We consider the dilation property of the modulation spaces $M^{p,q}$. Let $D_λ:f(t)\mapsto f(λt)$ be the dilation operator, and we consider the behavior of the operator norm $\|D_λ\|_{M^{p,q}\to M^{p,q}}$ with respect to $λ$. Our result determines the best order for it, and as an application, we establish the optimality of the inclusion relation between the modulation spaces and Besov space, which was proved by Toft.
We consider a dilation operator on Besov spaces $(B^s_{r,t}(K))$ over local fields and estimate an operator norm on such a field for $s > σ_r = \text{max}\big(\frac{1}{r} -1,~0\big)$ which depends on the constant $k$ unlike the case of Euclidean spaces. In $\mathbb{R}^n$, it is independent of constant. A constant $k$ appears for liming case $s=0$ and $s=σ_r$. In case of local fields, the limig case is still open. Further we also estimate the localization property of Besov spaces over local fields.
We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices $A \in {\rm GL}(d,\mathbb{R})$, with the goal of clarifying when two such matrices induce the same scale of Besov spaces. For this purpose, we first establish that anisotropic Besov spaces have an alternative description as decomposition spaces. This result allows to relate properties of function spaces to combinatorial properties of the underlying coverings. This principle is applied to the question of classifying dilation matrices. It turns out the scales of homogeneous and inhomogeneo…
In this paper, we present more regularity conditions which ensure the boundedness of dilation operators on Besov and Triebel-Lizorkin spaces equiped with general weights.
This paper considers coorbit spaces parametrized by mixed, weighted Lebesgue spaces with respect to the quasi-regular representation of the semi-direct product of Euclidean space and a suitable matrix dilation group. The class of dilation groups that we allow, the so-called integrably admissible dilation groups, contains the matrix groups yielding an irreducible, square-integrable quasi-regular representation as a proper subclass. The obtained scale of coorbit spaces extends therefore the well-studied wavelet coorbit spaces associated to discrete series representations. We show that for any in…
A proof using the theory of completely positive maps is given to the fact that if $A \in M_2$, or $A \in M_3$ has a reducing eigenvalue, then every bounded linear operator $B$ with $W(B) \subseteq W(A)$ has a dilation of the form $I \otimes A$. This gives a unified treatment for the different cases of the result obtained by researchers using different techniques.
Researchers have identified complex matrices $A$ such that a bounded linear operator $B$ acting on a Hilbert space will admit a dilation of the form $A \otimes I$ whenever the numerical range inclusion relation $W(B) \subseteq W(A)$ holds. Such an operator $A$ and the identity matrix will span a maximal operator system, i.e., every unital positive map from ${\rm span} \{I, A, A^*\}$ to ${\cal B}({\cal H})$, the algebra of bounded linear operators acting on a Hilbert space ${\cal H}$, is completely positive. In this paper, we identify $m$-tuple of matrices ${\bf A} = (A_1, \dots, A_m)$ such tha…
In this paper we give a sharp estimate on the norm of the scaling operator $U_λf(x)=f(λx)$ acting on the weighted modulation spaces $\M{p,q}{s,t}(\R^{d})$. In particular, we recover and extend recent results by Sugimoto and Tomita in the unweighted case. As an application of our results, we estimate the growth in time of solutions of the wave and vibrating plate equations, which is of interest when considering the well posedeness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.
We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in ${\rm L}^2(\mathbb{R}^d)$, but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of ${\rm GL}(\mathbb{R}^d)$, the so-called {\em dilation group}.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces s…
Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces $\dot{\mathbf{F}}^α_{p,q}$ for the endpoint case of $p = \infty$ and the full scale of parameters $α\in \mathbb{R}$ and $q \in (0,\infty]$. In particular, a Peetre-type characterization of the anisotropic Besov space $\dot{\mathbf{B}}^α_{\infty,\infty} = \dot{\mathbf{F}}^α_{\infty,\infty}$ is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in $\dot{\mathbf{F}}^α_{\infty,q}$.
In this paper, we show the inclusion and the density of the Schwartz space in Besov-Dunkl spaces and we prove an interpolation formula for these spaces by the real method. We give another characterization for these spaces by convolution. Finally, we establish further results concerning integrability of the Dunkl transform of function in a suitable Besov-Dunkl space.
