Yurii Lyubarskii

Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's (

We describe the complete interpolating sequences for the Paley-Wiener spaces Lpp (1 < p < 8) in terms of Muckenhoupts (Ap) condition. For p = 2, this description coincides with those given by Pavlov [9], Nikolskii [8] and Minkin [7] of the unconditional bases of complex exponentials in L2(-p,p). While the techniques of these authors are linked to the Hilbert space geometry of Lp2, our method of proof is based in turning the problem into one about boundedness of the Hilbert transform in certain weighted Lp spaces of functions and sequences.

A_p

Given a frame in Cn which satisfies a form of the uncertainty principle (as introduced by Candes and Tao), it is shown how to quickly convert the frame representation of every vector into a more robust Kashins representation whose coefficients all have the smallest possible dynamic range O(1/ √(n). The information tends to spread evenly among these coefficients. As a consequence, Kashins representations have a great power for reduction of errors in their coefficients, including coefficient losses and distortions.

) condition

a. This article studies approximation of subharmonic functions in the complex plane by logarithms of moduli of entire functions. For special classes of subharmonic functions this problem has been considered by a number of authors. For example we refer the reader to the classical works [15] and [3], in which such kind of approximation is one of the basic tools. The case when the Riesz measure of subharmonic function is located on a system of curves has been treated in [10, 9, 8], see also Sec. 10.5 in [7]. In [1] the problem was considered for general classes of subharmonic functions of finite order ρ > 0, i.e. subharmonic functions u such that, for each e > 0, 1 u(z) ≺ (1 + |z|)ρ+e, z ∈ C.

Uncertainty Principles and Vector Quantization

It is well known that Gabor expansions generated by a lattice of Nyquist density are numerically unstable, in the sense that they do not constitute frame decompositions. In this paper, we clarify exactly how “bad” such Gabor expansions are, we make it clear precisely where the edge is between “enough” and “too little,” and we find a remedy for their shortcomings in terms of a certain summability method. This is done through an investigation of somewhat more general sequences of points in the time-frequency plane than lattices (all of Nyquist density), which in a sense yields information about the uncertainty principle on a finer scale than allowed by traditional density considerations. An important role is played by certain Hilbert scales of function spaces, most notably by what we call the Schwartz scale and the Bargmann scale, and the intrinsically interesting fact that the Bargmann transform provides a bounded invertible mapping between these two scales. This permits us to turn the problems into interpolation problems in spaces of entire functions, which we are able to treat.

On approximation of subharmonic functions

In a scale of Fock spaces

Convergence and summability of Gabor expansions at the Nyquist density

\mathcal F_\varphi

Riesz bases of reproducing kernels in Fock-type spaces

with radial weights

Radial oscillation of harmonic functions in the Korenblum class

\varphi

Trace ideal criteria for embeddings and composition operators on model spaces

we study the existence of Riesz bases of (normalized) reproducing kernels. We prove that these spaces possess such bases if and only if

Bandlimited Lipschitz functions

\varphi(x)