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Jari Taskinen

Publications and source records attributed to Jari Taskinen.

35 records · Page 2Linked to original sources
arXiv2020

On the boundedness of Toeplitz operators with radial symbols over weighted sup-norm spaces of holomorphic functions

We prove sufficient conditions for the boundedness and compactness of Toeplitz operators $T_a$ in weighted sup-normed Banach spaces $H_v^\infty$ of holomorphic functions defined on the open unit disc $\mathbb{D}$ of the complex plane; both the weights $v$ and symbols $a$ are assumed to be radial functions on $\mathbb{D}$. In an earlier work by the authors it was shown that there exists a bounded, harmonic (thus non-radial) symbol $a$ such that $T_a$ is not bounded in any space $H_v^\infty$ with an admissible weight $v$. Here, we show that a mild additional assumption on the logarithmic decay rate of a radial symbol $a$ at the boundary of $\mathbb{D} $ guarantees the boundedness of $T_a$. The sufficient conditions for the boundedness and compactness of $T_a$, in a number of variations, are derived from the general, abstract necessary and sufficient condition recently found by the authors. The results apply for a large class of weights satisfying the so called condition$(B)$, which includes in addition to standard weight classes also many rapidly decreasing weights.

math.FA↗
arXiv2020

Surface waves in a channel with thin tunnels and wells at the bottom: non-reflecting underwater tomography

We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given.

math.AP↗
arXiv2019

Floquet Problem and Center Manifold Reduction for Ordinary Differential Operators with Periodic Coefficients in Hilbert Spaces

A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite dimensional system of ordinary differential equations with constant coefficients and an infinite dimensional part whose solutions have better properties in a certain sense. This complements the well-known asymptotic results for periodic hypoelliptic problems in cylinders (Kuchment) and for elliptic problems in quasicylinders (Nazarov). As an application we give a center manifold reduction for a class of non-linear ordinary differential equations in Hilbert spaces with periodic coefficients. This result generalizes the known case with constant coefficients (Mielke).

math.AP↗
arXiv2018

Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain

We consider the Robin Laplacian in the domains $Ω$ and $Ω^\varepsilon$, $\varepsilon >0$, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient $a$ is large enough, the spectrum of the problem in $Ω$ is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain $Ω^\varepsilon$ is discrete. However, our results reveal the strange behavior of the discrete spectrum as the blunting parameter $\varepsilon$ tends to 0: we construct asymptotic forms of the eigenvalues and detect families of "hardly movable" and "plummeting" ones. The first type of the eigenvalues do not leave a small neighborhood of a point for any small $\varepsilon > 0$ while the second ones move at a high rate $O(|\ln \varepsilon|)$ downwards along the real axis $\mathbb{R}$ to $ -\infty$. At the same time, any point $λ\in \mathbb{R}$ is a "blinking eigenvalue", i.e., it belongs to the spectrum of the problem in $Ω^\varepsilon$ almost periodically in the $|\ln \varepsilon|$-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

math.AP↗
arXiv2018

"Blinking eigenvalues" of the Steklov problem generate the continuous spectrum in a cuspidal domain

We study the Steklov spectral problem for the Laplace operator in a bounded domain $Ω\subset \mathbb{R}^d$, $d \geq 2$, with a cusp such that the continuous spectrum of the problem is non-empty, and also in the family of bounded domains $Ω^\varepsilon \subset Ω$, $\varepsilon > 0$, obtained from $Ω$ by blunting the cusp at the distance of $\varepsilon$ from the cusp tip. While the spectrum in the blunted domain $Ω^\varepsilon$ consists for a fixed $\varepsilon$ of an unbounded positive sequence $\{ λ_j^\varepsilon \}_{j=1}^\infty$ of eigenvalues, we single out different types of behavior of some eigenvalues as $\varepsilon \to +0$: in particular, stable, blinking, and gliding families of eigenvalues are found. We also describe a mechanism which transforms the family of the eigenvalue sequences into the continuous spectrum of the problem in $Ω$, when $\varepsilon \to +0$.

math.AP↗
arXiv2018

Schauder bases and the decay rate of the heat equation

We consider the classical Cauchy problem for the linear heat equation and integrable initial data in the Euclidean space $\mathbb{R}^N$. In the case $N=1$ we show that given a weighted $L^p$-space $L_w^p(\mathbb{R})$ with $1 \leq p < \infty$ and a fast growing weight $w$, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_ w^p(\mathbb{R})$ with the following property: given a positive integer $m $ there exists $n_m > 0$ such that, if the initial data $f$ belongs to the closed linear space of $e_n$ with $n \geq n_m$, then the decay rate of the solution of the heat equation is at least $t^{-m}$. The result is also generalized to the case $N >1$ with a slightly weaker formulation. The proof is based on a construction of a Schauder basis of $L_w^p( \mathbb{R}^N)$, which annihilates an infinite sequence of bounded functionals.

