arXiv subjects

Andrew Bakan

Publications and source records attributed to Andrew Bakan.

7 recordsLinked to original sources

arXiv2023

Hyperbolic Fourier series

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 $L^1 (\mathbb{R} )$ which is biorthogonal to the system $1$, $\exp ( i πn x)$, $\exp ( i πn/ x)$, $n \in \mathbb{Z} \setminus \{0\}$, and show that it is complete in $L^1 (\mathbb{R})$. We associate with each $f \in L^1 (\mathbb{R}, (1+x^2)^{-1} d x)$ its hyperbolic Fourier series $h_{0}(f) + \sum_{n \in \mathbb{Z}\setminus \{0\}}(h_{n}(f) e^{ i πn x} + m_{n}(f) e^{-i πn / x} )$ and prove that it converges to $f$ in the space of tempered distributions on the real line. Applied to the above mentioned biorthogonal system, the integral transform given by $U_φ (x, y):= \int_{\mathbb{R}} φ(t) \exp \left( i x t + i y / t \right) d t $, for $φ\in L^{1} (\mathbb{R})$ and $(x, y) \in \mathbb{R}^{2}$, supplies interpolating functions for the Klein-Gordon equation.

math.AP↗

arXiv2021

Exponential integral representations of theta functions

Let $Θ_{3} (z):= \sum_{n\in\mathbb{Z}} \exp (i πn^2 z)$ 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 $\logΘ_3(z)$ which is uniquely determined by the requirements that it should be holomorphic in $\mathbb{H}$ and real-valued on the positive imaginary axis. We derive an integral representation of $\logΘ_{3} (z)$ when $z$ belongs to the hyperbolic quadrilateral $\mathcal{F}^{||}_{\square}$, consisted of all those $z \in \mathbb{H}$ which satisfy $-1 \leq Re\, z \leq 1$, $|2 z - 1| > 1$ and $ |2 z + 1| > 1$. Since every point of $\mathbb{H}$ is equivalent to at least one point in $\mathcal{F}^{||}_{\square}$ under the theta subgroup of the modular group on the upper half-plane, this representation carries over in modified form to all of $\mathbb{H}$ via the identity recorded by Berndt. The logarithms of the related Jacobi theta functions $Θ_{4}$ and $Θ_{2}$ may be conveniently expressed in terms of $\logΘ_{3}$ via functional equations, and hence get controlled as well. Our approach is based on a study the logarithm of the Gauss hypergeometric function for a specific choice of the parameters. This connects with the study of the universally starlike mappings introduced by Ruscheweyh, Salinas, and Sugawa.

math.CA↗

arXiv2020

Fourier uniqueness in $\mathbb{R}^4$

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.

math.FA↗

arXiv2018

Universally starlike and Pick functions

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 $\mathcal{U}(Λ)$ be the family of functions $z\cdot f(z)$, where $f \in\mathcal{P}_{\log}$ and $f$ is holomorphic on $Λ:=\mathbb{C}\setminus [1, +\infty)$. Important examples of functions in $\mathcal{U}(Λ)$ are the classical polylogarithms $Li_α(z)$ $:=$ $\sum_{k=1}^{\infty} z^k / k^α$ for $α\geq 0$. In this paper we prove that every $φ\in \mathcal{U}(Λ)$ is universally starlike, i.e., $φ$ maps every circular domain in $Λ$ containing the origin one-to-one onto a starlike domain. Furthermore, we show that every non-constant function $f \in \mathcal{P}_{\log}$ belongs to the Hardy space $H_p$ on the upper half-plane for some constant $p=p(f) > 1$, unless $f$ is proportional to some function $(a-z)^{-θ}$ with $a \in \mathbb{R}$ and $0 < θ\leq 1$. Finally we derive a necessary and sufficient condition on a real-valued function $v$ for which there exists $f \in \mathcal{P}_{\log}$ such that $v (x) = \lim_{\varepsilon \to 0} \mathrm{Im} f (x + i \varepsilon)$ for almost all $x \in \mathbb{R}$.

