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Jari Taskinen

Publications and source records attributed to Jari Taskinen.

At least 19 recordsLinked to original sources
arXiv2026

On parameter-dependent inhomogeneous boundary-value problems in Sobolev spaces

We study a wide class of linear inhomogeneous boundary-value problems for $r$th order ODE-systems depending on a parameter $\mu$ belonging to a general metric space $\mathcal M$. The solutions belong to the Sobolev spaces $(W^{n+r}_p)^m$, $n\in\mathbb{N}\cup\{0\}$, $m, r \in \mathbb{N}$, $1\leq p\leq \infty$. The boundary conditions are of a most general form $By=c$, where $B$ is an arbitrary continuous operator from $(W^{n+r}_p)^m$ to $\mathbb{C}^{rm}$. Thus, they may contain derivatives of the unknown vector function of integer and/or fractional orders $\geq r$. We find necessary and sufficient conditions for the continuity of solutions with respect to the parameter $\mu$. We also prove that the solutions of the original problems can be approximated in the space $(W^{n+r}_p)^m$ by solutions of ODE-systems with polynomial coefficients, right-hand sides of the equation, and multipoint boundary conditions, which are independent of the original problem's right-hand sides.

math.CA↗
arXiv2026

Homogenization and operator estimates for Steklov problems in perforated domains

Let the set $\Omega_\varepsilon$ be obtained from the bounded domain $\Omega$ by removing a family of $\varepsilon$-periodically distributed identical balls. In $\Omega_\varepsilon$ one considers the Steklov spectral problem. It is known from [Girouard-Henrot-Lagac\'e, ARMA (2021)] that, if the radii of the holes shrink at a critical rate such that the surface area of a single hole is comparable to the volume of a periodicity cell, then, in the limit $\varepsilon \to 0$, the Steklov spectrum converges to the spectrum of the problem $-\Delta u=\lambda Q u$ on $\Omega$ with some weight $Q>0$. In the present work, we extend this result by proving, under fairly general assumptions on the locations and shapes of the holes, convergence of the associated resolvent operators in the operator norm topology, together with quantitative estimates for the Hausdorff distance between the spectra. The underlying domain $\Omega$ is not assumed to be bounded.

math.AP↗
arXiv2025

Parameter-dependent inhomogeneous boundary-value problems in Sobolev spaces

We study a wide class of linear inhomogeneous boundary-value problems for $r$th order ODE-systems depending on a parameter $\mu$ in a general metric space $\mathcal M$. The solutions belong to the Sobolev spaces $(W^{n+r}_p)^m$, $n\in\mathbb{N}\cup\{0\}$, $m, r \in \mathbb{N}$, $1\leq p\leq \infty$. The boundary conditions are of a most general form $By=c$, where $B$ is an arbitrary continuous operator from $(W^{n+r}_p)^m$ to $\mathbb{C}^{rm}$. They may thus contain derivatives of the unknown vector function of integer and/or fractional orders $\geq r$. We find necessary and sufficient conditions for the continuity of solutions with respect to the parameter $\mu$. We also prove that the solutions of the original problems can be approximated in the space $(W^{n+r}_p)^m$ by solutions of ODE-systems with polynomial coefficients and multipoint boundary conditions, which do not depend on the right-hand sides of the original problem.

math.CA↗
arXiv2025

Bergman Projections, Kernel $p$-Norm Estimates, and Toeplitz Operators with B\'{e}koll\'{e} and Bonami weights

In this paper, we establish entirely new $p$-norm estimates for reproducing kernels to characterize the bounded and compact Toeplitz operators $T_{\mu}$ acting between weighted B\'{e}koll\'{e}--Bonami Bergman spaces $A^p_u(\mathbb{D})$ and $A^q_u(\mathbb{D})$ for all positive exponents $0 < p, q < \infty$. These operator-theoretic properties are completely described in terms of generalized Berezin transforms, averaging functions, and Carleson measures. We introduce two explicit conditions on the weights to ensure the boundedness of the weighted Bergman projection $P_u$, generalizing results from Hilbert spaces to Banach spaces.Our work generalizes the main results of Tong, Li, and Arroussi \cite{TLA} from Hilbert spaces to the more general setting of Banach spaces.

