Schwartz Space Research Articles

Schwartz correspondence for the complex motion group on [formula omitted

If (G,K) is a Gelfand pair, with G a Lie group of polynomial growth and K a compact subgroup of G, the Gelfand spectrum Σ of the bi-K-invariant algebra L1(K﹨G/K) admits natural embeddings into Rℓ spaces as a closed subset.For any such embedding, define S(Σ) as the space of restrictions to Σ of Schwartz functions on Rℓ. We call Schwartz correspondence for (G,K) the property that the spherical transform is an isomorphism of S(K﹨G/K) onto S(Σ).In all the cases studied so far, Schwartz correspondence has been proved to hold true. These include all pairs with G=K⋉H and K abelian and a large number of pairs with G=K⋉H and H nilpotent.We prove Schwartz correspondence for the pair (U2⋉M2(C),U2), where M2(C) is the complex motion group and U2=K acts on it by conjugation. Our proof goes through a detailed analysis of (M2(C),U2) as a strong Gelfand pair and reduction of the problem to Schwartz correspondence for each K-type τ∈Kˆ with appropriate control of the estimates in terms of τ.

On the Method of Stationary Phase in Calculating the Propagator of the Coulomb–Kepler Problem

A theorem is proven on the method of stationary phase to be applied in the calculation of the propagator from the Green’s function of the Coulomb - Kepler problem, by means of Fourier transformation. The Green’s function is available in position representation in an exact compact form as ingeniously derived by Hostler (1963). The Fourier transform, at first, exists in the distributional sense. In order to avoid series representation in Schwartz space, the integrals are defined away from the real axis of the integration variable and by taking principal values. The theorem deals with integrands with a highly oscillating exponential term. It is shown that the method of stationary phase renders exact results in cases relevant to the problem, whereby the integrand can contain an analytic amplitude factor in addition to the exponential term.

Tauberian theorems for smooth functions that never vanish

It is well known that if a function has slow growth in the ordinary sense, then it has slow growth in the Cesàro sense. The converse holds if special extra conditions are assumed; results of this form are called Tauberian Theorems. In this paper, we use techniques from generalized functions to investigate this phenomenon, with particular emphasis on the space of multipliers of Schwartz functions. We then apply our results to study series of delta functions and the division problem for tempered distributions.

On the kernel of the -Generalized fourier transform

Abstract For the kernel $B_{\kappa ,a}(x,y)$ of the $(\kappa ,a)$ -generalized Fourier transform $\mathcal {F}_{\kappa ,a}$ , acting in $L^{2}(\mathbb {R}^{d})$ with the weight $|x|^{a-2}v_{\kappa }(x)$ , where $v_{\kappa }$ is the Dunkl weight, we study the important question of when $\|B_{\kappa ,a}\|_{\infty }=B_{\kappa ,a}(0,0)=1$ . The positive answer was known for $d\ge 2$ and $\frac {2}{a}\in \mathbb {N}$ . We investigate the case $d=1$ and $\frac {2}{a}\in \mathbb {N}$ . Moreover, we give sufficient conditions on parameters for $\|B_{\kappa ,a}\|_{\infty }>1$ to hold with $d\ge 1$ and any a. We also study the image of the Schwartz space under the $\mathcal {F}_{\kappa ,a}$ transform. In particular, we obtain that $\mathcal {F}_{\kappa ,a}(\mathcal {S}(\mathbb {R}^d))=\mathcal {S}(\mathbb {R}^d)$ only if $a=2$ . Finally, extending the Dunkl transform, we introduce nondeformed transforms generated by $\mathcal {F}_{\kappa ,a}$ and study their main properties.

