Gelfand-Shilov Spaces Research Articles

Progressive Gelfand-Shilov Spaces and Wavelet Transforms

We discuss progressive Gelfand-Shilov spaces consisting of analytic signals with almost exponential decay in time and frequency variables. It is shown that such signals enjoy an additional localization property. We define wavelet transform and inverse wavelet transform in (progressive) Gelfand-Shilov spaces and study their continuity properties. It is shown that with a slightly faster decay in domain we may control the decay of the wavelet transform independently in each variable.

The Bargmann transform on modulation and Gelfand–Shilov spaces, with applications to Toeplitz and pseudo-differential operators

We investigate mapping properties for the Bargmann transform on an extended family of modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation spaces to convenient Lebesgue spaces of analytic functions. We use this to prove that such modulation spaces fulfill most of the continuity properties which are valid for modulation spaces with moderate weights. Finally we use the results to establish continuity properties of Toeplitz and pseudo-differential operators on these modulation spaces, and on Gelfand–Shilov spaces.

Quasianalytic Gelfand-Shilov Spaces with Application to Localization Operators

Quasianalytic Gelfand-Shilov Spaces with Application to Localization Operators

Sub-Exponential Decay and Uniform Holomorphic Extensions for Semilinear Pseudodifferential Equations

The goal of the present paper is to derive a simultaneous description of the decay and the regularity properties for elliptic equations in ℝ n with coefficients admitting irregular decay at infinity of the type O(|x|−σ), σ > 0, filling the gap between the case of Cordes globally elliptic operators and the case of regular/Fuchs behavior at infinity. Representative examples in ℝ n are the equations where 0 < σ <2, ⟨x⟩ = (1 + |x|2)1/2, ω(x) a bounded smooth function, f given and F[u] a polynomial in u, and similar Schrödinger equations at the endpoint of the spectrum. Other relevant examples are given by linear and nonlinear ordinary differential equations with irregular type of singularity for x → ∞, admitting solutions y(x) with holomorphic extension in a strip and sub-exponential decay of type |y(x)| ≤Ce −ϵ|x| r ; 0 < r < 1. Sobolev estimates for the linear case are proved in the frame of a suitable pseudodifferential calculus; decay and uniform holomorphic extensions are then obtained in terms of Gelfand–Shilov spaces by an inductive technique. The same technique allows to extend the results to the semilinear case.

Structural theorems for Gelfand–Shilov spaces

In this article we give the structural theorem for Gelfand–Shilov spaces, and , of Roumieu and Beurling type in quasi-analytic and non-quasi-analytic case in an uniform way.

Coarse-scale representations and smoothed Wigner transforms

Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and/or semiclassical limits of wave equations. We derive explicit, closed formulations for the coarse-scale representation of the action of pseudodifferential operators. The resulting “smoothed operators” are in general of infinite order. The formulation of an appropriate framework, resembling the Gelfand–Shilov spaces, is necessary. Similarly we treat the “smoothed Wigner calculus”. In particular this allows us to reformulate any linear equation, as well as certain nonlinear ones (e.g., Hartree and cubic nonlinear Schrödinger), as coarse-scale phase-space equations (e.g., smoothed Vlasov), with spatial and spectral resolutions controlled by two free parameters. Finally, it is seen that the smoothed Wigner calculus can be approximated, uniformly on phase-space, by differential operators in the semiclassical regime. This improves the respective weak-topology approximation result for the Wigner calculus.

