On the modulus of continuity of solid derivatives of a Cauchy-type integral

We establish sufficient conditions for the existence of solid derivatives of a continuous extension of a Cauchy-type integral onto the closure of a domain and obtain an estimate for the moduli of continuity of these derivatives. We prove that the Newton-Leibniz formula is true for certain classes of Jordan rectifiable curves. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 476–484, April, 1998.     

An estimate for the modulus of continuity of a quaternion singular Cauchy integral

We establish an upper bound for the modulus of continuity of a quaternion singular Cauchy integral in terms of the modulus of continuity of the integrand and a metric characteristic of a curve.     

Estimates for the modulus of a Cauchy-type integral and its derivatives

In a domain bounded by a closed rectifiable Jordan curve, we obtain estimates for the modulus of a Cauchy-type integral and its derivatives in terms of the contour moduli of smoothness of the integrand and a metric characteristic of the curve. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 732–743, June, 1999.     

On the integral modulus of continuity of Fourier series

We study L ^p -integrability (1<p<??) of a sum ?? of trigonometric series under the assumptions that the sequence of coefficients of ?? belongs to the class $\overline{\mathrm{GM}}_{\theta}^{r}$ . Then we discuss the relations between the properties of ?? and the properties of the sequence (?? _n )??GM(??,r), and deduce an estimate for modulus of continuity of ?? in L ^p norm.     

Two-sided estimate for the modulus of continuity of a convolution

We study the differential properties of the convolution of functions with a generalized Bessel-Macdonald kernel. The integral properties of a function are characterized in terms of its decreasing permutation. The differential properties of the convolution are described in terms of its modulus of continuity of arbitrary order in the uniform norm. We obtain order-sharp estimates for the modulus of continuity of the convolution. By way of application, we present two-sided estimates for the modulus of continuity of the classical Bessel potential.     

Numerical implementations of Cauchy-type integral equations

In this paper, a good interpolation formulae are applied to the numerical solution of Cauchy integral equations of the first kind with using some Chebyshev quadrature rules. To demonstrate the effectiveness of the Radau-Chebyshev with respect to the olds, [6], [7], [8] and [12], some examples are given.     

Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations

Our main interest is an analog of a Cauchy-type integral for the theory of the Moisil-Theodoresco system of differential equations in the case of a piecewise-Lyapunov surface of integration. The topics of the paper concern theorems that cover basic properties of this Cauchy-type integral: the Sokhotskii-Plemelj theorem for it as well as a necessary and sufficient condition for the possibility of extending a given Hölder function from such a surface up to a solution of the Moisil-Theodoresco system of partial differential equations in a domain. A formula for the square of a singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between the theory of the Moisil-Theodoresco system of partial differential equations and some versions of quaternionic analysis.     

Estimate of the modulus of continuity of a cauchy-type integral in a domain and on its boundary

We estimate the modulus of continuity of a Cauchy-type integral in a closed domain and its limit values on the boundary in the case where the boundary of the domain is an arbitrary closed rectifiable Jordan curve.     

On the modulus of continuity for spectral measures in substitution dynamics

The paper gives first quantitative estimates on the modulus of continuity of the spectral measure for weak mixing suspension flows over substitution automorphisms, which yield information about the “fractal” structure of these measures. The main results are, first, a Hölder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function; second, a log-Hölder estimate for self-similar suspension flows; and, third, a Hölder asymptotic expansion of the spectral measure at zero for such flows. Our second result implies log-Hölder estimates for the spectral measures of translation flows along stable foliations of pseudo-Anosov automorphisms. A key technical tool in the proof of the second result is an “arithmetic-Diophantine” proposition, which has other applications. In Appendix A this proposition is used to derive new decay estimates for the Fourier transforms of Bernoulli convolutions.     

Some properties of the Cauchy-type integral for the time-harmonic Maxwell equations

We study the analog of the Cauchy-type integral for the theory of time-harmonic electromagnetic fields in case of a piece-wise Liapunov surface of integration and we prove the Sokhotski-Plemelj theorem for it as well as the necessary and sufficient condition for the possibility to extend a given pair of vector fields from such a surface up to a solution of the time-harmonic Maxwell equations in a domain. Formula for the square of the singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between time-harmonic solutions of the Maxwell equations and some versions of quaternionic analysis.     

The integral modulus of continuity of the Dirichlet kernel and the conjugate Dirichlet kernel

Asymptotic estimates for the integral modulus of continuity of order s of the Dirichlet kernel and the conjugate Dirichlet kernel are obtained. For example, if k/2, then _s (D _k ,)=2 ^s +1/²sin ^s k/2 log(1+k/s)+O(2 ^s sin ^s k/2)holds uniformly with respect to all the parameters.Translated from Matematicheskie Zametki, Vol. 54, No. 3, pp. 98–105, September, 1993.     

On a transplant operator and explicit construction of Cauchy-type integral representations for p-analytic functions

We present a new technique for explicit construction of Cauchy kernels and Cauchy integral representations for a class of generalized analytic functions and p-analytic functions.     

Estimates for the best approximation and integral modulus of continuity of a function in terms of its Fourier coefficients

In the integral metric, lower bounds are obtained for the best approximation and the modulus of continuity of a function in terms of its Fourier coefficients. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 496–503, April, 1998.