Advanced ordinary and fractional approximation by positive sublinear operators

Here we consider the ordinary and fractional approximation of functions by sublinear positive operators with applications to generalized convolution type operators expressed by sublinear integrals such as of Choquet and Shilkret ones. The fractional approximation is under fractional differentiability of Caputo, Canavati and Iterated-Caputo types. We produce Jackson type inequalities under basic initial conditions. So our way is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of ordinary and fractional derivatives of the function under approximation. We give also an application related to Picard singular integral operators.

INTRACAVITY LARGE-SCALE HETEROGENEITY AND THEIR INFLUENCE ON RADIATION PARAMETERS OF POWERFUL LASERS

Large‑scale inhomogeneities occur in the optical resonator of a high‑power laser due to uneven redistribution of heat and errors in the alignment of the circuit. Such inhomogeneities can significantly affect the characteristics of laser radiation. The paper analyzes the effect of large‑scale phase inhomogeneities associated with the separation and thermal deformation of unstable resonator mirrors on the parameters of the laser radiation – the power and angular divergence. The numerical simulation of the unstable resonator in the di‑fractional approximation is carried out, on the basis of which the quantitative effect of phase aberrations such as optical wedge (separation) and spherical lens (thermodeformation) on the power and divergence of laser radiation depending on the main characteristics of the unstable resonator‑the Fresnel number and the magnification coefficient is determined. The results of experimental studies of the effect of separation on the radiation parameters of a 100‑kilowatt CO2 gas‑dynamic laser are presented. The results of the work should be taken into account in the structural‑parametric optimization of the resonator.