math.FA↗
arXiv2017

Monomial basis in Korenblum type spaces of analytic functions

It is shown that the monomials $Λ=(z^n)_{n=0}^{\infty}$ are a Schauder basis of the Fréchet spaces $A_+^{-γ}, \ γ\geq 0,$ that consists of all the analytic functions $f$ on the unit disc such that $(1-|z|)^μ|f(z)|$ is bounded for all $μ> γ$. Lusky \cite{L} proved that $Λ$ is not a Schauder basis for the closure of the polynomials in weighted Banach spaces of analytic functions of type $H^{\infty}$. A sequence space representation of the Fréchet space $A_+^{-γ}$ is presented. The case of (LB)-spaces $A_{-}^{-γ}, \ γ> 0,$ that are defined as unions of weighted Banach spaces is also studied.

math.FA↗
arXiv2017

Solid hulls and cores of weighted $H^\infty$-spaces

We determine the solid hull and solid core of weighted Banach spaces $H_v^\infty$ of analytic functions $f$ such that $v|f|$ is bounded, both in the case of the holomorphic functions on the disc and on the whole complex plane, for a very general class of radial weights $v$. Precise results are presented for concrete weights on the disc that could not be treated before. It is also shown that if $H_v^\infty$ is solid, then the monomials are an (unconditional) basis of the closure of the polynomials in $H_v^\infty$. As a consequence $H_v^\infty$ does not coincide with its solid hull and core in the case of the disc. An example shows that this does not hold for weighted spaces of entire functions.

math.FA↗
arXiv2017

On generalized Toeplitz and little Hankel operators on Bergman spaces

We find a concrete integral formula for the class of generalized Toeplitz operators $T_a$ in Bergman spaces $A^p$, $1<p<\infty$, studied in an earlier work by the authors. The result is extended to little Hankel operators. We give an example of an $L^2$-symbol $a$ such that $T_{|a|} $ fails to be bounded in $A^2$, although $T_a : A^2 \to A^2$ is seen to be bounded by using the generalized definition. We also confirm that the generalized definition coincides with the classical one whenever the latter makes sense.

math.FA↗
arXiv2016

Solid hulls of weighted Banach spaces of entire functions

Given a continuous, radial, rapidly decreasing weight $v$ on the complex plane $\mathbf{C}$, we study the solid hull of its associated weighted space $H_v^\infty(\mathbf{C})$ of all the entire functions $f$ such that $v|f|$ is bounded. The solid hull is found for a large class of weights satisfying the condition (B) of Lusky. Precise formulations are obtained for weights of the form $v(r)=\exp(-ar^p), a>0, p>0$. Applications to spaces of multipliers are included.

math.FA↗
arXiv2014

A note about Volterra operators on weighted Banach spaces of entire functions

We characterize boundedness, compactness and weak compactness of Volterra operators acting between different weighted Banach spaces of entire functions with weighted sup-norms in terms of the symbol g. Thus we complement recent work by Bassallote, Contreras, Hernández-Mancera, Martín and Paul for spaces of holomorphic functions on the disc and by Constantin and Peláez for reflexive weighted Fock spaces.

math.FA↗
arXiv2013

Spectral gaps for the linear surface wave model in periodic channels

We consider the linear water-wave problem in a periodic channel which consists of infinitely many identical containers connected with apertures of width $ε$. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the continuous spectrum. We show that the for small apertures there exists a large number of gaps and also find asymptotic formulas for the position of the gaps as $ε\to 0$: the endpoints are determined within corrections of order $ε^{3/2}$. The width of the first bands is shown to be $O(ε)$. Finally, we give a sufficient condition which guarantees that the spectral bands do not degenerate into eigenvalues of infinite multiplicity.

math.SP↗
arXiv2004

Anomalous scaling for 3d Cahn-Hilliard fronts

We prove the stability of the one dimensional kink solution of the Cahn-Hilliard equation under d-dimensional perturbations for d > 2. We also establish a novel scaling behavior of the large time asymptotics of the solution. The leading asymptotics of the solution is characterized by a length scale proportional to the cubic root of t instead of the usual square root of t scaling typical to parabolic problems.

math-ph↗
arXiv1994

The subspace problem for weighted inductive limits of spaces of holomorphic functions

We construct a countable inductive limit of weighted Banach spaces of holomorphic functions, which is not a topological subspace of the corresponding weighted inductive limit of spaces of continuous functions. The main step of our construction, using a special sequence of outer holomorphic functions, shows that a certain sequence space is isomorphic to a complemented subspace of a weighted space of holomorphic functions in two complex variables. This example solves in the negative a well-known open problem raised by Bierstedt, Meise and Summers.

math.FA↗
arXiv1992

Structure of local Banach spaces of locally convex spaces

We show that a continuous bilinear mapping P: C(I) \times C(I) \to C(I) can be presented in the form P(f,g) = B((Af)(Ag)), where A and B are bounded linear operators on C(I) and multiplication is defined pointwise, if and only if for all t in I the bilinear form (f,g) -> P(f,g)(t) is integral on C(I) times C(I) and depends in a sense continuously on t. To this end we construct a continuous surjection phi : I \to I^2 admitting a regular averaging operator in the sense of Pelczynski.

math.FA↗