Gelfand-Shilov Spaces Research Articles

Iterates of composition operators on global spaces of ultradifferentiable functions

We analyze the behaviour 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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} for which the sequence of iterates is equicontinuous between two different Gelfand–Shilov spaces. For the particular case in which the weight ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} is equivalent to a power of the logarithm, the result obtained characterizes the polynomials ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} for which the composition operator Cψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_\\psi $$\\end{document} is power bounded in Sω(R).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {S}}}_\\omega ({{\\mathbb {R}}}).$$\\end{document} Unlike the composition operators in the Schwartz class, the Waelbroek spectrum of an operator Cψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_\\psi $$\\end{document}, ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} being a polynomial of degree greater than one lacking fixed points is never compact. We focus on the problem of the convergence of Neumann series. We deduce the continuity of the resolvent operator between two different Gelfand–Shilov classes for polynomials ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} 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′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d, d'$$\\end{document} that are necessary for the resolvent operator to be continuous between the Gelfand–Shilov classes Σd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma _d$$\\end{document} and Σd′.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma _{d'}.$$\\end{document}

Systems of differential operators in time-periodic Gelfand–Shilov spaces

Systems of differential operators in time-periodic Gelfand–Shilov spaces

Globally solvable time-periodic evolution equations in Gelfand–Shilov classes

AbstractIn this paper we consider a class of evolution operators with coefficients depending on time and space variables $$(t,x) \in {\mathbb {T}}\times {\mathbb {R}}^n$$ ( t , x ) ∈ T × R n , where $${\mathbb {T}}$$ T is the one-dimensional torus, and prove necessary and sufficient conditions for their global solvability in (time-periodic) Gelfand–Shilov spaces. The argument of the proof is based on a characterization of these spaces in terms of the eigenfunction expansions given by a fixed self-adjoint, globally elliptic differential operator on $${\mathbb {R}}^n$$ R n .

Ridgelet transform on Gelfand–Shilov‐type spaces

We construct Gelfand–Shilov spaces of type and discuss the continuity and boundedness of the ridgelet transform. We then extend the analysis to generalized functions and explore its application to the space of tempered ultradistributions. Finally, we extend our findings to Gevrey functions of compact support. We also discuss an application of the ridgelet transform in image processing.

On the non-triviality of anisotropic Roumieu Gelfand–Shilov spaces and inclusion between them

Abstract The purpose of this work is to prove the non-triviality of anisotropic Roumieu Gelfand–Shilov spaces S { N } { M } ⁢ ( ℝ n ) {S^{\{M\}}_{\{N\}}(\mathbb{R}^{n})} , and to establish the inclusion between them.

Fourier Characterizations and Non-triviality of Gelfand–Shilov Spaces, with Applications to Toeplitz Operators

We find growth estimates on functions and their Fourier transforms in the one-parameter Gelfand–Shilov spaces S_s, S^sigma , Sigma _s and Sigma ^sigma . We obtain characterizations for these spaces and their duals in terms of estimates of short-time Fourier transforms. We determine conditions on the symbols of Toeplitz operators under which the operators are continuous on the one-parameter spaces. Lastly, it is determined that Sigma _s^sigma is nontrivial if and only if s+sigma > 1.

Microlocal analysis for Gelfand–Shilov spaces

We introduce an anisotropic global wave front set of Gelfand–Shilov ultradistributions with different indices for regularity and decay at infinity. The concept is defined by the lack of super-exponential decay along power type curves in the phase space of the short-time Fourier transform. This wave front set captures the phase space behaviour of oscillations of power monomial type, a k a chirp signals. A microlocal result is proved with respect to pseudodifferential operators with symbol classes that give rise to continuous operators on Gelfand–Shilov spaces. We determine the wave front set of certain series of derivatives of the Dirac delta, and exponential functions.

A note on the barrelledness of weighted PLB-spaces of ultradifferentiable functions

In this note, we consider weighted PLB-spaces of ultradifferentiable functions defined via a weight function and a weight system, as introduced in our previous work (2022). We provide a complete characterization of when these spaces are ultrabornological and barrelled in terms of the defining weight system, thereby improving the main result of our above mentioned work. In particular, we obtain that the multiplier space of the Gelfand–Shilov space \Sigma^r_s(\mathbb{R}^d) of Beurling type is ultrabornological, whereas the one of the Gelfand–Shilov space \mathcal{S}^r_s(\mathbb{R}^d) of Roumieu type is not barrelled.

Uncertainty principles with error term in Gelfand–Shilov spaces

In this note, an alternative approach to establish observability for semigroups based on their smoothing properties is presented. The results discussed here reproduce some of those recently obtained in [arXiv:2112.01788], but the current proof allows to get rid of several technical assumptions by following the standard complex analytic approach established by Kovrijkine combined with an idea from [arXiv:2201.02370].

Uncertainty principles in Gelfand-Shilov spaces and null-controllability

We provide new uncertainty principles for functions in a general class of Gelfand-Shilov spaces. These results apply, in particular, with the classical Gelfand-Shilov spaces as well as for spaces of functions with weighted Hermite expansions. Thanks to these uncertainty principles, we derive null-controllability results for evolution equations with adjoint systems enjoying smoothing effects in specific Gelfand-Shilov spaces. More precisely, we consider control subsets which are thick with respect to a quasi linearly growing density and establish sufficient conditions on the growth of the density to ensure null-controllability of these evolution equations.

