Schwartz Space Research Articles

Superoscillating Sequences Towards Approximation in $${\mathscr {S}}$$ S or $${\mathscr {S}}'$$ S ′ -Type Spaces and Extrapolation

Aharonov–Berry superoscillations are band-limited sequences of functions that happen to oscillate asymptotically faster than their fastest Fourier component. In this paper we analyze in what sense functions in the Schwartz space \(\mathscr {S}(\mathbb {R},\mathbb {C})\) or in some of its subspaces, tempered distributions or also ultra-distributions, could be approximated over compact sets or relatively compact open sets (depending on the context) by such superoscillating sequences. We also show how one can profit of the existence of such sequences in order to extrapolate band-limited signals with finite energy from a given segment of the real line.

Boundary-value problems in a layer for evolutionary pseudo-differential equations with integral conditions

Boundary-value problems for evolutionary pseudo-differential equations with an integral condition are studied. Necessary and sufficient conditions of well-posedness are obtained for these problems in the Schwartz spaces. Existence of a well-posed boundary-value problem is proved for each evolutionary pseudo-differential equation.

О некоторых функциональных уравнениях в пространствах Шварца и их приложениях

О некоторых функциональных уравнениях в пространствах Шварца и их приложениях

The linear canonical wavelet transform on some function spaces

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space [Formula: see text] for convenience. As a generation of FT, it is necessary to detect the linear canonical transform (LCT) on a new space for obtaining the similar properties like FT on [Formula: see text]. Therefore, a space [Formula: see text] generalized from [Formula: see text] is introduced firstly, and further we prove that LCT is a homeomorphism from [Formula: see text] onto itself. The linear canonical wavelet transform (LCWT) is a newly proposed transform based on the convolution theorem in LCT domain. Moreover, we propose an equivalent definition of LCWT associated with LCT and further study some properties of LCWT on [Formula: see text]. Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of [Formula: see text] and [Formula: see text].

Multilinear Stockwell transforms

The main aim of this paper is to introduce multilinear versions of the Stockwell transforms (also named S-transforms) by using the fact that S-transforms can be written as convolution products. Further on we extend the multilinear S-transforms from the Schwartz class of rapidly decreasing functions to the space of tempered distributions. In the sequel we give a relation between multilinear S-transforms and multilinear pseudo-differential operators. We also state and prove some boundedness results regarding multilinear S-transforms on the Lebegue’s spaces Lp(Rn) and also on the Hörmander’s spaces Bp,k(Rn), where p ≥ 1 and k is a temperate weight function. In the end, a weak uncertainty principle for multilinear S-transforms and for its adjoint is also given.

Harmonic Analysis on the Space of $p$-adic Unitary Hermitian Matrices, Mainly for Dyadic Case

We are interested in the harmonic analysis on $p$-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space $X$ of unitary hermitian matrices of size $m$ over a ${\mathfrak p}$-adic field $k$ and give unified description including dyadic case, which is a continuation of our previous papers on non-dyadic case. The space becomes complicated when $e = v_\pi(2) > 0$. First we introduce a typical spherical function $\omega(x;z)$ on $X$, and study their functional equations, which depend on $m$ and $e$, we give an explicit formula for $\omega(x;z)$, where Hall-Littlewood polynomials of type $C_n$ appear as a main term with different specialization according as $m = 2n$ or $2n+1$, but independent of $e$. By spherical transform, we show the Schwartz space ${\mathcal S}(K \backslash X)$ is a free Hecke algebra ${\mathcal H}(G,K)$-module of rank $2^n$, and give parametrization of all the spherical functions on $X$ and the explicit Plancherel formula on ${\mathcal S}(K \backslash X)$. The Plancherel measure does not depend on $e$, but the normalization of $G$-invariant measure on $X$ depends.

Nilpotent Lie Groups: Fourier Inversion and Prime Ideals

We establish a Fourier inversion theorem for general connected, simply connected nilpotent Lie groups G= hbox {exp}({mathfrak {g}}) by showing that operator fields defined on suitable sub-manifolds of {mathfrak {g}}^* are images of Schwartz functions under the Fourier transform. As an application of this result, we provide a complete characterisation of a large class of invariant prime closed two-sided ideals of L^1(G) as kernels of sets of irreducible representations of G.

Dynamics and spectra of composition operators on the Schwartz space

In this paper we study the dynamics of the composition operators defined in the Schwartz space S(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.

Cusp forms for reductive symmetric spaces of split rank one

For reductive symmetric spaces G / H G/H of split rank one we identify a class of minimal parabolic subgroups for which certain cuspidal integrals of Harish-Chandra–Schwartz functions are absolutely convergent. Using these integrals we introduce a notion of cusp forms and investigate its relation with representations of the discrete series for G / H G/H .

Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line

We derive asymptotic formulas for the solution of the derivative nonlinear Schrödinger equation on the half-line under the assumption that the initial and boundary values lie in the Schwartz class. The formulas clearly show the effect of the boundary on the solution. The approach is based on a nonlinear steepest descent analysis of an associated Riemann–Hilbert problem.

A family of pseudo-differential operators on the Schwartz space associated with the fractional Fourier transform

Motivated essentially by their potential for applications in the mathematical, physical, and engineering sciences, the authors introduce and investigate various useful properties of a family of pseudo-differential operators on the Schwartz space S(ℝ) (i.e., the space of rapidly decreasing infinitely differentiable functions on ℝ), which are associated with the fractional Fourier transform of order α (0 < α ≦ 1). Relevant connections with several related recent developments on this subject are also pointed out.

