Andrew Gene Bakan

• Dr.Sc., Ph.D.
• Senior Researcher at National Academy of Sciences of Ukraine

In this paper we shall study Hardy spaces of analytic functions in a strip S. Our main result is on one hand an intrinsic characterization of the spaces and on the second that polynomials are dense.We also present an orthogonal (in H_2 (S) ) basis of polynomials.

A deep result of J. Lewis (1983) shows that the polylogarithms Li_{α}(z) :=\sum_{k≥1} z^{k}/k^{α} map the open unit disk D centered at the origin one-to-one onto convex domains for all α ≥ 0. In the present paper this result is generalized to the so-called universal convexity and universal starlikeness (with respect to the origin) in the slit-domai...

Denote by $\mathcal{P}_{\log}$ the set of all non-constant Pick functions $f$ whose logarithmic derivatives $f^{\, \prime}/f$ also belong to the Pick class. Let $\mathcal{U}(\Lambda)$ be the family of functions $z\cdot f(z)$, where $f \in\mathcal{P}_{\log}$ and $f$ is holomorphic on $\Lambda:=\mathbb{C}\setminus [1, +\infty)$. Important examples of...

Let $\Theta_{3} (z):= \sum_{n\in\mathbb{Z}} \exp (i \pi n^2 z)$ be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane $\mathbb{H}$, and takes positive values along the positive imaginary axis. We define its logarithm $\log\Theta_3(z)$ which is uniquely determined by the requirements that it should be holo...

In an effort to extend classical Fourier theory, Hedenmalm and Montes-Rodríguez (2011) found that the function system e m (x) = e iπmx , e † n (x) = e n (−1/x) = e −iπn/x is weak-star complete in L ∞ (R) when m, n range over the integers with n = 0. It turns out that the system can be used to provide unique representation of functions and more gene...

In this article we explain the essence of the interrelation described in [PNAS 118, 15 (2021)] on how to write explicit interpolation formula for solutions of the Klein-Gordon equation by using the recent Fourier pair interpolation formula of Viazovska and Radchenko from [Publ Math-Paris 129, 1 (2019)]. We construct explicitly the sequence in $L^1...

Significance We show an interrelation between the uniqueness aspect of the recent Fourier interpolation formula of D.R. and M.V. and the lattice-cross uniqueness set for the Klein–Gordon equation studied by H.H. and A.M.-R. With appropriate modifications, the approach applies in any even dimension ≥ 4 and is based on a sophisticated analysis of the...

Let \(\varTheta _{3} (z):= \sum \nolimits _{n\in \mathbb {Z}} \exp (\mathrm {i} \pi n^2 z)\) be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane \(\mathbb {H}:=\{z\in \mathbb {C}\,|\,\,\mathrm{{Im}}\, z>0\}\), and takes positive values along \( \mathrm {i} \mathbb {R}_{>0}\), the positive imaginary axis...

We show an interrelation between the uniqueness aspect of the recent Fourier interpolation formula of Radchenko and Viazovska and the Heisenberg uniqueness study for the Klein-Gordon equation and the lattice-cross of critical density, studied by Hedenmalm and Montes-Rodriguez. This has been known since 2017.

Denote by $\mathcal{P}_{\log}$ the set of all non-constant Pick functions $f$ whose logarithmic derivatives $f^{\, \prime}/f$ also belong to the Pick class. Let $\mathcal{U}(\Lambda)$ be the family of functions $z\cdot f(z)$, where $f \in\mathcal{P}_{\log}$ and $f$ is holomorphic on $\Lambda:=\mathbb{C}\setminus [1, +\infty)$. Important examples of...

In 1924 S.Bernstein asked for conditions on a uniformly bounded on $\mathbb{R}$ Borel function (weight) $w: \mathbb{R} \to [0, +\infty )$ which imply the denseness of algebraic polynomials ${\mathcal{P} }$ in the seminormed space $ C^{0}_{w} $ defined as the linear set $ \{f \in C (\mathbb{R}) \ | \ w (x) f (x) \to 0 \ \mbox{as} \ {|x| \to +\infty}...

