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Kristian Seip

Publications and source records attributed to Kristian Seip.

At least 19 recordsLinked to original sources
arXiv2026

Fourier-invariant functions with dense zero sets

For every $0\leq\beta\leq1/2$, we construct a nonzero real-valued continuous function $f_\beta$ in $L^1(\mathbb R)\cap L^2(\mathbb R)$ such that $\widehat {f}_\beta=f_\beta$ and $f_\beta(\sqrt{n}/[\log(e+n)]^{\beta})=0$ for all $n\geq 0$. The case $\beta=0$ settles in the negative a question raised by Radchenko and Viazovska regarding their Fourier interpolation formula. The construction uses a scale of reproducing kernel Hilbert spaces generated by the Fourier-invariant Hermite functions. Applying the Mehler formula, we identify the reproducing kernels of these spaces. By suitable estimates of these kernels, we show that $(\sqrt{n}/[\log(e+n)]^{\beta})$, with one auxiliary point added to it, is a universal interpolating sequence for at least one of the Hilbert spaces under consideration. However, this result fails when $\beta>1/2$.

math.CA↗
arXiv2025

The basis functions of Fourier interpolation

The basis functions of the Fourier interpolation formula of Radchenko and Viazovska, constructed by means of weakly holomorphic modular forms for the Hecke theta group, are entire functions of order $2$ having interesting time-frequency properties. We give precise size estimates and study the distribution of zeros of these functions. We give in particular asymptotic estimates for the location and the number of extraneous zeros on or close to the real line. This result reveals the surprising existence of Fourier nonuniqueness pairs whose apparent ``excess'' compared to the Fourier uniqueness pair of Radchenko and Viazovska may be made arbitrarily large. Our estimates also show that the basis functions fail to yield a Riesz basis in the Hilbert space used by Kulikov, Nazarov, and Sodin in their recent study of Fourier uniqueness pairs. Some numerical data are presented, suggesting additional fine scale properties.

math.NT↗
arXiv2025

Contractive Hardy--Littlewood inequalities in the Dirichlet range

The class $A_\alpha^p$ consists of those analytic functions $f$ in the unit disc such that \[\|f\|_{\alpha,p}^p := |f(0)|^p+\int_0^1 \left(\frac{d}{dr} M_p^p(r,f)\right) (1-r^2)^{\alpha-1} \,dr < \infty,\] where $M_p^p(r,f)$ is the radial integral mean of $|f|^p$ and $0<\alpha, p <\infty$. For $\alpha>1$, $A_\alpha^p$ is the standard weighted Bergman space, and $A_1^p=H^p$. We consider $A_\alpha^p$ for $0<\alpha<1$ and show that (weighted) isometric conformal invariance extends to this range, and we also clarify the relation between $A_{\alpha}^p$ and the classical Besov spaces. Our main result is the contractive inequality $\|f\|_{\beta,q} \leq \|f\|_{\alpha,p}$, valid when $0<\alpha<\beta<\infty$ and $\alpha/p=\beta/q$. We also identify the functions for which equality is attained. We thus extend recent results of the second-named author ($1\leq \alpha<\beta$) and Llinares ($\beta=1$ and $p=2$). The extension of results from the classical range $1\leq \alpha < \infty$ to the Dirichlet range $0<\alpha <1$ uses arguments relying on analytic continuation.

math.CV↗
arXiv2025

The H\"{o}rmander--Bernhardsson extremal function

We characterize the function $\varphi$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $\pi$ for which $f(0)=1$. This function, studied by H\"{o}rmander and Bernhardsson in 1993, has only real zeros $\pm \tau_n$, $n=1,2, \ldots$. Starting from the fact that $n+\frac12-\tau_n$ is an $\ell^2$ sequence, established in an earlier paper of ours, we identify $\varphi$ in the following way. We factor $\varphi(z)$ as $\Phi(z)\Phi(-z)$, where $\Phi(z)= \prod_{n=1}^\infty(1+(-1)^n\frac{z}{\tau_n})$ and show that $\Phi$ satisfies a certain second order linear differential equation along with a functional equation either of which characterizes $\Phi$. We use these facts to establish an odd power series expansion of $n+\frac12-\tau_n$ in terms of $(n+\frac12)^{-1}$ and a power series expansion of the Fourier transform of $\varphi$, as suggested by the numerical work of H\"{o}rmander and Bernhardsson. The dual characterization of $\Phi$ arises from a commutation relation that holds more generally for a two-parameter family of differential operators, a fact that is used to perform high precision numerical computations.