This article describes how the ideas promoted by the fundamental papers published by M. Frazier and B. Jawerth in the eighties have influenced subsequent developments related to the theory of atomic decompositions and Banach frames for function spaces such as the modulation spaces and Besov-Triebel-Lizorkin spaces. Both of these classes of spaces arise as special cases of two different, general constructions of function spaces: coorbit spaces and decomposition spaces. Coorbit spaces are defined by imposing certain decay conditions on the so-called voice transform of the function/distribution u…
In this article we construct affine systems that provide a simultaneous atomic decomposition for a wide class of functional spaces including the Lebesgue spaces $L^p(\Rdst)$, $1<p<+\infty$. The novelty and difficulty of this construction is that we allow for non-lattice translations. We prove that for an arbitrary expansive matrix $A$ and any set $Λ$ - satisfying a certain spreadness condition but otherwise irregular- there exists a smooth window whose translations along the elements of $Λ$ and dilations by powers of $A$ provide an atomic decomposition for the whole range of the anisotropic Tr…
The inclusions between the Besov spaces $B^q$, the Bloch space $\mathcal{B}$ and the standard weighted Bergman spaces $A^p_α$ are completely understood, but the norms of the corresponding inclusion operators are in general unknown. In this work, we compute or estimate asymptotically the norms of these inclusions.
Within the new concept of a local iterated function system (local IFS), we consider a class of attractors of such IFSs, namely those that are graphs of functions. These new functions are called local fractal functions and they extend and generalize those that are currently found in the fractal literature. For a class of local fractal functions, we derive explicit conditions for them to be elements of Besov and Triebel--Lizorkin spaces. These two scales of functions spaces play an important role in interpolation theory and for certain ranges of their defining parameters describe many classical …
This paper is concerned with the study of Besov-type decomposition spaces, which are scales of spaces associated to suitably defined coverings of the euclidean space $\mathbb{R}^d$, or suitable open subsets thereof. A fundamental problem in this domain, that is currently not well understood, is deciding when two different coverings give rise to the same scale of decomposition spaces. In this paper, we establish a coarse geometric approach to this problem, and show how it specializes for the case of wavelet coorbit spaces associated to a particular class of matrix groups $H < GL(\mathbb{R}^d)$ …
This paper is concerned with the characterization of $α$-modulation spaces by Banach frames, i.e., stable and redundant non-orthogonal expansions, constituted of functions obtained by a suitable combination of translation, modulation and dilation of a mother atom. In particular, the parameter $α\in [0,1]$ governs the dependence of the dilation factor on the frequency. The result is achieved by exploiting intrinsic properties of localization of such frames. The well-known Gabor and wavelet frames arise as special cases ($α= 0$) and limiting case ($ α\to 1)$, to characterize respectively modulat…
We define and investigate $α$-modulation spaces $M_{p,q}^{s,α}(G)$ associated to a step two stratified Lie group $G$ with rational structure constants. This is an extension of the Euclidean $α$-modulation spaces $M_{p,q}^{s,α}(\mathbb{R}^n)$ that act as intermediate spaces between the modulation spaces ($α= 0$) in time-frequency analysis and the Besov spaces ($α= 1$) in harmonic analysis. We will illustrate that the the group structure and dilation structure on $G$ affect the boundary cases $α= 0,1$ where the spaces $M_{p,q}^{s}(G)$ and $\mathcal{B}_{p,q}^{s}(G)$ have non-standard translation …
First, we consider some fundamental properties including dual spaces, complex interpolations of $α$-modulation spaces $M^{s,α}_{p,q}$ with $0<p,q \le \infty$. Next, necessary and sufficient conditions for the scaling property and the inclusions between $α_1$-modulation and $α_2$-modulation spaces are obtained. Finally, we give some criteria for $α$-modulation spaces constituting multiplication algebra. As a by-product, we show that there exists an $α$-modulation space which is not an interpolation space between modulation and Besov spaces.
An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V from H to K such that C=V^*TV. A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp factor $\vartheta(d)$, of all d-by-d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space. An analytic formula for $\vartheta(d)$ is derived, which as a by-product gives new probabilistic results for the binomial and beta distributions. Dilating to commuting operators has consequences…
Let $\mathcal{H}$ be a space of analytic functions on the unit ball $\mathbb B_d$ in $\mathbb C^d$ with multiplier algebra $\mathrm{Mult}(\mathcal{H})$. A function $f\in \mathcal{H}$ is called cyclic if the set $[f]$, the closure of $\{φf:φ\in \mathrm{Mult}(\mathcal{H})\}$, equals $\mathcal{H}$. For multipliers we also consider a weakened form of the cyclicity concept. Namely for $n\in \mathbb N_0$ we consider the classes $$\mathcal{C}_n(\mathcal{H})=\{φ\in \mathrm{Mult}(\mathcal H):φ\ne 0, [φ^n]=[φ^{n+1}]\}.$$ Many of our results hold for $N$:th order radially weighted Besov spaces on $\mathb…