Laplacians on smooth distributions

Let be a compact smooth manifold equipped with a positive smooth density and let be a smooth distribution endowed with a fibrewise inner product . We define the Laplacian associated with and prove that it gives rise to an unbounded self-adjoint operator in . Then, assuming that generates a singular foliation , we prove that, for any function in the Schwartz space , the operator is a smoothing operator in the scale of longitudinal Sobolev spaces associated with . The proofs are based on pseudodifferential calculus on singular foliations, which was developed by Androulidakis and Skandalis, and on subelliptic estimates for . Bibliography: 35 titles.

Nilpotent Gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs

It has been shown [1,2,9,10] that for several nilpotent Gelfand pairs (N,K) (i.e., with N a nilpotent Lie group, K a compact group of automorphisms of N and the algebra L1(N)K commutative) the spherical transform establishes a 1-to-1 correspondence between the space S(N)K of K-invariant Schwartz functions on N and the space S(Σ) of functions on the Gelfand spectrum Σ of L1(N)K which extend to Schwartz functions on Rd, once Σ is suitably embedded in Rd. We call this property (S).We present here a general bootstrapping method which allows to establish property (S) to new nilpotent pairs (N,K), once the same property is known for a class of quotient pairs of (N,K) and a K-invariant form of Hadamard formula holds on N.We finally show how our method can be recursively applied to prove property (S) for a significant class of nilpotent Gelfand pairs.

Operator algebra in the space of images

A consistent description of images on the disk and of their transformations is given as elements of a vector space and of an operators algebra. The vector space of images on the disk \U0001d53b is the Hilbert space L2(\U0001d53b) that has as a basis the Zernike functions. To construct the operator algebra that transforms the images, L2(\U0001d53b) must be complemented and the full rigged Hilbert space RHS(\U0001d53b) considered. Only this rigged Hilbert space allows indeed to write the operators of different cardinality we need to build the ladder operators on the Zernike functions that by inspection, belong to the representation of the algebra su(1, 1) ⊕ su(1, 1). Consequently the transformations of images are operators contained inside the universal enveloping algebra UEA[su(1, 1) ⊕ su(1, 1)]. Because of limited precision of experimental measures, physical states can be always described by vectors of the Schwartz space \U0001d54a(\U0001d53b), dense in the L2(\U0001d53b) space where the manipulation of images is performed.

Short-time asymptotics of the regularizing effect for semigroups generated by quadratic operators

We study accretive quadratic operators with zero singular spaces. These degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in the Schwartz space for any positive time. In this work, we study the short-time asymptotics of the regularizing effect induced by these semigroups. We show that these short-time asymptotics of the regularizing effect depend on the directions of the phase space, and that this dependence can be nicely understood through the structure of the singular space. As a byproduct of these results, we derive sharp subelliptic estimates for accretive quadratic operators with zero singular spaces pointing out that the loss of derivatives with respect to the elliptic case also depends on the phase space directions according to the structure of the singular space. Some applications of these results are then given to the study of degenerate hypoelliptic Ornstein–Uhlenbeck operators and degenerate hypoelliptic Fokker–Planck operators.

Spectral synthesis for the differentiation operator in the Schwartz space

We consider the spectral synthesis problem for the differentiation operator D=d/dt in the Schwartz space E(a;b) and the dual problem of local description for closed submodules in a special module of entire functions.

The multiplier algebra of the noncommutative Schwartz space

We describe the multiplier algebra of the noncommutative Schwartz space. This multiplier algebra can be seen as the largest ${}^*$-algebra of unbounded operators on a separable Hilbert space with the classical Schwartz space of rapidly decreasing functions as the domain. We show in particular that it is neither a $\mathcal{Q}$-algebra nor $m$-convex. On the other hand, we prove that classical tools of functional analysis, for example, the closed graph theorem, the open mapping theorem or the uniform boundedness principle, are still available.

Spatial asymptotic expansions in the incompressible Euler equation

In this paper we prove that the Euler equation describing the motion of an ideal fluid in $\R^d$ is well-posed in a class of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. These asymptotic expansions can involve log terms and lead to a family of conservation laws. Typically, the solutions of the Euler equation with initial data in the Schwartz class develop non-trivial spatial asymptotic expansions of the type considered here.

A restriction theorem for oscillatory integral operator with certain polynomial phase

We consider the following oscillatory integral operator \begin{equation}\label{opera-defi-1} T_{\alpha,m}f(x)=\int_{\mathbb R^n}e^{i(x_1^{\alpha_1} y_1^m+\cdots+x_n^{\alpha_n} y_n^m)}f(y)dy, \end{equation} where the function $f$ is a Schwartz function. In this paper, the restriction theorem on $\mathbb{S}^{n-1}$ for this operator is obtained. Moreover, we obtain a necessary condition which ensures the restriction theorem hold.

H-distributions with unbounded multipliers

We investigate H-distributions for sequences in the dual pairs of Bessel spaces, \((H^q_s , H^{p}_{-s}), s\in \mathbb {R}, q>1\) and \(q=p/(p-1),\) by the use of unbounded multipliers, with the finite regularity, as test functions. The results relating weak convergence, H-distributions and strong convergence are applied in the analysis of strong convergence for a sequence of approximated solutions to a class of differential equations \(P(x,D)u_n=f_n\), where P(x, D) is a differential operator of order k with coefficients in the Schwartz class and \((f_n)\) is a strongly convergent sequence in an appropriate Bessel potential space.