math.CA↗
arXiv2024

The H\"ormander--Bernhardsson extremal function: A preliminary study

We study the function $\varphi_1$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $\pi$ for which $f(0)=1$. This function, first studied by H\"{o}rmander and Bernhardsson in 1993, has only real zeros $\pm \tau_n$, $n=1,2, \ldots$, and the sequence $(\tau_n-n-\frac12)$ has $\ell^2$ norm bounded by $0.13$. The zeros $\tau_n$ can be computed by means of a fixed point iteration.

math.FA↗
arXiv2024

Critical exponents of the Riesz projection

Let $\mathfrak{p}_d(q)$ denote the critical exponent of the Riesz projection from $L^q(\mathbb{T}^d)$ to the Hardy space $H^p(\mathbb{T}^d)$, where $\mathbb{T}$ is the unit circle. We present the state-of-the-art on the conjecture that $\mathfrak{p}_1(q) = 4(1-1/q)$ for $1 \leq q \leq \infty$ and prove that it holds in the endpoint case $q = 1$. We then extend the conjecture to \[\mathfrak{p}_d(q) = 2+\cfrac{2}{d+\cfrac{2}{q-2}}\] for $d\geq1$ and $\frac{2d}{d+1} \leq q \leq \infty$ and establish that if the conjecture holds for $d=1$, then it also holds for $d=2$. When $d=2$, we verify that the conjecture holds in the endpoint case $q = 4/3$.

math.FA↗
arXiv2023

A dichotomy for extreme values of zeta and Dirichlet L-functions

We exhibit large values of the Dedekind zeta function of a cyclotomic field on the critical line. This implies a dichotomy whereby one either has improved lower bounds for the maximum of the Riemann zeta function, or large values of Dirichlet $L$-functions on the level of the Bondarenko--Seip bound.

math.NT↗
arXiv2023

Maximal norm Hankel operators

A Hankel operator $\mathbf{H}_φ$ on the Hardy space $H^2$ of the unit circle with analytic symbol $φ$ has minimal norm if $\|\mathbf{H}_φ\|=\|φ\|_2$ and maximal norm if $\|\mathbf{H}_φ\| = \|φ\|_\infty$. The Hankel operator $\mathbf{H}_φ$ has both minimal and maximal norm if and only if $|φ|$ is constant almost everywhere on the unit circle or, equivalently, if and only if $φ$ is a constant multiple of an inner function. We show that if $\mathbf{H}_φ$ is norm-attaining and has maximal norm, then $\mathbf{H}_φ$ has minimal norm. If $|φ|$ is continuous but not constant, then $\mathbf{H}_φ$ has maximal norm if and only if the set at which $|φ|=\|φ\|_{\infty}$ has nonempty intersection with the spectrum of the inner factor of $φ$. We obtain further results illustrating that the case of maximal norm is in general related to "irregular" behavior of $\log |φ|$ or the argument of $φ$ near a "maximum point" of $|φ|$. The role of certain positive functions coined apical Helson--Szegő weights is discussed in the former context.

math.FA↗
arXiv2022

Fourier interpolation with zeros of zeta and $L$-functions

We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other $L$-functions. We establish a duality principle for Fourier interpolation bases in terms of certain kernels of general Dirichlet series with variable coefficients. Such kernels admit meromorphic continuation, with poles at a sequence dual to the sequence of frequencies of the Dirichlet series, and they satisfy a functional equation. Our construction of concrete bases relies on a strengthening of Knopp's abundance principle for Dirichlet series with functional equations and a careful analysis of the associated Dirichlet series kernel, with coefficients arising from certain modular integrals for the theta group.