math.CV↗
arXiv2025

Elliptic functions, Floquet transform and Bergman spaces on doubly periodic domains

We study Bergman spaces A^2(D), their kernels and Toeplitz operators on unbounded, doubly periodic domains D in the complex plane. We establish the mapping properties of the Floquet transform operator defined in A^2(D) and derive a general formula connecting the Bergman kernel and projection of the domain D to a kernel and projection on the bounded periodic cell B. As an application, we prove, for Toeplitz operators T_a with doubly periodic symbols, a spectral band formula, which describes the spectrum and essential spectrum of T_a in terms of the spectra of a family of Toeplitz-type operators on the cell B. Technical challenges arise from the fact that double quasiperiodic boundary conditions have to be taken into account in the definitions of the spaces and operators on the periodic cell B. This requires novel operator theoretic tools, which are based on modifications of certain elliptic functions, e.g. the Weierstrass p-function.

math.CV↗
arXiv2025

Quasiconformal symbols and projected composition operators

We study projected composition operators K_g with quasiconformal symbols g on weighted Bergman spaces on the open unit disc D. If the symbol were conformal, i.e.a M\"obius transform of D, the corresponding composition operator would be automatically invertible at least in standard weighted spaces. We show that the invertibility remains, if the Beltrami coefficient is small enough, in particular, it satisfies a certain vanishing condition at the boundary of the disc. We also consider the invertibility of K_g for symbols g which are conformal in an annulus { R < |z| < 1 }. The weight classes in our considerations include both standard and exponentially decreasing weights.

math.FA↗
arXiv2025

Pointwise lower bounds in growth spaces with little o conditions

Pointwise lower bounds on the open unit disc $\bbD$ for the sum of the moduli of two analytic functions $f$ and $g$ (or their derivatives) are known in several cases, like $f,g$ belonging to the Bloch space $\cB$, $BMOA$ or the weighted Hardy space $H_\omega^\infty$. We find complementary results of Ramey-Ullrich and Abakumov-Doubtsov for functions with little o conditions.

math.CV↗
arXiv2024

On Bergman-Toeplitz operators in periodic planar domains

We study spectra of Toeplitz operators $T_a $ with periodic symbols in Bergman spaces $A^2(\Pi)$ on unbounded periodic planar domains $\Pi$, which are defined as the union of infinitely many copies of the translated, bounded periodic cell $\varpi$. We introduce Floquet-transform techniques and prove a version of the band-gap-spectrum formula, which is well-known in the framework of periodic elliptic spectral problems and which describes the essential spectrum of $T_a$ in terms of the spectra of a family of Toepliz-type operators $T_{a,\eta}$ in the cell $\varpi$, where $\eta$ is the so-called Floquet variable. As an application, we consider periodic domains $\Pi_h$ containing thin geometric structures and show how to construct a Toeplitz operator $T_{\sf a}: A^2(\Pi_h) \to A^2(\Pi_h)$ such that the essential spectrum of $T_{\sf a}$ contains disjoint components which approximatively coincide with any given finite set of real numbers. Moreover, our method provides a systematic and illustrative way how to construct such examples by using Toeplitz operators on the unit disc $\mathbb{D}$ e.g. with radial symbols. Using a Riemann mapping one can then find a Toeplitz operator $T_a : A^2(\mathbb{D}) \to A^2(\mathbb{D})$ with a bounded symbol and with the same spectral properties as $T_{\sf a}$.

math.FA↗
arXiv2024

A geometric condition for the invertibility of Toeplitz operators on the Bergman space

