arXiv subjects
Publications and source records attributed to Antonio Galbis.
We study composition operators whose symbols are suitable perturbations of the identity and which act between different weighted modulation classes. We consider both modulation spaces formed by tempered distributions and those whose elements are ultradistributions defined in terms of a subadditive weight.
We analyze the behavior of the iterates of composition operators defined by polynomials acting on global classes of ultradifferentiable functions of Beurling type and being invariant under Fourier transform. We characterize the polynomials $ψ$ for which the sequence of iterates is equicontinuous between two different Gelfand-Shilov spaces. For the particular case in which the weight $ω$ is equivalent to a power of the logarithm, the result obtained characterizes the polynomials $ψ$ for which the composition operator $C_ψ$ is power bounded in ${\mathcal S}_ω({\mathbb R}).$ Unlike the composition operators in Schwartz class, the Waelbroek spectrum of an operator $C_ψ$, being $ψ$ a polynomial of degree greater than one lacking fixed points is never compact. We focus on the problem of convergence of Neumann series. We deduce the continuity of the resolvent operator between two different Gelfand-Shilov classes for polynomials $ψ$ lacking fixed points. Concerning polynomials of second degree the most interesting case is the one in which the polynomial only has one fixed point: we provide some restrictions on the indices $d, d'$ that are necessary for the resolvent operator to be continuous between the Gelfand-Shilov classes $Σ_d$ and $Σ_{d'}.$
The usefulness of time-frequency analysis methods in the study of quasicrystals was pointed out in a previous paper, where we proved that a tempered distribution $μ$ on ${\mathbb R}^d$ whose Wigner transform is a measure supported on the cartesian product of two uniformly discrete sets in ${\mathbb R}^d$ is a Fourier quasicrystal. In this paper we go further in this direction using the matrix-Wigner transforms to detect quasicrystal structures. The results presented here cover essentially all the most important quadratic time-frequency distributions, and are obtained considering two different (disjoint) classes of matrix-Wigner transforms, discussed respectively in Theorems 1 and 2. The transforms considered in Theorem 1 include the classical Wigner transform, as well as all the time-frequency representations of matrix-Wigner type belonging to the Cohen class. On the other hand Theorem 2, which does not apply to the classical Wigner, has, as main example, the Ambiguity function. In this second case we only suppose that the support of the matrix-Wigner transform of $μ$ is contained in the cartesian product of two discrete sets, obtaining that both the support and the spectrum of $μ$ are uniformly discrete.
We obtain a complete characterization of the bounded Hausdorff operators acting on a Fock space $F^p_α$ and taking its values into a larger one $F^q_α,\ 0 < p \leq q \leq \infty,$ as well as some necessary or sufficient conditions for a Hausdorff operator to transform a Fock space into a smaller one. Some results are written in the context of mixed norm Fock spaces. Also the compactness of Hausdorff operators on a Fock space is characterized. The compactness result for Hausdorff operators on the Fock space $F^\infty_α$ is extended to more general Banach spaces of entire functions with weighted sup norms defined in terms of a radial weight and conditions for the Hausdorff operators to become $p$-summing are also included.
We study composition operators on global classes of ultradifferentiable functions of Beurling type invariant under Fourier transform. In particular, for the classical Gelfand-Shilov classes $Σ_d,\ d > 1,$ we prove that a necessary condition for the composition operator $f\mapsto f\circ ψ$ to be well defined is the boundedness of $ψ'.$ We find the optimal index $d'$ for which $C_ψ(Σ_d({\mathbb R}))\subset Σ_{d'}({\mathbb R})$ holds for any non-constant polynomial $ψ.$
Quasicrystals are tempered distributions $μ$ which satisfy symmetric conditions on $μ$ and $\widehat μ$. This suggests that techniques from time-frequency analysis could possibly be useful tools in the study of such structures. In this paper we explore this direction considering quasicrystals type conditions on time-frequency representations instead of separately on the distribution and its Fourier transform. More precisely we prove that a tempered distribution $μ$ on ${\mathbb R}^d$ whose Wigner transform, $W(μ)$, is supported on a product of two uniformly discrete sets in ${\mathbb R}^d$ is a quasicrystal. This result is partially extended to a generalization of the Wigner transform, called matrix-Wigner transform which is defined in terms of the Wigner transform and a linear map $T$ on ${\mathbb R}^{2d}$.
