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Kristian Seip

Publications and source records attributed to Kristian Seip.

At least 55 records · Page 3Linked to original sources
arXiv2014

GCD sums and complete sets of square-free numbers

It is proved that \[ \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N\exp\left(C\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right) \] holds for arbitrary integers $1\le n_1<\cdots < n_N$. This bound is essentially better than that found in a recent paper of Aistleitner, Berkes, and Seip and can not be improved by more than possibly a power of $1/\sqrt{\log\log\log N}$. The proof relies on ideas from classical work of Gál, the method of Aistleitner, Berkes, and Seip, and a certain completeness property of extremal sets of square-free numbers.

math.NT↗
arXiv2013

Decay rates for approximation numbers of composition operators

A general method for estimating the approximation numbers of composition operators on $\Ht$, using finite-dimensional model subspaces, is studied and applied in the case when the symbol of the operator maps the unit disc to a domain whose boundary meets the unit circle at just one point. The exact rate of decay of the approximation numbers is identified when this map is sufficiently smooth at the point of tangency; it follows that a composition operator with any prescribed slow decay of its approximation numbers can be explicitly constructed. Similarly, an asymptotic expression for the approximation numbers is found when the mapping has a sharp cusp at the distinguished boundary point. Precise asymptotic estimates in the intermediate cases, including that of maps with a corner at the distinguished boundary point, are also established.

math.FA↗
arXiv2013

GCD sums from Poisson integrals and systems of dilated functions

Upper bounds for GCD sums of the form [\sum_{k,{\ell}=1}^N\frac{(\gcd(n_k,n_{\ell}))^{2α}}{(n_k n_{\ell})^α}] are proved, where $(n_k)_{1 \leq k \leq N}$ is any sequence of distinct positive integers and $0<α\le 1$; the estimate for $α=1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $α=1/2$. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish a Carleson--Hunt-type inequality for systems of dilated functions of bounded variation or belonging to $\lip12$, a result that in turn settles two longstanding problems on the a.e.\ behavior of systems of dilated functions: the a.e. growth of sums of the form $\sum_{k=1}^N f(n_k x)$ and the a.e.\ convergence of $\sum_{k=1}^\infty c_k f(n_kx)$ when $f$ is 1-periodic and of bounded variation or in $\lip12$.

math.NT↗
arXiv2012

Zeros of functions in Hilbert spaces of Dirichlet series

The Dirichlet--Hardy space $\Ht$ consists of those Dirichlet series $\sum_n a_n n^{-s}$ for which $\sum_n |a_n|^2<\infty$. It is shown that the Blaschke condition in the half-plane $\operatorname{Re} s>1/2$ is a necessary and sufficient condition for the existence of a nontrivial function $f$ in $\Ht$ vanishing on a given bounded sequence. The proof implies in fact a stronger result: every function in the Hardy space $H^2$ of the half-plane $\operatorname{Re} s>1/2$ can be interpolated by a function in $\Ht$ on such a Blaschke sequence. Analogous results are proved for the Hilbert space $\Da$ of Dirichlet series $\sum_n a_n n^{-s}$ for which $\sum_n |a_n|^2[d(n)]^α<\infty$; here $d(n)$ is the divisor function and $α$ a positive parameter. In this case, the zero sets are related locally to the zeros of functions in weighted Dirichlet spaces of the half-plane $\operatorname{Re} s>1/2$. Partial results are then obtained for the zeros of functions in $\Hp$ ($L^p$ analogues of $\Ht$) for $2<p<\infty$, based on certain contractive embeddings of $\Da$ in $\Hp$.

math.CV↗
arXiv2011

Interpolation and sampling in small Bergman spaces

Carleson measures and interpolating and sampling sequences for weighted Bergman spaces on the unit disk are described for weights that are radial and grow faster than the standard weights $(1-|z|)^{-α}$, $0<α<1$. These results make the Hardy space $H^2$ appear naturally as a "degenerate" endpoint case for the class of Bergman spaces under study.

math.CV↗
arXiv2011

A lower bound in Nehari's theorem on the polydisc

By theorems of Ferguson and Lacey (d=2) and Lacey and Terwilleger (d>2), Nehari's theorem is known to hold on the polydisc D^d for d>1, i.e., if H_ψis a bounded Hankel form on H^2(D^d) with analytic symbol ψ, then there is a function ϕin L^\infty(\T^d) such that ψis the Riesz projection of ϕ. A method proposed in Helson's last paper is used to show that the constant C_d in the estimate \|ϕ\|_\infty\le C_d \|H_ψ\| grows at least exponentially with d; it follows that there is no analogue of Nehari's theorem on the infinite-dimensional polydisc.

math.CV↗
arXiv2011

Hankel operators on Fock spaces and related Bergman kernel estimates

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on $\cn$. The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with Hörmander's $L^2$ estimates for the $\bar{\partial}$ operator are key ingredients in the proof.

math.FA↗
arXiv2010

$L^\infty$ to $L^p$ constants for Riesz projections

The norm of the Riesz projection from $L^\infty(\T^n)$ to $L^p(\T^n)$ is considered. It is shown that for $n=1$, the norm equals $1$ if and only if $p\le 4$ and that the norm behaves asymptotically as $p/(πe)$ when $p\to \infty$. The critical exponent $p_n$ is the supremum of those $p$ for which the norm equals $1$. It is proved that $2+2/(2^n-1)\le p_n <4$ for $n>1$; it is unknown whether the critical exponent for $n=\infty$ exceeds $2$.

