Gelfand-Shilov Spaces Research Articles

Directional Short-Time Fourier Transform of Ultradistributions

We define and analyze the k-directional short-time Fourier transform and its synthesis operator over Gelfand–Shilov spaces \(\mathcal {S}^\alpha _{\beta }(\mathbb {R}^n)\) and \(\mathcal {S}^\alpha _{\beta }({\mathbb {R}}^{k+n})\), respectively, and their duals. Also, we investigate directional regular sets and their complements—directional wave fronts, for elements of \(\mathcal {S}^{\prime \alpha }_{\alpha }(\mathbb {R}^n)\).

Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators A_a^{varphi _1,varphi _2} with symbols a in the wide weighted modulation space M^infty _{w}({mathbb {R}^{2d}}), and windows varphi _1, varphi _2 in the Gelfand–Shilov space mathcal {S}^{left( 1right) }(mathbb {R}^d). If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of A_a^{varphi _1,varphi _2} have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space mathcal {S}^{(gamma )} (mathbb {R^{d}}) , where the parameter gamma ge 1 is related to the growth of the considered weight. An important role is played by tau -pseudodifferential operators Op_{tau } (sigma ). In that direction we show convenient continuity properties of Op_{tau } (sigma )when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of Op_{tau } (sigma ) when the symbol sigma belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.

Ornstein-Uhlenbeck Semigroup on the dual space of Gelfand-Shilov Spaces of Beurling type

We use a previously obtained topological characterization of Gelfand-Shilov spaces of Beurling type to characterize its dual using Riesz representation theorem. Using the characterization of the dual space equipped with the weak topology, we study the action of Ornstein-Uhlenbeck Semigroup on the dual space.

Sequence space representations for spaces of entire functions with rapid decay on strips

We obtain sequence space representations for a class of Fréchet spaces of entire functions with rapid decay on horizontal strips. In particular, we show that the projective Gelfand-Shilov spaces Σν1 and Σ1ν are isomorphic to Λ∞(n1/(ν+1)) for ν>0.

Topological properties of convolutor spaces via the short-time Fourier transform

We discuss the structural and topological properties of a general class of weighted $L^1$ convolutor spaces. Our theory simultaneously applies to weighted $\mathcal{D}'_{L^1}$ spaces as well as to convolutor spaces of the Gelfand-Shilov spaces $\mathcal{K}\{M_p\}$. In particular, we characterize the sequences of weight functions $(M_p)_{p \in \mathbb{N}}$ for which the space of convolutors of $\mathcal{K}\{M_p\}$ is ultrabornological, thereby generalizing Grothendieck's classical result for the space $\mathcal{O}'_{C}$ of rapidly decreasing distributions. Our methods lead to the first direct proof of the completeness of the space $\mathcal{O}_{C}$ of very slowly increasing smooth functions.

The Zak Transform on Gelfand\u2013Shilov and Modulation Spaces with Applications to Operator Theory

We characterize Gelfand–Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by Zak transforms.

Estimates of pseudo-differential operators associated with fractional Hankel transform on the space of type S

ABSTRACT In this article, we present some boundedness results of fractional Hankel transform on Gelfand–Shilov spaces of type S. Certain inequalities are obtained for pseudo-differential operators involving the fractional Hankel transform on Gelfand–Shilov spaces of type S. This article made further discussion on the continuity properties of fractional Hankel transform and pseudo-differential operator on some ultradifferentiable function spaces.

Structure, boundedness, and convergence in the dual of a Hankel-K{Mp } space

ABSTRACT Hankel- spaces, as introduced by the second-named author, play the same role in the theory of the Hankel transformation as the Gelfand-Shilov spaces in the theory of the Fourier transformation. For , under suitable restrictions on the weights , the topology of a Hankel- space E can be generated by norms of -type involving the Bessel operator . In this paper, adapting a technique due to Kamiński, the elements of the dual space are represented as -distributional derivatives of a single continuous function. Corresponding characterizations of boundedness and convergence in (the weak, weak*, strong topologies of) are obtained.

