fand-Shilov Spaces Research Articles

On the anti-Wick symbol as a Gelfand-Shilov generalized function

The purpose of this article is to prove that the anti-Wick symbol of an operator mapping S ( R n ) \mathcal {S}(\mathbb {R}^n) into S ′ ( R n ) \mathcal {S}’(\mathbb {R}^n) , which is generally not a tempered distribution, can still be defined as a Gel′fand-Shilov generalized function. This result relies on test function spaces embeddings involving the Schwartz and Gel′fand-Shilov spaces. An additional embedding concerning Schwartz and Gevrey spaces is also given.

The fractional wavelet transform on spaces of type W

In this paper, the fractional Fourier transform and the fractional wavelet transform are studied on certain Gel'fand–Shilov spaces of type W. The continuity of the fractional wavelet transform on some spaces of type W is shown.

The S-transform on spaces of type W

The aim of this paper is to study the S-transform on Gel'fand Shilov spaces of type W. It is shown that the S-transform with the Gaussian window is a continuous linear map of the spaces of type W into the space of type [Wtilde] defined on ℝ×(ℝ\\{0}), ℂ×(ℝ\\{0}).

Generalized Weyl correspondence and Moyal multiplier algebras

We show that the Weyl correspondence and the concept of a Moyal multiplier can be naturally extended to generalized function classes that are larger than the class of tempered distributions. This generalization is motivated by possible applications to noncommutative quantum field theory. We prove that under reasonable restrictions on the test function space E ⊂ L2, any operator in L2 with a domain E and continuous in the topologies of E and L2 has a Weyl symbol, which is defined as a generalized function on the Wigner-Moyal transform of the projective tensor square of E. We also give an exact characterization of the Weyl transforms of the Moyal multipliers for the Gel’fand-Shilov spaces Sββ.

Asymptotic Expansions of Solutions to the Heat Equations with Initial Value in the Dual of Gel'fand-Shilov Spaces

We will derive the asymptotic expansions of the solutions $U(x,t)$ to the heat equation with $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$, $r\geq 1/2$, initial value, where $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$ is the dual space of the Gel'fand-Shilov space $\mathcal{S}^r_r(\mathbf{R}^d$. Moreover, we show that, when $1/2\leq r\leq 1$, these asymptotic expansions satisfy the strong asymptotic condition on some circle $D_R=\{t\in\mathbf{C}\ |\ \mathrm{Re}\ t^{-1}>R^{-1}\}$. Therefore, we find that these asymptotic series for $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$ initial value are Borel summable by means of A. D. Sokal's result on the Borel summability. As an application, we show the asymptotic expansions of the Weyl transform with Planck's constant $\hbar$ in some state, which are refinement of a classical limit of the quantum mechanical expectation values expressed by the Weyl transform.

Moyal multiplier algebras of the test function spaces of type S

The Gel'fand-Shilov spaces of type S are considered as topological algebras with respect to the Moyal star product and their corresponding algebras of multipliers are defined and investigated. In contrast to the well-studied case of Schwartz's space S, these multipliers are allowed to have nonpolynomial growth or infinite order singularities. The Moyal multiplication is thereby extended to certain classes of ultradistributions, hyperfunctions, and analytic functionals. The main theorem of the paper characterizes those elements of the dual of a given test function space that are the Moyal multipliers of this space. The smallest nontrivial Fourier-invariant space in the scale of S-type spaces is shown to play a special role, because its corresponding Moyal multiplier algebra contains the largest algebra of functions for which the power series defining their star products are absolutely convergent. Furthermore, it contains analogous algebras associated with cone-shaped regions, which can be used to formulate a causality condition in quantum field theory on noncommutative space-time.

Entire extensions and exponential decay for semilinear elliptic equations

We consider semilinear partial differential equations in ℝ n of the form $$ \sum\limits_{\frac{{|\alpha |}} {m} + \frac{{|\beta |}} {k} \leqslant 1} {c_{\alpha \beta } x^\beta D_x^\alpha u = F(u)} , $$ where k and m are given positive integers. Relevant examples are semilinear Schrodinger equations $$ - \Delta u + V(x)u = F(u), $$ , where the potential V(x) is given by an elliptic polynomial. We propose techniques, based on anisotropic generalizations of the global ellipticity condition of M. Shubin and multiparameter Picard type schemes in spaces of entire functions, which lead to new results for entire extensions and asymptotic behaviour of the solutions. Namely, we study solutions (eigenfunctions and homoclinics) in the framework of the Gel’fand-Shilov spaces S µ (ℝ n ). Critical thresholds are identified for the indices µ and ν, corresponding to analytic regularity and asymptotic decay, respectively. In the one-dimensional case −u″ + V(x)u = F(u), our results for linear equations link up with those given by the classical asymptotic theory and by the theory of ODE in the complex domain, whereas for homoclinics, new phenomena concerning analytic extensions are described.