math.FA↗
arXiv2010

Unitary discrete Hilbert transforms

Weighted discrete Hilbert transforms $(a_n)_n \mapsto \big(\sum_n a_n v_n/(λ_j-γ_n)\big)_j$ from $\ell^2_v$ to $\ell^2_w$ are considered, where $Γ=(γ_n)$ and $Λ=(λ_j)$ are disjoint sequences of points in the complex plane and $v=(v_n)$ and $w=(w_j)$ are positive weight sequences. It is shown that if such a Hilbert transform is unitary, then $Γ\cupΛ$ is a subset of a circle or a straight line, and a description of all unitary discrete Hilbert transforms is then given. A characterization of the orthogonal bases of reproducing kernels introduced by L. de Branges and D. Clark is implicit in these results: If a Hilbert space of complex-valued functions defined on a subset of $\CC$ satisfies a few basic axioms and has more than one orthogonal basis of reproducing kernels, then these bases are all of Clark's type.

math.CV↗
arXiv2009

Discrete Hilbert transforms on sparse sequences

Weighted discrete Hilbert transforms $(a_n)_n \mapsto \sum_n a_n v_n/(z-γ_n)$ from $\ell^2_v$ to a weighted $L^2$ space are studied, with $Γ=(γ_n)$ a sequence of distinct points in the complex plane and $v=(v_n)$ a corresponding sequence of positive numbers. In the special case when $|γ_n|$ grows at least exponentially, bounded transforms of this kind are described in terms of a simple relative to the Muckenhoupt $(A_2)$ condition. The special case when $z$ is restricted to another sequence $Λ$ is studied in detail; it is shown that a bounded transform satisfying a certain admissibility condition can be split into finitely many surjective transforms, and precise geometric conditions are found for invertibility of such two weight transforms. These results can be interpreted as statements about systems of reproducing kernels in certain Hilbert spaces of which de Branges spaces and model subspaces of $H^2$ are prime examples. In particular, a connection to the Feichtinger conjecture is pointed out. Descriptions of Carleson measures and Riesz bases of normalized reproducing kernels for certain "small" de Branges spaces follow from the results of this paper.

math.CV↗
arXiv2009

The Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive

The Bohnenblust--Hille inequality says that the $\ell^{\frac{2m}{m+1}}$-norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\C^n$ is bounded by $\| P\|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ denotes the supremum norm on the polydisc $\D^n$. The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be $C^m$ for some $C>1$. Combining this improved version of the Bohnenblust--Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc $\D^n$ behaves asymptotically as $\sqrt{(\log n)/n}$ modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies $\bigl\{\log n: n \text{a positive integer} \le N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$ as $N\to \infty$.

math.CV↗
arXiv2009

The Sidon constant for homogeneous polynomials

The Sidon constant for the index set of nonzero m-homogeneous polynomials P in n complex variables is the supremum of the ratio between the l^1 norm of the coefficients of P and the supremum norm of P in D^n. We present an estimate which gives the right order of magnitude for this constant, modulo a factor depending exponentially on m. We use this result to show that the Bohr radius for the polydisc D^n is bounded from below by a constant times sqrt((log n)/n).

math.CV↗
arXiv2008

Phase space distribution of Gabor expansions

We present an example of a complete and minimal Gabor system consisting of time-frequency shifts of a Gaussian, localized at the coordinate axes in the time-frequency plane (phase space). Asymptotically, the number of time-frequency shifts contained in a disk centered at the origin is only 2/pi times the number of points from the von Neumann lattice found in the same disk. Requiring a certain regular distribution in phase space, we show that our system has minimal density among all complete and minimal systems of time-frequency shifts of a Gaussian.

math.FA↗
arXiv2007

Integral means and boundary limits of Dirichlet series

We study the boundary behavior of functions in the Hardy spaces HD^p for ordinary Dirichlet series. Our main result, answering a question of H. Hedenmalm, shows that the classical F. Carlson theorem on integral means does not extend to the imaginary axis for functions in HD^\infty, i.e., for ordinary Dirichlet series in H^\infty of the right half-plane. We discuss an important embedding problem for HD^p, the solution of which is only known when p is an even integer. Viewing HD^p as Hardy spaces of the infinite-dimensional polydisc, we also present analogues of Fatou's theorem.

math.CV↗
arXiv2007

The perfect lens on a finite bandwidth

The resolution associated with the so-called perfect lens of thickness $d$ is $-2πd/\ln(|χ+2|/2)$. Here the susceptibility $χ$ is a Hermitian function in $H^2$ of the upper half-plane, i.e., a $H^2$ function satisfying $χ(-ω)=\bar{χ(ω)}$. An additional requirement is that the imaginary part of $χ$ be nonnegative for nonnegative arguments. Given an interval $I$ on the positive half-axis, we compute the distance in $L^\infty(I)$ from a negative constant to this class of functions. This result gives a surprisingly simple and explicit formula for the optimal resolution of the perfect lens on a finite bandwidth.

physics.optics↗
arXiv2006

Local interpolation in Hilbert spaces of Dirichlet series

We denote by $\Hp$ the Hilbert space of ordinary Dirichlet series with square-summable coefficients. The main result is that a bounded sequence of points in the half-plane $σ>1/2$ is an interpolating sequence for $\Hp$ if and only if it is an interpolating sequence for the Hardy space $H^2$ of the same half-plane. Similar local results are obtained for Hilbert spaces of ordinary Dirichlet series that relate to Bergman and Dirichlet spaces of the half-plane $σ>1/2$.

math.CV↗
arXiv2005

Bounds for the refractive indices of metamaterials

The set of realizable refractive indices as a function of frequency is considered. For passive media we give bounds for the refractive index variation in a finite bandwidth. Special attention is given to the loss and index variation in the case of left-handed materials.

physics.optics↗