Coherent Schwartz distributions of the Heisenberg algebra and inverted oscillator

An unusual family of generalized coherent states of the Heisenberg algebra is constructed. It consists of two families of functionals on the Schwartz space. Each of them has the key properties of ordinary coherent states: it intertwines the representations of a given algebra, has the completeness property, and minimizes the product of variances in the Heisenberg relation. However, unlike the usual coherent states that are eigenstates for the annihilation operator, the constructed distributions belong to the continuous spectrum of Hermitian generators of the Heisenberg algebra. The inner product of functionals from different families has the main properties of the overlap function of ordinary coherent states: it is continuous in parameters, satisfies the reproducing identity, and has the corresponding geometric meaning. The application of unusual coherent states of the continuous spectrum is demonstrated by an example of solving the spectral problem for an inverted oscillator.

Self-adjointness of Toeplitz operators on the Segal-Bargmann space

We prove a new criterion that guarantees self-adjointness of Toeplitz operators with unbounded operator-valued symbols. Our criterion applies, in particular, to symbols with Lipschitz continuous derivatives, which is the natural class of Hamiltonian functions for classical mechanics. For this we extend the Berger-Coburn estimate to the case of vector-valued Segal-Bargmann spaces. Finally, we apply our result to prove self-adjointness for a class of (operator-valued) quadratic forms on the space of Schwartz functions in the Schrödinger representation.

Sequence Space Representations for Translation-Modulation Invariant Function and Distribution Spaces

We provide sequence space representations for the test function space $\mathcal{D}_{E}$ and the distribution space $\mathcal{D}^{\prime}_{E}$ associated to a Banach space $E$ belonging to a broad class of translation-modulation invariant Banach spaces of distributions. The spaces $\mathcal{D}_{E}$ and $\mathcal{D}^{\prime}_{E}$ generalize the classical Schwartz spaces $\mathcal{D}_{L^p}$ and $\mathcal{D}^{\prime}_{L^p}$, respectively. Our proof is based on Gabor frame characterizations of $\mathcal{D}_{E}$ and $\mathcal{D}^{\prime}_{E}$, which are also established here and are of independent interest. We recover in a unified way some known sequence space representations as well as obtain several new ones.

Sharp superlevel set estimates for small cap decouplings of the parabola

We prove sharp bounds for the size of superlevel sets \{x \in \mathbb{R}^2\colon |f(x)| >\alpha\} , where \alpha>0 and f\colon\mathbb{R}^2\to\mathbb{C} is a Schwartz function with Fourier transform supported in an R^{-1} -neighborhood of the truncated parabola \mathbb{P}^1 . These estimates imply the small cap decoupling theorem for \mathbb{P}^1 of Demeter, Guth, and Wang (2020) and the canonical decoupling theorem for \mathbb{P}^1 of Bourgain and Demeter (2015). New (\ell^q,L^p) small cap decoupling inequalities also follow from our sharp level set estimates.

Universal optimality of the $E_8$ and Leech lattices and interpolation formulas

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions eight and twenty-four, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function $f$ from the values and radial derivatives of $f$ and its Fourier transform $\hat{f}$ at the radii $\sqrt{2n}$ for integers $n\ge 1$ in $\mathbb{R}^8$ and $n\ge 2$ in $\mathbb{R}^{24}$. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska's work on sphere packing and placing it in the context of a more conceptual theory.

Harmonic and anharmonic oscillators on the Heisenberg group

In this article, we present a notion of the harmonic oscillator on the Heisenberg group Hn, which, under a few reasonable assumptions, forms the natural analog of a harmonic oscillator on Rn: a negative sum of squares of operators on Hn, which is essentially self-adjoint on L2(Hn) with purely discrete spectrum and whose eigenvectors are Schwartz functions forming an orthonormal basis of L2(Hn). The differential operator in question is determined by the Dynin–Folland group—a stratified nilpotent Lie group—and its generic unitary irreducible representations, which naturally act on L2(Hn). As in the Euclidean case, our notion of harmonic oscillator on Hn extends to a whole class of so-called anharmonic oscillators, which involve left-invariant derivatives and polynomial potentials of order greater or equal 2. These operators, which enjoy similar properties as the harmonic oscillator, are in one-to-one correspondence with positive Rockland operators on the Dynin–Folland group. The latter part of this article is concerned with spectral multipliers. We obtain useful Lp-Lq-estimates for a large class of spectral multipliers of the sub-Laplacian LHn,2 and, in fact, of generic Rockland operators on graded groups. As a by-product, we obtain explicit hypoelliptic heat semigroup estimates and recover the continuous Sobolev embeddings on graded groups, provided 1 < p ≤ 2 ≤ q < ∞.