math.NT↗
arXiv2022

Point evaluation in Paley--Wiener spaces

We study the norm of point evaluation at the origin in the Paley--Wiener space $PW^p$ for $0 < p < \infty$, i. e., we search for the smallest positive constant $C$, called $\mathscr{C}_p$, such that the inequality $|f(0)|^p \leq C \|f\|_p^p$ holds for every $f$ in $PW^p$. We present evidence and prove several results supporting the following monotonicity conjecture: The function $p\mapsto \mathscr{C}_p/p$ is strictly decreasing on the half-line $(0,\infty)$. Our main result implies that $\mathscr{C}_p p/2$ for $1 \leq p < 2$. We also estimate the asymptotic behavior of $\mathscr{C}_p$ as $p \to \infty$ and as $p \to 0^+$. Our approach is based on expressing $\mathscr{C}_p$ as the solution of an extremal problem. Extremal functions exist for all $0<p<\infty$; they are real entire functions with only real zeros, and the extremal functions are known to be unique for $1\leq p < \infty$. Following work of H\"{o}rmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau--Pollak--Slepian operator of time--frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range $1<p<\infty$, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to $\mathscr{C}_p$ and the extremal functions.

math.CA↗
arXiv2021

Idempotent Fourier multipliers acting contractively on $H^p$ spaces

We describe the idempotent Fourier multipliers that act contractively on $H^p$ spaces of the $d$-dimensional torus $\mathbb{T}^d$ for $d\geq 1$ and $1\leq p \leq \infty$. When $p$ is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on $L^p$ spaces, which in turn can be described by suitably combining results of Rudin and Andô. When $p=2(n+1)$, with $n$ a positive integer, contractivity depends in an interesting geometric way on $n$, $d$, and the dimension of the set of frequencies associated with the multiplier. Our results allow us to construct a linear operator that is densely defined on $H^p(\mathbb{T}^\infty)$ for every $1 \leq p \leq \infty$ and that extends to a bounded operator if and only if $p=2,4,\ldots,2(n+1)$.

math.FA↗
arXiv2021

Hilbert points in Hardy spaces

A Hilbert point in $H^p(\mathbb{T}^d)$, for $d\geq1$ and $1\leq p \leq \infty$, is a nontrivial function $φ$ in $H^p(\mathbb{T}^d)$ such that $\| φ\|_{H^p(\mathbb{T}^d)} \leq \|φ+ f\|_{H^p(\mathbb{T}^d)}$ whenever $f$ is in $H^p(\mathbb{T}^d)$ and orthogonal to $φ$ in the usual $L^2$ sense. When $p\neq 2$, $φ$ is a Hilbert point in $H^p(\mathbb{T})$ if and only if $φ$ is a nonzero multiple of an inner function. An inner function on $\mathbb{T}^d$ is a Hilbert point in any of the spaces $H^p(\mathbb{T}^d)$, but there are other Hilbert points as well when $d\geq 2$. We investigate the case of $1$-homogeneous polynomials in depth and obtain as a byproduct a new proof of the sharp Khintchin inequality for Steinhaus variables in the range $2 4$ but only numerically for $1\leq p<4$.

math.FA↗
arXiv2021

Riesz projection and bounded mean oscillation for Dirichlet series

We prove that the norm of the Riesz projection from $L^\infty(\Bbb{T}^n)$ to $L^p(\Bbb{T}^n)$ is $1$ for all $n\ge 1$ only if $p\le 2$, thus solving a problem posed by Marzo and Seip in 2011. This shows that $H^p(\Bbb{T}^{\infty})$ does not contain the dual space of $H^1(\Bbb{T}^{\infty})$ for any $p>2$. We then note that the dual of $H^1(\Bbb{T}^{\infty})$ contains, via the Bohr lift, the space of Dirichlet series in $\operatorname{BMOA}$ of the right half-plane. We give several conditions showing how this $\operatorname{BMOA}$ space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on $\Bbb{T}$, we compute its $L^p$ norm when $1<p<\infty$, and we use this result to show that the $L^\infty$ norm of the $N$th partial sum of a bounded Dirichlet series over $d$-smooth numbers is of order $\log\log N$.