Invertibility of Toeplitz operators on the Bergman space and the related Douglas problem are long standing open problems. In this paper we study the invertibility problem under the novel geometric condition on the image of the symbols, which relaxes the standard positivity condition. We show that under our geometric assumption, the Toeplitz operator $T_\varphi$ is invertible if and only if the Berezin transform of $|\varphi|$ is invertible in $L^{\infty}$. It is well known that the Douglas problem is still open for harmonic functions. We study a class of rather general harmonic polynomials and characterize the invertibility of the corresponding Toeplitz operators. We also give a number of related results and examples.

math.FA↗
arXiv2024

Bergman projection induced by radial weight acting on growth spaces

Let $\omega$ be a radial weight on the unit disc of the complex plane $\mathbb{D}$ and denote $\omega_x =\int_0^1 s^x \omega(s)\,ds$, $x\ge 0$, for the moments of $\omega$ and $\widehat{\omega}(r)=\int_r^1 \omega(s)\,ds$ for the tail integrals. A radial weight $\omega$ belongs to the class $\widehat{\mathcal{D}}$ if satisfies the upper doubling condition $$\sup_{0<r<1}\frac{\widehat{\omega}(r)}{\widehat{\omega}\left(\frac{1+r}{2}\right)}<\infty.$$ If $\nu$ or $\omega$ belongs to $\widehat{\mathcal{D}}$, it is described the boundedness of the Bergman projection $P_\omega$ induced by $\omega$ on the growth space $L^\infty_{\widehat{\nu}} =\{ f: \|f\|_{\infty,v}={ esssup}_{z\in\mathbb{D}} |f(z)|\widehat{\nu}(z)<\infty\}$ in terms of neat conditions on the moments and/or the tail integrals of $\omega$ and $\nu$. Moreover, it is solved the analogous problem for $P_\omega$ from $L^\infty_{\widehat{\nu}}$ to the Bloch type space $B^\infty_{\widehat{\nu}}$ of analytic functions such that $\sup_{z\in \mathbb{D}}(1-|z|)\widehat{\nu}(z) |f'(z)|<\infty.$ We also study similar questions for exponentially decreasing radial weights.

math.CV↗
arXiv2023

Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layer

We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some symmetries such as the so-called Fichera layer. The geometry we consider depends on two parameters gathered in some vector $\kappa=(\kappa_1,\kappa_2)$ which characterizes the domain at the edges. By exchanging the axes and/or modifying their orientations if necessary, it is sufficient to restrict the analysis to the cases $\kappa_1\ge0$ and $\kappa_2\in[-\kappa_1,\kappa_1]$. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to $\kappa$. In particular, we show that for a given $\kappa_1>0$, there is some $h(\kappa_1)>0$ such that discrete spectrum exists for $\kappa_2\in[-\kappa_1,0)\cup(h(\kappa_1),\kappa_1]$ whereas it is empty for $\kappa_2\in[0,h(\kappa_1)]$. The proofs rely on classical arguments of spectral theory such as the max-min principle. The main originality lies rather in the delicate use of the features of the geometry.

math.SP↗
arXiv2022

Acoustic waveguide with a dissipative inclusion

We consider the propagation of acoustic waves in a waveguide containing a penetrable dissipative inclusion. We prove that as soon as the dissipation, characterized by some coefficient $\eta$, is non zero, the scattering solutions are uniquely defined. Additionally, we give an asymptotic expansion of the corresponding scattering matrix when $\eta\to0^+$ (small dissipation) and when $\eta\to+\infty$ (large dissipation). Surprisingly, at the limit $\eta\to+\infty$, we show that no energy is absorbed by the inclusion. This is due to the so-called skin-effect phenomenon and can be explained by the fact that the field no longer penetrates into the highly dissipative inclusion. These results guarantee that in monomode regime, the amplitude of the reflection coefficient has a global minimum with respect to $\eta$. The situation where this minimum is zero, that is when the device acts as a perfect absorber, is particularly interesting for certain applications. However it does not happen in general. In this work, we show how to perturb the geometry of the waveguide to create 2D perfect absorbers in monomode regime. Asymptotic expansions are justified by error estimates and theoretical results are supported by numerical illustrations.