On localization properties of Fourier transforms of hyperfunctions

In [A.G. Smirnov, Fourier transformation of Sato's hyperfunctions, Adv. Math. 196 (2005) 310–345] the author introduced a new generalized function space U ( R k ) which can be naturally interpreted as the Fourier transform of the space of Sato's hyperfunctions on R k . It was shown that all Gelfand–Shilov spaces S α ′ 0 ( R k ) ( α > 1 ) of analytic functionals are canonically embedded in U ( R k ) . While the usual definition of support of a generalized function is inapplicable to elements of S α ′ 0 ( R k ) and U ( R k ) , their localization properties can be consistently described using the concept of carrier cone introduced by Soloviev [M.A. Soloviev, Towards a generalized distribution formalism for gauge quantum fields, Lett. Math. Phys. 33 (1995) 49–59; M.A. Soloviev, An extension of distribution theory and of the Paley–Wiener–Schwartz theorem related to quantum gauge theory, Comm. Math. Phys. 184 (1997) 579–596]. In this paper, the relation between carrier cones of elements of S α ′ 0 ( R k ) and U ( R k ) is studied. It is proved that an analytic functional u ∈ S α ′ 0 ( R k ) is carried by a cone K ⊂ R k if and only if its canonical image in U ( R k ) is carried by K.

Star product algebras of test functions

We prove that the Gelfand-Shilov spaces $S^\beta_\alpha$ are topological algebras under the Moyal star product if and only if $\alpha\ge\beta$. These spaces of test functions can be used in quantum field theory on noncommutative spacetime. The star product depends on the noncommutativity parameter continuously in their topology. We also prove that the series expansion of the Moyal product converges absolutely in $S^\beta_\alpha$ if and only if $\beta<1/2$.

Linear continuous operators for the Stieltjes moment problem in Gelfand–Shilov spaces

We obtain linear continuous operators providing a solution to the Stieltjes moment problem in the framework of Gelfand–Shilov spaces of rapidly decreasing smooth functions. The construction rests on an interpolation procedure due to R. Estrada for general rapidly decreasing smooth functions, and adapted by S.-Y. Chung, D. Kim and Y. Yeom to the case of Gelfand–Shilov spaces. It requires a linear continuous version of the so-called Borel–Ritt–Gevrey theorem in asymptotic theory.

Localization properties of highly singular generalized functions

We study the localization properties of generalized functions defined on a broad class of spaces of entire analytic test functions. This class, which includes all Gelfand-Shilov spaces S α β (R k ) with β < 1, provides a convenient language for describing quantum fields with a highly singular infrared behavior. We show that the carrier cone notion, which replaces the support notion, can be correctly defined for the considered analytic functionals. In particular, we prove that each functional has a uniquely determined minimal carrier cone.

Decomposition theorems and kernel theorems for a class of functional spaces

We prove new theorems about properties of generalized functions defined on Gelfand-Shilov spaces with . For each open cone we define a space which is related to and consists of entire analytic functions rapidly decreasing inside and having order of growth outside the cone. Such sheaves of spaces arise naturally in non-local quantum field theory, and this motivates our investigation. We prove that the spaces are complete and nuclear and establish a decomposition theorem which implies that every continuous functional defined on has a unique minimal closed carrier cone in . We also prove kernel theorems for spaces over open and closed cones and elucidate the relation between the carrier cones of multilinear forms and those of the generalized functions determined by these forms.

On uncertainty bounds and growth estimates for fractional fourier transforms

Gelfand–Shilov spaces are spaces of entire functions defined in terms of a rate of growth in one direction and a concomitant rate of decay in an orthogonal direction. Gelfand and Shilov proved that the Fourier transform is an isomorphism among certain of these spaces. In this article we consider mapping properties of fractional Fourier transforms on Gelfand–Shilov spaces. Just as the Fourier transform corresponds to rotation through a right angle in the phase plane, fractional Fourier transforms correspond to rotations through intermediate angles. Therefore, the aim of fractional Fourier estimates is to set up a correspondence between growth properties in the complex plane versus decay properties in phase space.