The Ramanujan sequence $ \{\theta_{n}\}_{n \geq 0}$, defined as $$ \theta_{0}= \frac{1}{2} \ , \ \ \ \theta_{n} = \left(\ \ \frac{e^{n}}{2} - \sum_{k=0}^{n-1} \frac{n^{k}}{k !} \ \ \right) \cdot \frac{n !}{n^{n}} \ , \ \ n \geq 1 \ ,$$ has been studied on many occasions and in many different contexts. J.Adell and P.Jodra (2008) showed that the sequ...

To each nonzero sequence $s:= \{s_{n}\}_{n \geq 0}$ of real numbers we associate the Hankel determinants $D_{n} = \det H_{n}$ of the Hankel matrices $H_{n}:= (s_{i + j})_{i, j = 0}^{n}$, $n \geq 0$, and the nonempty set $N_{s}:= \{n \geq 1 | D_{n-1} \neq 0 \}$. We also define the Hankel determinant polynomials $P_0:=1$, and $P_n$, $n\geq 1$ as the...

Let 1≤p<∞. We show that ‘positive polynomial approximation property’ holds in the space L_p (R,dμ) (or C^0_w ) if and only if the algebraic polynomials are dense in L_{2p} (R,dμ) (or C^0_{\sqrt{w}} ). If μ is not a 2p-singular measure (or w is not a singular weight), this also implies the more general ‘oscillation-diminishing polynomial approximati...

We prove that the theorem on the incompleteness of polynomials in the space C^0_w established by de Branges in 1959 is not true for the space L_p (ℝ, dμ) if the support of the measure μ is sufficiently dense.

This paper is devoted to application of the results about entire and meromorphic functions obtained by A.Baernstein II (1974), I.V. Ostrowski (1970) and W.K. Hayman (1989) for weakening of the validity conditions for partial fraction decomposition of the reciprocal of an entire function which have earlier been found by M.G. Krein (1947), B.Ja.Levin...

The known theorem of A. Borichev and M. Sodin [J. Anal. Math. 76, 219–264] on the description of sets of elements of the second line of the Nevanlinna matrices which correspond to indefinite moment problems is extended to the sets of elements of the first line of these matrices.

It has been proved that algebraic polynomials P are dense in the space L(p)(R, d mu), p is an element of (0, infinity), i. the measure mu is representable as d mu = w(p) dv with a finite non-negative Borel measure v and an upper semi-continuous function w : R --> R(+) := [0, infinity) such that P is a dense subset of the space C(w)(0) := {f is an e...

We describe all zero-diminishing sequences (over the real-valued polynomials on R) which additionally satisfy a Carleman condition and show that they are of the same kind as those in E. Laguerre’s theorem from 1884.

For the sets Mp* (R), 1 ≤ p < ∞, of positive finite Borel measures μ on the real axis with the set of algebraic polynomials P dense in Lp (R, d μ), we establish a majorization principle of their "boundaries," i.e. for every μ ∈ Mp* (R) there exists ν ∈ Mp* (R) {set minus} {n-ary union}q > p Mq* (R) such that d μ / d ν ≤ 1. A corresponding principle...

We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed con...

We give characterisations of certain positive finite Borel measures with unbounded support on the real axis so that the algebraic polynomials are dense in all spacesL p (R,dμ),p≥1. These conditions apply, in particular, to the measures satisfying the classical Carleman conditions.

An addition to S.Mergelyan'’s theorem about polynomial denseness in the space C_0^w has been obtained when algebraic polynomials are dense in the space C_0^w. In that case a complete description of all functions which can be approximated by algebraic polynomials in the semi­-norm ||.||_w has been found.

In the criterion for polynomial denseness in the space C^0_w established by de Branges in 1959, we replace the requirement of the existence of an entire function by an equivalent requirement of the existence of a polynomial sequence. We introduce the notion of strict compactness of polynomial sets and establish sufficient conditions for a polynomia...