math.FA↗
arXiv2020

A converse to the Schwarz lemma for planar harmonic maps

A sharp version of a recent inequality of Kovalev and Yang on the ratio of the $(H^1)^\ast$ and $H^4$ norms for certain polynomials is obtained. The inequality is applied to establish a sharp and tractable sufficient condition for the Wirtinger derivatives at the origin for harmonic self-maps of the unit disc which fix the origin.

math.CV↗
arXiv2019

A footnote to a theorem of Halász

We study multiplicative functions $f$ satisfying $|f(n)|\le 1$ for all $n$, the associated Dirichlet series $F(s):=\sum_{n=1}^{\infty} f(n) n^{-s}$, and the summatory function $S_f(x):=\sum_{n\le x} f(n)$. Up to a possible trivial contribution from the numbers $f(2^k)$, $F(s)$ may have at most one zero or one pole on the one-line, in a sense made precise by Halász. We estimate $\log F(s)$ away from any such point and show that if $F(s)$ has a zero on the one-line in the sense of Halász, then $|S_f(x)|\le (x/\log x) \exp\big(c\sqrt{\log \log x}\big)$ for all $c>0$ when $x$ is large enough. This bound is best possible.

math.NT↗
arXiv2019

Universality and distribution of zeros and poles of some zeta functions

This paper studies zeta functions of the form $\sum_{n=1}^{\infty} χ(n) n^{-s}$, with $χ$ a completely multiplicative function taking only unimodular values. We denote by $σ(χ)$ the infimum of those $α$ such that the Dirichlet series $\sum_{n=1}^{\infty} χ(n) n^{-s}$ can be continued meromorphically to the half-plane $\operatorname{Re} s>α$, and denote by $ζ_χ(s)$ the corresponding meromorphic function in $\operatorname{Re} s>σ(χ)$. We construct $ζ_χ(s)$ that have $σ(χ)\le 1/2$ and are universal for zero-free analytic functions on the half-critical strip $1/2<\operatorname{Re} s <1$, with zeros and poles at any discrete multisets lying in a strip to the right of $\operatorname{Re} s =1/2$ and satisfying a density condition that is somewhat stricter than the density hypothesis for the zeros of the Riemann zeta function. On a conceivable version of Cramér's conjecture for gaps between primes, the density condition can be relaxed, and zeros and poles can also be placed at $β+i γ$ with $β\le 1-λ\log\log |γ|/\log |γ|$ when $λ>1$. Finally, we show that there exists $ζ_χ(s)$ with $σ(χ) \le 1/2$ and zeros at any discrete multiset in the strip $1/2<\operatorname{Re} s \le 39/40$ with no accumulation point in $\operatorname{Re} s >1/2$; on the Riemann hypothesis, this strip may be replaced by the half-critical strip $1/2 < \operatorname{Re} s < 1$.

math.NT↗
arXiv2019

Linear space properties of $H^p$ spaces of Dirichlet series

We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the range $0<p< \infty$. We begin by showing that two natural ways to define $\mathcal{H}^p$ coincide. We then proceed to study some linear space properties of $\mathcal{H}^p$. More specifically, we study linear functionals generated by fractional primitives of the Riemann zeta function; our estimates rely on certain Hardy--Littlewood inequalities and display an interesting phenomenon, called contractive symmetry between $\mathcal{H}^p$ and $\mathcal{H}^{4/p}$, contrasting the usual $L^p$ duality. We next deduce general coefficient estimates, based on an interplay between the multiplicative structure of $\mathcal{H}^p$ and certain new one variable bounds. Finally, we deduce general estimates for the norm of the partial sum operator $\sum_{n=1}^\infty a_n n^{-s}\mapsto \sum_{n=1}^N a_n n^{-s}$ on $\mathcal{H}^p$ with $0< p \le 1$, supplementing a classical result of Helson for the range $1<p<\infty$. The results for the coefficient estimates and for the partial sum operator exhibit the traditional schism between the ranges $1\le p \le \infty$ and $0<p<1$.

math.FA↗