Modulation Spaces, Gelfand-Shilov Spaces and Pseudodifferential Operators

Modulation Spaces, Gelfand-Shilov Spaces and Pseudodifferential Operators

Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients

We show that all eigenfunctions of linear partial differential operators in R n with polynomial coefficients of Shubin type are extended to entire functions in C n of finite exponential type 2 and decay like exp ( − | z | 2 ) for | z | → ∞ in conic neighbourhoods of the form | Im z | ⩽ γ | Re z | . We also show that under semilinear polynomial perturbations all nonzero homoclinics keep the super-exponential decay of the above type, whereas a loss of the holomorphicity occurs, namely we show holomorphic extension into a strip { z ∈ C n | | Im z | ⩽ T } for some T > 0 . The proofs are based on geometrical and perturbative methods in Gelfand–Shilov spaces. The results apply in particular to semilinear Schrödinger equations of the form (∗) − Δ u + | x | 2 u − λ u = F ( x , u , ∇ u ) . Our estimates on homoclinics are sharp. In fact, we exhibit examples of solutions of (∗) with super-exponential decay, which are meromorphic functions, the key point of our argument being the celebrated great Picard theorem in complex analysis.

Two classes of generalized functions used in nonlocal field theory

We elucidate the relation between the two ways of formulating causality in nonlocal quantum field theory: using analytic test functions belonging to the space $S^0$ (which is the Fourier transform of the Schwartz space $\mathcal D$) and using test functions in the Gelfand-Shilov spaces $S^0_\alpha$. We prove that every functional defined on $S^0$ has the same carrier cones as its restrictions to the smaller spaces $S^0_\alpha$. As an application of this result, we derive a Paley-Wiener-Schwartz-type theorem for arbitrarily singular generalized functions of tempered growth and obtain the corresponding extension of Vladimirov's algebra of functions holomorphic on a tubular domain.

Spaces of Test Functions via the STFT

We characterize several classes of test functions, among them Björck′s ultra‐rapidly decaying test functions and the Gelfand‐Shilov spaces of type S, in terms of the decay of their short‐time Fourier transform and in terms of their Gabor coefficients.

Equivalence of the Gelfand–Shilov Spaces

We prove that there is a one to one correspondence between the Gelfand–Shilov spaceWMΩof typeWand the spaceSMpNpof generalized typeS. As an application we prove the equalityWM∩WΩ=WMΩ, which is a generalization of the equalitySr∩Ss=Srsfound by I. M. Gelfand and G. E. Shilov (“Generalized Functions, II, III,” Academic Press, New York/London, 1967).

An Extension of Distribution Theory and of the Paley-Wiener-Schwartz Theorem Related to Quantum Gauge Theory

We show that a considerable part of the theory of (ultra)distributions and hyperfunctions can be extended to more singular generalized functions, starting from an angular localizability notion introduced previously. Such an extension is needed to treat gauge quantum field theories with indefinite metric in a generic covariant gauge. Prime attention is paid to the generalized functions defined on the Gelfand-Shilov spaces $S_\alpha^0$ which gives the widest framework for construction of gauge-like models. We associate a similar test function space with every open and every closed cone, show that these spaces are nuclear and obtain the required formulas for their tensor products. The main results include the generalization of the Paley--Wiener--Schwartz theorem to the case of arbitrary singularity and the derivation of the relevant theorem on holomorphic approximation.

Characterizations of the Gelfand-Shilov spaces via Fourier transforms

We give symmetric characterizations, with respect to the Fourier transformation, of the Gelfand-Shilov spaces of (generalized) type S S and type W W . These results explain more clearly the invariance of these spaces under the Fourier transformations.

Nonlocalizability and asymptotical commutativity

The mathematical formalism commonly used in treating nonlocal highly singular interactions is revised. The notion of support cone is introduced which replaces that of support for nonlocalizable distributions. Such support cones are proven to exist for distributions defined on the Gelfand-Shilov spaces $S^\beta$, where $0<\beta <1$ . This result leads to a refinement of previous generalizations of the local commutativity condition to nonlocal quantum fields. For string propagators, a new derivation of a representation similar to that of K\"{a}llen-Lehmann is proposed. It is applicable to any initial and final string configurations and manifests exponential growth of spectral densities intrinsic in nonlocalizable theories.