The Müntz-Szász problem

Mathematical Notes -     

Approximation of the Müntz-Szász type in weight spaces L p and zeroes of functions of Bergman classes in a half-plane

We study the completeness of the system of exponents exp(?λ _n t), Re λ _n > 0, in spaces L ^p with the power weight on the semiaxis ?₊. We prove a sufficient condition for the completeness; one can treat it as a modification of the well-known Szász condition. With p = 2 it is unimprovable (in a sense). The proof is based on the results (which are also obtained in this paper) on the distribution of zeroes of functions of the Bergman classes in a half-plane.     

Entire functions of bernstein's class that are not fourier-stieltjes transforms

We consider certain subclasses of the class of entire functions of exponential type bounded on the real axis. We construct functions that belong to these subclasses but are not Fourier-Stieltjes transforms. Particular attention is given to the distribution of zeros of such functions. The results obtained allow us to study the stability of completeness of systems of exponentials inC andL ^p under small perturbations of the exponents. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 367–380, March, 1997. Translated by M. A. Shishkova     

Bases of exponentials in weighted spaces generated by zeros of functions of sine type

If ω is an A _p -weight with some additional condition and (λ) is a separated sequence of all zeros of a sine-type function possessing a certain multiplier (in the sense of Fourier transforms) property, then the corresponding system of exponentials $(e^{i\lambda _n t} )$ constitutes a basis in the weighted space L ^p ((?π, π), ω(t)dt), 1 < π < ∞.     

A periodic in the mean extension and bases of exponential functions in LP(−π, π)

We develop sufficiency conditions for: 1) periodic-in-the-mean extendability of functions from LP; 2) a system of exponential functions to be a basis in LP(–, ).Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 37–42, July, 1972.     

Complete and incomplete systems of exponentials in spaces with a power weight on a half-line

We essentially widen the class of sequences λ _n for which the completeness (incompleteness) of system of exponentials $e^{ - \lambda _n t}$ , Reλ _n > 0, is proved in the spaces L ^p (?₊, t ^α dt), α > ?1. The proof uses the invariance of completeness relative to the change of the weight t ^α by the weight (1 + t) ^α ; this fact is also proved here.     

Localization of small zeros of sine and cosine Fourier transforms of a finite positive nondecreasing function

Let a function f be integrable, positive, and nondecreasing in the interval (0, 1). Then by Polya’s theorem all zeros of the corresponding cosine and sine Fourier transforms are real and simple; in this case positive zeros lie in the intervals (π(n−1/2), π(n+1/2)), (πn, π(n+1)), n ∈ ℕ, respectively. In the case of sine transforms it is required that f cannot be a stepped function with rational discontinuity points. In this paper, zeros of the function with small numbers are included into intervals being proper subsets of the corresponding Polya intervals. A localization of small zeros of the Mittag-Leffler function E _1/2(−z ²; μ), μ ∈ (1, 2) ∪ (2, 3) is obtained as a corollary.