norm as well as for continuity envelope function of generalized Bessel potentials. Such estimates admit sharp embedding theorems into a Calderon space and imply estimates for the approximation numbers of the embedding operator.     

Some properties of the modulus of continuity

Necessary and sufficient conditions are derived for the continuity and semiadditivity of the modulus of continuity of a functionf(x) given on a compact in n-dimensional euclidean space.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 495–500, May, 1971.     

Extension of functions preserving the modulus of continuity

The problem of the extension of a real-valued function from a subset of a metric space to the entire space is treated. An extension operator preserving the modulus of continuity of a function is proposed and its properties are studied. An application to the problem of the trace of a locally Lipschitz function on a compact subset of a metric space is given. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 236–245, February, 1997. Translated by N. K. Kulman     

Some sharp inequalities for n-monotone functions

Summary Some sharp inequalities involving n-monotone functions and their derivatives are obtained. In particular, the following generalization of the Favard-Berwald inequality is established here: \emph{If\/ is non-negative and -f is -monotone, then      

Fourier coefficients of functions with a given modulus of continuity

In this paper, we study the role of the convexity condition for the modulus of continuity in the problem of finding an upper bound for the Fourier coefficients taken over the class of functions with a given modulus of continuity. Also, we solve the problem of the Fourier coefficients for the Rademacher system.     

Uniform-convergence factors for Fourier series of functions with a given modulus of continuity

It is proved that a sequence of factors { _v }, which are Fourier-Stieitjes coefficients, converts the Fourier series of any function whose modulus of continuity does not exceed a given modulus of continuity () into a uniformly convergent series, if and only if The Sufficiency of this condition is known.Translated from Matematicheskii Zametki, Vol. 10, No. 1, pp. 33–40, July, 1971.     

Some sharp estimates of the constants of Landau and Lebesgue

We establish several sharp two-sided inequalities involving the constants of Landau and Lebesgue, which occur and play important roles in two related extremal problems in complex analysis and in Fourier analysis respectively. The main results improves essentially the Watson asymptotic expansion formulas for these constants.     

Quasiconvex uniform-convergence factors for Fourier series of functions with a given modulus of continuity

It is proved that a quasiconvex sequence _v of convergence factors transforms Fourier series of functions whose moduli of continuity do not exceed a given modulus of continuity(gd) into uniformly convergent series if and only if _n (1/n) log n 0 for n . The sufficiency of this condition is already known.Translated from Matematicheskie Zametki, Vol. 8, No. 5,pp. 619–623, November, 1970.     

On rational approximation of convex functions with a given modulus of continuity

Доказывается, что для наименьших равномер ных рациональных уклоне нийR _n(f) выпуклой на [0,1] функции с модулем непрерывно сти, не превосходящемω(δ), сп раведлива оценка $$R_n (f) \leqq c\frac{{\ln ^2 n}}{{n^2 }}\mathop {\max }\limits_{e^{ - n} \leqq \theta< 1} \left\{ {\omega (\theta )\ln \frac{1}{\theta }} \right\},$$ гдес — абсолютная по стоянная.