Semigroups for quadratic evolution equations acting on Shubin–Sobolev and Gelfand–Shilov spaces

We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is the Weyl quantization of a homogeneous quadratic form with non-negative definite real part. The solution semigroup is shown to be strongly continuous on several spaces: the Shubin-Sobolev spaces, the Schwartz space, the tempered distributions, the equal index Beurling type Gelfand-Shilov spaces and their dual ultradistribution spaces.

The Cauchy problem for 3-evolution equations with data in Gelfand–Shilov spaces

We consider the Cauchy problem for a third-order evolution operator P with (t, x)-depending coefficients and complex-valued lower-order terms. We assume the initial data to be Gevrey regular with an exponential decay at infinity, that is, the data belong to some Gelfand–Shilov space of type {mathscr {S}}. Under suitable assumptions on the decay at infinity of the imaginary parts of the coefficients of P we prove the existence of a solution with the same Gevrey regularity of the data and we describe its behavior for |x| rightarrow infty .

SPECTRAL INEQUALITIES FOR COMBINATIONS OF HERMITE FUNCTIONS AND NULL-CONTROLLABILITY FOR EVOLUTION EQUATIONS ENJOYING GELFAND–SHILOV SMOOTHING EFFECTS

AbstractThis work is devoted to the study of uncertainty principles for finite combinations of Hermite functions. We establish some spectral inequalities for control subsets that are thick with respect to some unbounded densities growing almost linearly at infinity, and provide quantitative estimates, with respect to the energy level of the Hermite functions seen as eigenfunctions of the harmonic oscillator, for the constants appearing in these spectral estimates. These spectral inequalities allow us to derive the null-controllability in any positive time for evolution equations enjoying specific regularizing effects. More precisely, for a given index $\frac {1}{2} \leq \mu <1$ , we deduce sufficient geometric conditions on control subsets to ensure the null-controllability of evolution equations enjoying regularizing effects in the symmetric Gelfand–Shilov space $S^{\mu }_{\mu }(\mathbb {R}^{n})$ . These results apply in particular to derive the null-controllability in any positive time for evolution equations associated to certain classes of hypoelliptic non-self-adjoint quadratic operators, or to fractional harmonic oscillators.

Application of the dual space of Gelfand-Shilov spaces of Beurling type

Using a previously obtained structure theorem of Gelfand-Shilov spaces $\Sigma _{\alpha }^{\beta }$ of Beurling type of ultradistributions, we prove that these ultradistributions can be represented as an initial values of solutions of the heat equation by describing the action of the Gauss-Weierstrass semigroup on the dual space $(\Sigma _{\alpha }^{\beta})^{\prime }.$

Time-periodic Gelfand-Shilov spaces and global hypoellipticity on [formula omitted

We introduce a class of time-periodic Gelfand-Shilov spaces of functions on T×Rn, where T∼R/2πZ is the one-dimensional torus. We develop a Fourier analysis inspired by the characterization of the Gelfand-Shilov spaces in terms of the eigenfunction expansions given by a fixed normal, globally elliptic differential operator on Rn. In this setting, as an application, we characterize the global hypoellipticity for a class of linear differential evolution operators on T×Rn.

Weighted (PLB)-spaces of ultradifferentiable functions and multiplier spaces

We study weighted (PLB)-spaces of ultradifferentiable functions defined via a weight function (in the sense of Braun, Meise and Taylor) and a weight system. We characterize when such spaces are ultrabornological in terms of the defining weight system. This generalizes Grothendieck’s classical result that the space $${\mathcal {O}}_M$$ of slowly increasing smooth functions is ultrabornological to the context of ultradifferentiable functions. Furthermore, we determine the multiplier spaces of Gelfand-Shilov spaces and, by using the above result, characterize when such spaces are ultrabornological. In particular, we show that the multiplier space of the space of Fourier ultrahyperfunctions is ultrabornological, whereas the one of the space of Fourier hyperfunctions is not.

Surjectivity of the asymptotic Borel map in Carleman–Roumieu ultraholomorphic classes defined by regular sequences

We study the surjectivity of, and the existence of right inverses for, the asymptotic Borel map in Carleman–Roumieu ultraholomorphic classes defined by regular sequences in the sense of E. M. Dyn’kin. We extend previous results by J. Schmets and M. Valdivia, by V. Thilliez, and by the authors, and show the prominent role played by an index, associated with the sequence, that was introduced by V. Thilliez. The techniques involve regular variation, integral transforms and characterization results of A. Debrouwere in a half-plane, stemming from his study of the surjectivity of the moment mapping in general Gelfand–Shilov spaces.

Inclusion Theorems for the Moyal Multiplier Algebras of Generalized Gelfand–Shilov Spaces

We prove that the Moyal multiplier algebras of the generalized Gelfand–Shilov spaces of type S contain Palamodov spaces of type \({\mathscr {E}}\) and the inclusion maps are continuous. We also give a direct proof that the Palamodov spaces are algebraically and topologically isomorphic to the strong duals of the spaces of convolutors for the corresponding spaces of type S. The obtained results provide a general and efficient way to describe the algebraic and continuity properties of pseudodifferential operators with symbols having an exponential or super-exponential growth at infinity.

Continuity of the fractional Hankel wavelet transform on Gelfand–Shilov spaces

We give characterization results of the fractional Hankel transform as well its inverse on some Gelfand–Shilov spaces of type W. Furthermore, we derive the boundedness properties of wavelet transforms involving the fractional Hankel transform on certain suitably constructed spaces of type W.

Pseudo-differential operator related to Hankel transform on Gelfand–Shilov spaces of type W

Pseudo-differential operator related to Hankel transform on Gelfand–Shilov spaces of type W