The nuclearity of Gelfand–Shilov spaces and kernel theorems

We study the nuclearity of the Gelfand-Shilov spaces $\mathcal{S}^{(\mathfrak{M})}_{(\mathscr{W})}$ and $\mathcal{S}^{\{\mathfrak{M}\}}_{\{\mathscr{W}\}}$, defined via a weight (multi-)sequence system $\mathfrak{M}$ and a weight function system $\mathscr{W}$. We obtain characterizations of nuclearity for these function spaces that are counterparts of those for K\"{o}the sequence spaces. As an application, we prove new kernel theorems. Our general framework allows for a unified treatment of the Gelfand-Shilov spaces $\mathcal{S}^{(M)}_{(A)}$ and $\mathcal{S}^{\{M\}}_{\{A\}}$ (defined via weight sequences $M$ and $A$) and the Beurling-Bj\"orck spaces $\mathcal{S}^{(\omega)}_{(\eta)}$ and $\mathcal{S}^{\{\omega\}}_{\{\eta\}}$ (defined via weight functions $\omega$ and $\eta$). Our results cover anisotropic cases as well.

Characterization of the Moyal Multiplier Algebras for the Generalized Spaces of Type S

We investigate the properties of the Moyal multiplier algebras for the generalized Gelfand-Shilov spaces $$S_{{a_k}}^{{b_n}}$$ . We prove that these algebras contain Palamodov spaces of type $${\mathscr{E}}$$ , and establish continuity properties of the operators with Weyl symbols in this class. Analogous results are obtained for the projective version of the spaces of type S and are extended to the multiplier algebras for various translation-invariant star products.

Multiresolution expansions and wavelets in Gelfand–Shilov spaces

We study approximation properties generated by highly regular scaling functions and orthonormal wavelets. These properties are conveniently described in the framework of Gelfand-Shilov spaces. Important examples of multiresolution analyses for which our results apply arise in particular from Dziuba\'{n}ski-Hern\'{a}ndez construction of band-limited wavelets with subexponential decay. Our results are twofold. Firstly, we obtain approximation properties of multiresolution expansions of Gelfand-Shilov functions and (ultra)distributions. Secondly, we establish convergence of wavelet series expansions in the same regularity framework.

Solution to the Stieltjes moment problem in Gelfand–Shilov spaces

We characterize the surjectivity and the existence of a continuous linear right inverse of the Stieltjes moment mapping on Gelfand-Shilov spaces, both of Beurling and Roumieu type, in terms of their defining weight sequence. As a corollary, we obtain some new results about the Borel-Ritt problem in spaces of ultraholomorphic functions on the upper half-plane.

Spaces of Type S and Deformation Quantization

We study the properties of the Gelfand-Shilov spaces $$S_{a_k}^{b_n}$$ in the context of deformation quantization. Our main result is a characterization of their corresponding multiplier algebras with respect to a twisted convolution, which is given in terms of the inclusion relation between these algebras and the duals of the spaces of pointwise multipliers with an explicit description of these functional spaces. The proof of the inclusion theorem essentially uses the equality $$S_{a_k}^{b_n}=S^{b_n}\cap{S_{a_k}}$$.

Spaces of Type S as Topological Algebras under Twisted Convolution and Star Product

The properties of the generalized Gelfand-Shilov spaces $$S_{{b_n}}^{{a_k}}$$ are studied from the viewpoint of deformation quantization. We specify the conditions on the defining sequences (ak) and (bn) under which $$S_{{b_n}}^{{a_k}}$$ is an algebra with respect to the twisted convolution and, as a consequence, its Fourier transformed space $$S_{{a_k}}^{{b_n}}$$ is an algebra with respect to the Moyal star product. We also consider a general family of translation-invariant star products. We define and characterize the corresponding algebras of multipliers and prove the basic inclusion relations between these algebras and the duals of the spaces of ordinary pointwise and convolution multipliers. Analogous relations are proved for the projective counterpart of the Gelfand-Shilov spaces. A key role in our analysis is played by a theorem characterizing those spaces of type S for which the function exp(iQ(x)) is a pointwise multiplier for any real quadratic form Q.

