Andrew Bakan

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 $1$, $\exp ( i \pi n x)$, $\exp ( i \pi n/ x)$, $n \in \mathbb{Z} \setminus \{0\}$, 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 $h_{0}(f) + \sum_{n \in \mathbb{Z}\setminus \{0\}}(h_{n}(f) e^{ i \pi n x} + m_{n}(f) e^{-i \pi n / x} )$ 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_{\varphi} (x, y):= \int_{\mathbb{R}} \varphi (t) \exp \left( i x t + i y / t \right) d t $, for $\varphi \in L^{1} (\mathbb{R})$ and $(x, y) \in \mathbb{R}^{2}$, supplies interpolating functions for the Klein-Gordon equation.

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 holomorphic in $\mathbb{H}$ and real-valued on the positive imaginary axis. We derive an integral representation of $\log\Theta_{3} (z)$ 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 $\Theta_{4}$ and $\Theta_{2}$ may be conveniently expressed in terms of $\log\Theta_{3}$ 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.

In 1924 S. Bernstein [Bull. Soc. Math. France 52 (1924), 399-410] asked for conditions on a uniformly bounded <!-- MATH $\mathbb{R}$ --> Borel function (weight) <!-- MATH $w: \mathbb{R} \to [0, +\infty )$ --> which imply the denseness of algebraic polynomials <!-- MATH ${\mathcal {P} }$ --> in the seminormed space <!-- MATH $C^{\,0}_{w}$ --> defined as the linear set <!-- MATH $\{f \in C (\mathbb{R}) \ | \ w (x) f (x) \to 0 \ \mbox {as} \ {|x| \to +\infty }\}$ --> equipped with the seminorm <!-- MATH $\|f\|_{w} := \sup \nolimits _{x \in {\mathbb{R}}} w(x)| f( x )|$ --> . In 1998 A. Borichev and M. Sodin [J. Anal. Math 76 (1998), 219-264] completely solved this problem for all those weights for which <!-- MATH ${\mathcal {P} }$ --> is dense in <!-- MATH $C^{\,0}_{w}$ --> but for which there exists a positive integer such that <!-- MATH ${\mathcal {P} }$ --> is not dense in <!-- MATH $C^{\,0}_{(1+x^{2})^{n} w}$ --> . In the present paper we establish that if <!-- MATH ${\mathcal {P} }$ --> is dense in <!-- MATH $C^{\,0}_{(1+x^{2})^{n} w}$ --> for all , then for arbitrary <!-- MATH $\varepsilon > 0$ --> 0$"> there exists a weight <!-- MATH $W_{\varepsilon } \in C^{\infty } (\mathbb{R})$ --> such that <!-- MATH ${\mathcal {P}}$ --> is dense in <!-- MATH $C^{\,0}_{(1+x^{2})^{n} W_{\varepsilon }}$ --> for every and <!-- MATH $W_{\varepsilon } (x) \geq w (x) + \mathrm {e}^{- \varepsilon |x|}$ --> for all <!-- MATH $x\in \mathbb{R}$ --> .

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 $x\in \mathbb{R}$.

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) and S. Koumandos (2013) showed, respectively, that the sequences $\{\theta_{n}\}_{n \geq 0}$ and $\{4/135 - n \cdot (\theta_{n}- 1/3 )\}_{n \geq 0}$ are completely monotone. In the present paper we establish that the sequence $\{(n+1)(\theta_{n}- 1/3 )\}_{n \geq 0}$ is also completely monotone. Furthermore, we prove that the analytic function $(\theta_{1}- 1/3 )^{-1} \sum_{n=1}^{\infty} (\theta_{n}- 1/3 ) \cdot z^{n} / n^{\alpha} $ is universally starlike for every $ \alpha \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 $(\theta_{1}- 1/3 )^{-1} \sum_{n=1}^{\infty} (\theta_{n}- 1/3 ) \cdot z^{n} $ is universally convex.

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 $n\geq r$.

A deep result of J. Lewis (1983) shows that the polylogarithms <!-- MATH $Li_\alpha (z)$ --> <!-- MATH $\sum _{k=1}^{\infty }z^k/k^\alpha$ --> map the open unit disk <!-- MATH $\mathbb{D}$ --> centered at the origin one-to-one onto convex domains for all <!-- MATH $\alpha \geq 0$ --> . 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-domain <!-- MATH $\Lambda := \mathbb{C}\setminus [1,\infty )$ --> , introduced by S. Ruscheweyh, L. Salinas and T. Sugawa (2009). This settles a conjecture made in that work and proves, in particular, that <!-- MATH $Li_\alpha (z)$ --> maps an arbitrary open disk or half-plane in one-to-one onto a convex domain for every <!-- MATH $\alpha \geq 1$ --> .