math.CA↗

arXiv2017

Smoothing of weights in the Bernstein approximation problem

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}\}$ equipped with the seminorm $\|f\|_{w} := \sup_{x \in {\mathbb{R}}} w(x)| f( x )|$. In 1998 A.Borichev and M.Sodin completely solved this problem for all those weights $w$ for which ${\mathcal{P} }$ is dense in $ C^{0}_{w} $ but there exists a positive integer $n=n(w)$ such that $\mathcal{P}$ is not dense in $ C^{0}_{(1+x^{2})^{n} w}$. In the present paper we establish that if $\mathcal{P}$ is dense in $ C^{0}_{(1+x^{2})^{n} w}$ for all $n \geq 0$ then for arbitrary $\varepsilon > 0$ there exists a weight $W_{\varepsilon} \in C^{\infty} (\mathbb{R})$ such that ${\mathcal{P}}$ is dense in $C^{\,0}_{(1+x^{2})^{n} W_{\varepsilon}}$ for every $n \geq 0$ and $W_{\varepsilon} (x) \geq w (x) + \mathrm{e}^{- \varepsilon |x|}$ for all $x\in \mathbb{R}$.

math.FA↗

arXiv2016

More properties of the Ramanujan sequence

The Ramanujan sequence $ \{θ_{n}\}_{n \geq 0}$, defined as $$ θ_{0}= \frac{1}{2} \ , \ \ \ θ_{n} = \left(\ \ \frac{e^{n}}{2} - \sum_{k=0}^{n-1} \frac{n^{k}}{k !} \ \ \right) \cdot \frac{n !}{n^{n}} \ , \ \ n \geq 1 \ ,$$ has been studied on many occasions and in many different contexts. J.Adell and P.Jodra (2008) and S. Koumandos (2013) showed, respectively, that the sequences $\{θ_{n}\}_{n \geq 0}$ and $\{4/135 - n \cdot (θ_{n}- 1/3 )\}_{n \geq 0}$ are completely monotone. In the present paper we establish that the sequence $\{(n+1)(θ_{n}- 1/3 )\}_{n \geq 0}$ is also completely monotone. Furthermore, we prove that the analytic function $(θ_{1}- 1/3 )^{-1} \sum_{n=1}^{\infty} (θ_{n}- 1/3 ) \cdot z^{n} / n^α $ is universally starlike for every $ α\geq 1 $ in the slit domain $ \mathbb{C} \setminus [1,\infty)$. This seems to be the first result putting the Ramanujan sequence into the context of analytic univalent functions and is a step towards a previous stronger conjecture, proposed by S.Ruscheweyh, L.Salinas and T.Sugawa in 2009, namely that the function $(θ_{1}- 1/3 )^{-1} \sum_{n=1}^{\infty} (θ_{n}- 1/3 ) \cdot z^{n} $ is universally convex.

math.CA↗

arXiv2016

Solvability of the Hankel determinant problem for real sequences

To each nonzero sequence $s:= \{s_{n}\}_{n \geq 0}$ of real numbers we associate the Hankel determinants $D_{n} = \det \mathcal{H}_{n}$ of the Hankel matrices $\mathcal{H}_{n}:= (s_{i + j})_{i, j = 0}^{n}$, $n \geq 0$, 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 determinant of the Hankel matrix $\mathcal H_n$ modified by replacing the last row by the monomials $1, x, \ldots, x^n$. Clearly $P_n$ is a polynomial of degree at most $n$ and of degree $n$ if and only if $n\in N_s $. Kronecker established in 1881 that if $N_s $ is finite then rank $\mathcal{H}_{n} = r$ for each $n \geq r-1$, where $r := \max N_s $. By using an approach suggested by I.S.Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence $\{t_n\}_{n\geq 0}$ to be of the form $t_n=D_n$, $n\geq 0$ for a real sequence $\{s_n\}_{n\geq 0}$. This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial $ P_n $ satisfying deg$P_n = n\geq 1$ is preceded by a nonzero polynomial $P_{n-1}$ whose degree can be strictly less than $n-1$ and which has no common zeros with $ P_n $. As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that $D_0 > 0, \ldots, D_{r-1} > 0 $ and $D_n=0$ for all $n\geq r$.

math.CA↗