The present paper deals with some geometric classes of analytic functions, such as starlike, convex and bounded turning property in a complex domain. These classes are defined by a new linear operator in the normalized space of analytic functions. The linear operator is introduced by a convolution of Szego function involving parametric coefficients type Laguerre polynomial, with the normalized ...

We prove two formulae that express the Alexander polynomial 1C of several variables of a plane curve singularity C in terms of the ring OC of germs of analytic functions on the curve. One of them expresses1C in terms of dimensions of some factors corresponding to a (multi-indexed) filtration on the ring OC . The other one gives the coefficients of the Alexander polynomial 1C as Euler characteri...

We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as sometimes used in machine learning, certain new nonlinearly factorizable kernels and a kernel which is closely related to the Gaussian. Each such kernel reproduces in a certain 'native' Hilbert space of multivariate analytic functions. If functions from this space are interpola...

We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1, 1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1, 1) random polynomial of degree N are concentrated in a narrow annulus of the order of N around the unit circle on the complex plane, and we find ...

We construct polynomial approximations of Dzjadyk type (in terms of the k -th modulus of continuity, k ≥ 1 ) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E . Furthermore, the error in this approximation is decaying as e−cn α strictly inside E , where c and α are positive constants independent of the degree n of the app...

We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1, 1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1, 1) random polynomial of degree N are concentrated in a narrow annulus of the order of N around the unit circle on the complex plane, and we find ...

We define the notion of rational presentation of a complete metric space, in order to study metric spaces from the algorithmic complexity point of view. In this setting, we study some representations of the space C[0, 1] of uniformly continuous real functions over [0,1] with the usual norm : ‖ f ‖∞= Sup{| f(x) |; 0 ≤ x ≤ 1}. This allows us to have a comparison of global kind between complexity ...