An estimate for the norm of selfadjoint Toeplitz operators with a radial, bounded and integrable symbol is obtained. This emphasizes the fact that the norm of such operator is strictly less than the supremum norm of the symbol. Consequences for time-frequency localization operators are also given.
We describe the proper closed invariant subspaces of the integration operator when it acts continuously on countable intersections and countable unions of weighted Banach spaces of holomorphic functions on the unit disc or the complex plane. Applications are given to the case of Korenblum type spaces and Hörmander algebras of entire functions.
We show that a smooth function $f$ on $\mathbf R^d$ belongs to the Pilipovi{ć} space $\mathcal H_{\flat _σ}(\mathbf R^d)$ or the Pilipovi{ć} space $\mathcal H_{0,\flat _σ}(\mathbf R^d)$, if and only if the $L^p$ norm of $H_d^Nf$ for $N\ge 0$, satisfy certain types of estimates. Here $H_d=|x|^2-Δ_x$ is the harmonic oscillator.
We study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum reduces to 0, while the spectrum of any non mean ergodic composition operator with a polynomial always contains the closed unit disc except perhaps the origen. We obtain a complete description of the spectrum of the composition operator with a quadratic polynomial or a cubic polynomial with positive leading coefficient.
Using the matrix representation of Fourier integral operators with respect to a Gabor frame, we study their compactness on weighted modulation spaces. As a consequence, we recover and improve some compactness results for pseudodifferential operators.
In this paper we study the dynamics of the composition operators defined in the Schwartz space $\mathcal{S}(\mathbb{R})$ of rapidly decreasing functions. We prove that such an operator is never supercyclic and, for monotonic symbols, it is power bounded only in trivial cases. For a polynomial symbol $φ$ of degree greater than one we show that the operator is mean ergodic if and only if it is power bounded and this is the case when $φ$ has even degree and lacks fixed points. We also discuss the spectrum of composition operators.
We prove that every unconditionally summable sequence in a Hilbert space can be factorized as the product of a square summable scalar sequence and a Bessel sequence. Some consequences on the representation of unconditionally convergent multipliers are obtained, thus providing positive answers to a conjecture by Balazs and Stoeva in some particular cases.
We study composition operators on the Schwartz space of rapidly decreasing functions. We prove that such a composition operator is never a compact operator and we obtain necessary or sufficient conditions for the range of the composition operator to be closed. These conditions are expressed in terms of multipliers for the Schwartz class and the closed range property of the corresponding operator considered in the space of smooth functions.
We consider the counter images $\maclJ (\rr d)$ and $\maclJ _0(\rr d)$ of entire functions with exponential and almost exponential bounds, respectively, under the Bargmann transform, and we characterize them by estimates of powers of the harmonic oscillator. We also consider the Pilipovi{ć} spaces $\bsycalS _s(\rr d)$ and $\bsySig _s(\rr d)$ when $0<s<1/2$ and deduce their images under the Bargmann transform.
We give a construction of Gabor type frames for suitable separable subspaces of the non-separable Hilbert spaces $AP_2({\mathbb R})$ of almost periodic functions of one variable. Furthermore we determine a non-countable generalized frame for the whole space $AP_2({\mathbb R}).$ We show furthermore that Bessel-type estimates hold for the $AP$ norm with respect to a countable Gabor system using suitable almost periodic norms of sequencies.
Frames and Bessel sequences in Fréchet spaces and their duals are defined and studied. Their relation with Schauder frames and representing systems is analyzed. The abstract results presented here, when applied to concrete spaces of analytic functions, give many examples and consequences about sampling sets and Dirichlet series expansions.
Let $v$ be a submultiplicative weight. Then we prove that $v$ satisfies GRS-condition, if and only if $v\cdot e^{-\ep |\cdo |}$ is bounded for every positive $\ep$. We use this equivalence to establish identification properties between weighted Lebesgue spaces, and between certain modulation spaces and Gelfand-Shilov spaces.