arXiv subjects

Kristian Seip

Publications and source records attributed to Kristian Seip.

61 records · Page 4Linked to original sources

arXiv2005

An extremal problem related to negative refraction

We solve an extremal problem that arises in the study of the refractive indices of passive metamaterials. The problem concerns Hermitian functions in $H^2$ of the upper half-plane, i.e., $H^2$ functions satisfying $f(-x)=\bar{f(x)}$. An additional requirement is that the imaginary part of $f$ be nonnegative for nonnegative arguments. We parameterize the class of such functions whose real part is constant on an interval, and solve the problem of minimizing the imaginary part on the interval on which the function's real part takes a given constant value.

math.CV↗

arXiv2004

On Fourier frames

We solve the problem of Duffin and Schaeffer (1952) of characterizing those sequences of real frequencies which generate Fourier frames. Equivalently, we characterize the sampling sequences for the Paley-Wiener space. The key step is to connect the problem with de Branges' theory of Hilbert spaces of entire functions. We show that our description of sampling sequences permits us to obtain a classical inequality of H. Landau as a consequence of Pavlov's description of Riesz bases of complex exponentials and the John-Nirenberg theorem. Finally, we discuss how to transform our description into a working condition by relating it to an approximation problem for subharmonic functions. By this approach, we determine the critical growth rate of a nondecreasing function $ψ$ such that the sequence $\{\lamdda_k\}){k\in \Bbb Z}$ defined by $λ_k + ψ(λ_k)=k$ is a sampling.

math.CV↗

arXiv2003

Harmonic measure and uniform densities

We study two problems concerning harmonic measure on "champagne subdomains" of the unit disk. These domains are obtained by removing from the unit disk little disks around sequences of points with a uniform distribution with respect to the pseudohyperbolic metric of the unit disk. We find (I) a necessary and sufficient condition on the decay of the radii of the little disks for the exterior boundary to have positive harmonic measure, and (II) describe sampling and interpolating sequences for Bergman spaces in terms of the harmonic measure on such ``champagne subdomains''.

math.CV↗

arXiv2003

The constant of interpolation

We prove that a suitably adjusted version of Peter Jones' formula for interpolation by bounded holomorphic functions gives a sharp upper bound for what is known as the constant of interpolation. We show how this leads to precise and computable numerical bounds for this constant.

math.CV↗

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

Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's $(A_p)$ condition

We describe the complete interpolating sequences for the Paley-Wiener spaces $L^p_π$ ($1<p<\infty$) in terms of Muckenhoupt's $(A_p)$ condition. For $p=2$, this description coincides with those given by Pavlov (1979), Nikol'skii (1980), and Minkin (1992) of the unconditional bases of complex exponentials in $L^2(-π,π)$. While the techniques of these authors are linked to the Hilbert space geometry of $L^2_π$, our method of proof is based on turning the problem into one about boundedness of the Hilbert transform in certain weighted $L^p$ spaces of functions and sequences.

math.FA↗

arXiv1992

Density theorems for sampling and interpolation in the Bargmann-Fock space

We give a complete description of sampling and interpolation in the Bargmann-Fock space, based on a density concept of Beurling. Roughly speaking, a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von Neumann lattice, and similarly, a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von Neumann lattice.

math.CV↗