math.AP↗
arXiv2022

On solid cores and hulls of weighted Bergman spaces $A_{\mu}^1$

We consider weighted Bergman spaces $A_\mu^1$ on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces we characterize the solid core of $A_\mu^1$. Also, as a consequence of a characterization of solid $A_\mu^1$-spaces we show that, in the case of entire functions, there indeed exist solid $A_\mu^1$-spaces. The second part of the paper is restricted to the case of the unit disc and it contains a characterization of the solid hull of $A_\mu^1$, when $\mu $ equals the weighted Lebesgue measure with weight $v$. The results are based on a duality relation of weighted $A^1$- and $H^\infty$-spaces, the validity of which requires the assumption that $- \log v$ belongs to the class $\mathcal{W}_0$, studied in a number of publications; moreover, $v$ has to satisfy condition $(b)$, introduced by the authors. The exponentially decreasing weight $v(z) = \exp( -1 /(1-|z|)$ provides an example satisfying both assumptions.

math.FA↗
arXiv2021

On the Bergman projection and kernel in periodic planar domains

We study Bergman kernels $K_\Pi$ and projections $P_\Pi$ in unbounded planar domains $\Pi$, which are periodic in one dimension. In the case $\Pi$ is simply connected we write the kernel $K_\Pi$ in terms of a Riemann mapping $\varphi$ related to the bounded periodic cell $\varpi$ of the domain $\Pi$. We also introduce and adapt to the Bergman space setting the Floquet transform technique, which is a standard tool for elliptic spectral problems in periodic domains. We investigate the boundedness properties of the Floquet transform operators in Bergman spaces and derive a general formula connecting $P_\Pi$ to a projection on a bounded domain. We show how this theory can be used to reproduce the above kernel formula for $K_\Pi$. Finally, we consider weighted $L^p$-estimates for $P_\Pi$ in periodic domains.

math.CV↗
arXiv2021

Weak BMO and Toeplitz operators on Bergman spaces

Inspired by our previous work on the boundedness of Toeplitz operators, we introduce weak BMO and VMO type conditions, denoted by BWMO and VWMO, respectively, for functions on the open unit disc of the complex plane. We show that the average function of a function $f$ in BWMO is boundedly oscillating, and the analogous result holds for $f$ in VWMO. The result is applied for generalizations of known results on the essential spectra and norms of Toeplitz operators. Finally, we provide examples of functions satisfying the VWMO condition which are not in the classical VMO or even in BMO.

math.FA↗
arXiv2020

Toeplitz operators on the unit ball with locally integrable symbols

We study the boundedness of Toeplitz operators $T_\psi$ with locally integrable symbols on weighted harmonic Bergman spaces over the unit ball of $\mathbb{R}^n$. Generalizing earlier results for analytic function spaces, we derive a general sufficient condition for the boundedness of $T_\psi$ in terms of suitable averages of its symbol. We also obtain a similar "vanishing" condition for compactness. Finally, we show how these results can be transferred to the setting of the standard weighted Bergman spaces of analytic functions.

math.FA↗
arXiv2020

Band-gap structure of the spectrum of the water-wave problem in a shallow canal with a periodic family of deep pools

We consider the linear water-wave problem in a periodic channel $\Pi^h \subset \mathbb{R}^3$, which is shallow except for a periodic array of deep potholes in it. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the essential spectrum in the linear water-wave system, which includes the spectral Steklov boundary condition posed on the free water surface. We apply methods of asymptotic analysis, where the most involved step is the construction and analysis of an appropriate boundary layer in a neighborhood of the joint of the potholes with the thin part of the channel. Consequently, the existence of a spectral gap for small enough $h$ is proven.

math.AP↗