arXiv subjects

Håkan Hedenmalm

Publications and source records attributed to Håkan Hedenmalm.

3 recordsLinked to original sources

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↗

arXiv1995

A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$

For a function $φ$ in $L^2(0,1)$, extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates $φ(nx)$, $n=1,2,3,\ldots$, constitutes a Riesz basis or a complete sequence in $L^2(0,1)$. The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space $\cal H$ of Dirichlet series $f(s)=\sum_n a_nn^{-s}$, where the coefficients $a_n$ are square summable. It proves useful to model $\cal H$ as the $H^2$ space of the infinite-dimensional polydisk, or, which is the same, the $H^2$ space of the character space, where a character is a multiplicative homomorphism from the positive integers to the unit circle. For given $f$ in $\cal H$ and characters $χ$, $f_χ(s)=\sum_na_nχ(n)n^{-s}$ is a vertical limit function of $f$. We study certain probabilistic properties of these vertical limit functions.

math.FA↗

arXiv1995

Wiener's Tauberian theorem in L^1(G//K) and harmonic functions in the unit disk

Our main result is to give necessary and sufficient conditions, in terms of Fourier transforms, on a closed ideal $I$ in $\loneg$, the space of radial integrable functions on $G=SU(1,1)$, so that $I=\loneg$ or $I=\lonez$---the ideal of $\loneg$ functions whose integral is zero. This is then used to prove a generalization of Furstenberg's theorem which characterizes harmonic functions on the unit disk by a mean value property and a ``two circles" Morera type theorem (earlier announced by Agranovski\uı).

math.CA↗