Subsequences of zeros for Bernstein spaces and the completeness of systems of exponentials in spaces of functions on an interval

Abstract

Let σ > 0. The symbol B ∞ σ denotes the space of all entire functions of exponential type not exceeding σ that are bounded on the real axis. Various exact descriptions of uniqueness sequences for the Bernstein spaces B ∞ σ are given in terms of σ and the Poisson and Hilbert transformations. These descriptions lead to completeness criteria for systems of exponentials (up to one or two members) in various classical function spaces on an interval (closed or open) of length d.

Affine Balayage of Measures and Distribution of Riesz Measures of Subharmonic Functions

“…{x} is called a Jensen potential on O with pole at x ∈ O if this function V satisfies conditions (6.17) [18,3], [1], [37], [58], [39, Definition 8], [52, IIIC], [41], [47], [6]. The class of all Jensen potential on O with pole at x ∈ O denote by P J x (O) ⊂ AS x (O).…”

“…In connection with the quadrature domains, see very informative overview [22,3] and bibliography in it. Example 5.1 ([18, 3], [51], [52], [14], [15], [11], [23], [27]- [43], [47], [6]). If a measure µ ∈ Meas + cmp (O) is a balayage of the Dirac measure δ x for sbh(O), where x ∈ O, then this measure µ is called a Jensen measure for x.…”

Poisson-Jensen Formulas and Balayage of Measures

“…(2.11) If x ∈ O and δ x sbh(O) ω, then this measure ω is called a Jensen measure on O at x [15, 3], [43], [44], [12], [13], [51], [10], [20], [21], [24], [42], [7], [38]. The class of these measures is denoted by J x (O), and properties (2.11) are supplemented by the positivity property pt ω−δx ≥ 0 on R d ∞ \x for all measures ω ∈ J x (O) ⊂ AS x (O).…”

“…A positive Arens -Singer potential is called a Jensen potential on O with pole at x ∈ O [15, 3], [1], [35], [47], [36, Definition 8], [44, IIIC], [37], [42], [7]. We denote by JP x (O) the class of all Jensen potentials on O with pole at x ∈ O.…”

Classical Balayage of Charges and Measures

“…These rules are described in detail in L. Schwartz's monograph [22, Vol. I,Ch.IV, § 6] and we do not dwell on them here, although here interesting questions arise, for example, for the Bernstein -Paley -Wiener -Mary Cartwright classes of entire functions [11], [16], [1], [15] etc.…”