Test functions space in noncommutative quantum field theory

It is proven that the -product of field operators implies that the space of test functions in the Wightman approach to noncommutative quantum field theory is one of the Gel'fand-Shilov spaces Sβ with β < 1/2. This class of test functions smears the noncommutative Wightman functions, which are in this case generalized distributions, sometimes called hyperfunctions. The existence and determination of the class of the test function spaces in NC QFT is important for any rigorous treatment in the Wightman approach.

The wavelet transform on spaces of type S

The continuous wavelet transform is studied on certain Gel'fand–Shilov spaces of type S. It is shown that, for wavelets belonging to the one type of S-space defined on R, the wavelet transform is a continuous linear map of the other type of the S-space into a space of the same type (latter type) defined on R × R+. The wavelet transforms of certain ultradifferentiable functions are also investigated.

Asymptotic hyperfunctions, tempered hyperfunctions, and asymptoticexpansions

We introduce new subclasses of Fourier hyperfunctions of mixed type, satisfying polynomial growth conditions at infinity, and develop their sheaf and duality theory. We use Fourier transformation and duality to examine relations of these asymptotic and tempered hyperfunctions to known classes of test functions and distributions, especially the Gel′fand‐Shilov spaces. Further it is shown that the asymptotic hyperfunctions, which decay faster than any negative power, are precisely the class that allows asymptotic expansions at infinity. These asymptotic expansions are carried over to the higher‐dimensional case by applying the Radon transformation for hyperfunctions.

Every Stieltjes moment problem has a solution in Gel'fand-Shilov spaces

We prove that every Stieltjes problem has a solution in Gel'fand-Shilov spaces Sβ for every β>1. In other words, for an arbitrary sequence {μn} there exists a function ϕ in the Gel'fand-Shilov space Sβ with support in the positive real line whose moment ∫0∞xnϕ(x)dx=μn for every nonnegative integer n. This improves the result of A. J. Duran in 1989 very much who showed that every Stieltjes moment problem has a solution in the Schwartz space S, since the Gel'fand-Shilov space is much a smaller subspace of the Schwartz space. Duran's result already improved the result of R. P. Boas in 1939 who showed that every Stieltjes moment problem has a solution in the class of functions of bounded variation. Our result is optimal in a sense that if β≤1 we cannot find a solution of the Stieltjes problem for a given sequence.

Laguerre Expansions of Gel′fand-Shilov Spaces

The subspaces G α, G β, and G β α (α, β ≥ 0)of Schwartz′ space S + in (0, + ∞) are associated with the Hankel transform in the same way as the Gel′fand-Shilov spaces S α, S β, and S β α are associated with the Fourier transform. Indeed, if we consider the Hankel transform H γ (γ < −1) defined by H γ(ƒ)( t) = 1 2 ∫ ∞ 0 ( xt) −γ/2 x γ J γ([formula]) ƒ( x) dx then H γ is an isomorphism from G α, G β, and G β α onto G α, G β, and G α β respectively. So. the spaces G α α are invariant for H γ. In this paper, we characterize the spaces G α α (α > 1) in terms of their Fourier-Laguerre coefficients. Also, we characterize the range of the Fourier-Laplace operator F D defined by F D (ƒ)( w) = ∫ ∞ 0 ƒ( t) e −(1/2)((1 + w)/(1 − w)) t for w ∈ D = { w ∈ C : | w| ≤ 1} when it acts on the space G α α.

On Hankel transform

In this paper, using some results of the author on Hankel transform in the Schwartz and Gel’fand-Shilov spaces, we characterize the integral operators of Hankel type which are isomorphisms between the spaces H μ {H_\mu } of Zemanian. As a particular case, we obtain the classical Zemanian results on Hankel transform, some results of Mendez, and improve some results of Lee. Finally, we use these results to characterize the functions f f of the Schwartz space which satisfy ∫ 0 ∞ t α + n f ( t ) d t = 0 \smallint _0^\infty {t^{\alpha + n}}f(t)dt = 0 for all n ≥ 0 n \geq 0 and α &gt; − 1 \alpha &gt; - 1 .

A Characterization of the Spaces $\mathfrak{S}_{{1/{k + 1}}}^{{k /{k + 1}}} $ by Means of Holomorphic Semigroups

The Gel’fand–Shilov spaces $\mathfrak{S}_\alpha ^\beta ,{\alpha = 1}/ {(k + 1)},{\beta = k} /{(k + 1)}$, are special cases of a general type of test function spaces introduced by de Graaf. We give a self-adjoins operator so that the test functions in those $\mathfrak{S}_\alpha ^\beta $ spaces can be expanded in terms of the eigenfunctions of that self-adjoins operator.