Anisotropic Shubin operators and eigenfunction expansions in Gelfand-Shilov spaces

We derive new results on the characterization of Gelfand-Shilov spaces \(\mathcal{S}_{\nu}^{\mu}(\mathbb{R}^n)\), μ, ν > 0, μ + ν ≥ 1 byGevrey estimates of the L2 norms of iterates of (m, k) anisotropic globally elliptic Shubin (or Γ) type operators, (- Δ)m/2 + |x>k with m, k ∈ 2ℕ being a model operator, and on the decay of the Fourier coefficients in the related eigenfunction expansions. Similar results are obtained for the spaces \(\Sigma_{\nu}^{\mu}(\mathbb{R}^n)\), μ, ν > 0, μ + ν > 1, cf. (1.2). In contrast to the symmetric case μ = ν and k = m (classical Shubin operators) we encounter resonance type phenomena involving the ratio κ:= μ/ν; namely we obtain a characterization of \(\mathcal{S}_{\nu}^{\mu}(\mathbb{R}^n)\) and \(\Sigma_{\nu}^{\mu}(\mathbb{R}^n)\) in the case μ = kt/(k + m), ν = mt/(k + m), t ≥ 1, that is, when κ = k/m ∈ ℚ.

Injectivity and surjectivity of the Stieltjes moment mapping in Gelfand–Shilov spaces

The Stieltjes moment problem is studied in the framework of general Gelfand-Shilov spaces defined via weight sequences. We characterize the injectivity and surjectivity of the Stieltjes moment mapping, sending a function to its sequence of moments, in terms of growth conditions for the defining weight sequence. Finally, a related moment problem at the origin is studied.

On weighted inductive limits of spaces of ultradifferentiable functions and their duals

AbstractIn the first part of this paper we discuss the completeness of two general classes of weighted inductive limits of spaces of ultradifferentiable functions. In the second part we study their duals and characterize these spaces in terms of the growth of convolution averages of their elements. This characterization gives a canonical way to define a locally convex topology on these spaces and we give necessary and sufficient conditions for them to be ultrabornological. In particular, our results apply to spaces of convolutors for Gelfand–Shilov spaces.

Null-controllability of hypoelliptic quadratic differential equations

We study the null-controllability of parabolic equations associated to a general class of hypoelliptic quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. We consider in this work the class of accretive quadratic operators with zero singular spaces. These possibly degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in specific Gelfand-Shilov spaces for any positive time. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that parabolic equations associated to these operators are null-controllable in any positive time from control regions, for which null-controllability is classically known to hold in the case of the heat equation on the whole space. Some applications of this result are then given to the study of parabolic equations associated to hypoelliptic Ornstein-Uhlenbeck operators acting on weighted $L^2$ spaces with respect to invariant measures. By using the same strategy, we also establish the null-controllability in any positive time from the same control regions for parabolic equations associated to any hypoelliptic Ornstein-Uhlenbeck operator acting on the flat $L^2$ space extending in particular the known results for the heat equation or the Kolmogorov equation on the whole space.

Wavelet transforms on Gelfand-Shilov spaces and concrete examples

In this paper, we study the continuity properties of wavelet transforms in the Gelfand-Shilov spaces with the use of a vanishing moment condition. Moreover, we also compute the Fourier transforms and the wavelet transforms of concrete functions in the Gelfand-Shilov spaces.

Factorizations and Singular Value Estimates of Operators with Gelfand\u2013Shilov and Pilipovi\u0107 kernels

We prove that any linear operator with kernel in a Pilipović or Gelfand–Shilov space can be factorized by two operators in the same class. We also give links on numerical approximations for such compositions. We apply these composition rules to deduce estimates of singular values and establish Schatten–von Neumann properties for such operators.