The criterion for the denseness of polynomials in the space L_2 ( R , d μ ) established by Hamburger in 1921 is extended to the spaces L_p ( R , d μ ) , 1 ≤ p < ∞ .

A calculation formula is established for the codimension of the polynomial subspace in L_2 (R; dμ) with discrete indeterminate measure μ. We clarify how much the masspoint of the n-canonical solution of an indeterminate Hamburger moment problem differs from the masspoint of the corresponding N-extremal solution at a given point of the real axis.

Generalized Pólya frequency functions are introduced through inverse Mellin transformations of the reciprocals of real entire functions with all zeros in sectors A^\phi and -A^\phi for 0≤φ≤π/4, where A^φ := { z \in C:|argz|≤φ}. It is shown that the constant π/4 is best possible in this context.

For an arbitrary function w: R → [0, 1], we determine the general form of a linear continuous functional on the space C^0_w. The criterion for denseness of polynomials in the space L_2 (R, dμ) established by Hamburger in 1921 is extended to the spaces C^0_w. .

A long standing open problem, known as the Karlin-Laguerre problem, in the study of the distribution of real zeros of a polynomial is to characterize all real sequences T={γ_k}_{k=0}^∞ such that they satisfy the property Z_c (T[p(x)])≤Z_c (p(x)), where Z_c(p(x)) denotes the number of non-real zeros of the real polynomial p(x)=∑_{k=0}^{n} a_k x_k an...

For a positive Borel measure µ on ℝ with all finite moments and unbounded support it has been proved that algebraic polynomials are dense in the space \( {L_p}(\mathbb{R},d\mu ),1 \leqslant p < \infty \)if and only if the measure can be represented in the following form:\( d\mu (x) = w{(x)^p}dv(x)\)where v is some finite positive Borel measure on ℝ...

The following problem, suggested by Laguerre's theorem (1884), remains open:characterize all real sequences {μ_k}_{k=0}^∞ which have the zero-diminishing property; that is, if p(x)=\sum_{k=0}^n a_k x^k is any real polynomial, then \sum_{k=0}^n μ_k a_k x^k has no more real zeros than p(x). In this paper this problem is solved under the additional as...

A complete description is given for the sequences {λ_k}_{k=0}^∞ such that, for an arbitrary real polynomial f(t)=\sum_{k=0}^n a_k t^k and an arbitrary A \in (0,+∞), the number of roots of the polynomial (Tf)t)=\sum_{k=0}^n λ_k a_k t^k on [0, AC] for some fixed C \in (0,+∞) does not exceed the number of roots of f(t) on [0,A]. Translated from Ukrain...

It is proved that the composition of a polynomial with a multiplier sequence of the first kind may lead to a diminishing of the number of real roots of this polynomial and that the reciprocals of the moments of a nonnegative function on [0, 1] need not form a multiplier sequence of the first kind. On the basis of these facts one establishes the ina...

We prove that convex sets A and B in a locally convex space X are properly separable if and only if there exist x,y∈Aff (A∪B) and a convex neighborhood of zero U in X such that lim_{ε↓0}ε^{−1} [ρ_{U} (A^{ε}_x, B)+ρ_{U} (A,B^{ε}_y) ]>0, where Aff(A):=a+lin(A−a), a∈A; linA is the least closed subspace containing the set A⊆X; A^{ε}_x:={tx+(1−t)a: a∈A,...

The paper examines conditions under which the Moreau-Rockafellar formula (1) ∂(p+q)=∂p+∂q holds for the subdifferential of the sum of two sublinear functionals p and q defined on a locally convex space. It is shown that this question is a particular case of a more general problem of the dual description of the fact of closedness of the union of a f...

Let X be a Hausdorff locally convex space, and α an ordinal number of the second class. For a pair of convex cones K_1 and K_2 in the space X, the notion of normality of order α is defined. Let N_α (X) denote the set of all normal pairs of cones of order α. It is proved earlier that \cup_{α} N_α (X) coincides with the set of pairs of cones for whic...