Mathematics

We introduce à la Vasilevski the weighted poly-Bergman spaces in the unit disc and provide concrete orthonormal basis and give close expression of their reproducing kernel. The main tool in the description if these spaces is the so-called disc polynomials that form an orthogonal basis of the whole weighted Hilbert space.

This article contains a complete proof of Gabrielov's rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung Theorem. We include, furthermore, new extensions of the rank Theorem (concerning the Zariski main Theorem and elimination theory) to commutative algebra.

For Fatou's interpolation theorem of 1906 we suggest a new elementary proof.

Assume that f lies in the class of starlike functions of order α∈[0,1) , that is, which are regular and univalent for |z|<1 and such that Re( z f ′ (z) f(z) )>α for |z|<1. In this paper we show that for each α∈[0,1) , the following sharp inequality holds: |f(r e iθ ) | −1 ∫ r 0 | f ′ (u e iθ )|du≤ Γ( 1 2 )Γ(2−α) Γ( 3 2 −α) for every r<1 and θ. This settles the conjecture of Hall (1980).

We establish a type of the Picard's theorem for entire curves in P n (C) whose spherical derivative vanishes on the inverse images of hypersurface targets. Then, as a corollary, we prove that there is an union D of finite number of hypersurfaces in the complex projective space P n (C) such that for every entire curve f in P n (C) , if the spherical derivative f # of f is bounded on f −1 (D) , then f # is bounded on the entire complex plane, and hence, f is a Brody curve.

We provide characterizations of Carleson measures on a certain class of bounded pseudoconvex domains. An example of a vanishing Carleson measure whose Berezin transform does not vanish on the boundary is given in the class of the Hartogs triangles H k :={( z 1 , z 2 )∈ C 2 : | z 1 | k <| z 2 |<1},k∈ Z + .

We present a simple short proof of the Fundamental Theorem of Algebra, without complex analysis and with a minimal use of topology. It can be taught in a first year calculus class.

We consider smooth deformations of the CR structure of a smooth 2 -pseudoconcave compact CR submanifold M of a reduced complex analytic variety X outside the intersection D∩M with the support D of a Cartier divisor of a positive line bundle F X . We show that nearby structures still admit projective CR embeddings. Special results are obtained under the additional assumptions that X is a projective space or a Fano variety.

We show that if a reproducing kernel Hilbert space H K , consisting of functions defined on E, enjoys Double Boundary Vanishing Condition (DBVC) and Linear Independent Condition (LIC), then for any preset natural number n, and any function f∈ H K , there exists a set of n parameterized multiple kernels K ~ w 1 ,⋯, K ~ w n , w k ∈E,k=1,⋯,n, and real (or complex) constants c 1 ,⋯, c n , giving rise to a solution of the optimization problem ∥f− ∑ k=1 n c k K ~ w k ∥=inf{∥f− ∑ k=1 n d k K ~ v k ∥ | v k ∈E, d k ∈R (or C),k=1,⋯,n}. By applying the theorem of this paper we show that the Hardy space and the Bergman space, as well as all the weighted Bergman spaces in the unit disc all possess n -best approximations. In the Hardy space case this gives a new proof of a classical result. Based on the obtained results we further prove existence of n -best spherical Poisson kernel approximation to functions of finite energy on the real-spheres.

In the last few years, the notion of optimal polynomial approximant has appeared in the mathematics literature in connection with Hilbert spaces of analytic functions of one or more variables. In the 70s, researchers in engineering and applied mathematics introduced least-squares inverses in the context of digital filters in signal processing. It turns out that in the Hardy space H 2 these objects are identical. This paper is a survey of known results about optimal polynomial approximants. In particular, we will examine their connections with orthogonal polynomials and reproducing kernels in weighted spaces and digital filter design. We will also describe what is known about the zeros of optimal polynomial approximants, their rates of decay, and convergence results. Throughout the paper, we state many open questions that may be of interest.