Fractional Jacobi-Dunkl transform: properties and application

The aim of this paper is to develop a novel harmonic analysis in the Jacobi-Dunkl setting. We propose the fractional Jacobi-Dunkl transform in order to generalize the fractional Fourier transform. Firstly, we investigate the associated harmonic analysis and we derive some of its basic properties, such as inversion formula and Parseval identity. In sequel, we derive the fractional Jacobi-Dunkl transform on the corresponding Schwartz space and tempered distribution. Finally, we extend the scope of our work by investigating the fractional Jacobi-Dunkl transform for heat equation.

Continuity of the integral kernel of Schrödinger propagator

In this paper, we prove that the integral kernel of e−it(−Δ+V) is bounded and jointly continuous in (t,x,y) for a class of potentials which need not to be smooth. This result gives a partial affirmative answer for Simon's conjecture ([21, p. 482]). As a byproduct, we prove a short time asymptotic estimate for the integral kernel of e−it(−Δ+V). Moreover, when V belongs to Schwartz function classes, we can write e−it((−Δ)λ2+V) as Fourier integral operators and give expansions of the symbols.

Translation and modulation invariant Banach spaces of tempered distributions satisfy the metric approximation property

In this paper, we establish the validity of the so-called Bounded Approximation Property (BAP) for a comprehensive class of translation and modulation invariant Banach spaces (B, ∥ · ∥B) of tempered distributions on the Euclidean space [Formula: see text]. In fact, such spaces have a double module structure, over some Beurling algebra with respect to convolution, and with respect to pointwise multiplication over some Fourier Beurling algebra. Combining this double module structure with functional analytic arguments which describe the approximation of convolution operators by discrete convolutions we are able to verify the BAP, in fact, for most cases even the Metric Approximation Property (MAP) for such Banach space. The family of spaces under consideration is very rich and contains virtually all the classical function spaces relevant for mathematical analysis, as long as the Schwartz space [Formula: see text] is dense in (B, ∥ · ∥B). In particular, all the reflexive spaces in this family are included. Moreover, this family of Banach spaces is closed with respect to intersections, sums and various interpolation methods.

The generalized Saint Venant operator and integral moment transforms

In this article, we work with a generalized Saint Venant operator introduced by Vladimir Sharafutdinov [Inverse and ill-posed problems series, VSP, Utrecht, 1994] to describe the kernel of the integral moment transforms over symmetric m m -tensor fields in n n -dimensional Euclidean space. We also provide an equivalence between the injectivity question for the integral moment transforms and generalized Saint Venant operator over symmetric tensor fields of Schwartz class.

Baroclinic tidal conversion: note on a paper of L.R.M. Maas

Maas (J. Fluid Mech., vol. 684, 2011, pp. 5–24) showed that, for an oscillating two-dimensional barotropic tide flowing over sub-critical topography of compact support, some topographic forms existed that produced non-radiating baroclinic disturbances. The problem is related to ‘stealth’ and ‘cloaking’ problems. Here Maas's result is derived using a simpler approach, not involving complicated mappings, but formally restricted to perturbation topography. Wider results come from the discussion of nearly compact support topographic disturbances provided by Schwartz functions with weak high-wavenumber radiation and by exploiting both a known functional equation formulation and Fourier methods. The problem is extended to disturbances on uniform slopes. A variety of non-radiating topographies can be found, although they are mathematically delicate and unlikely to be found in nature. Topography with weak radiation at high wavenumber is a much wider class of structures. Application of these solutions would lie with the ability to estimate dissipation over and near the topography from motions observed at a distance.

Cauchy Integral and Boundary Value for Vector-Valued Tempered Distributions

Using the historically general growth condition on scalar-valued analytic functions, which have tempered distributions as boundary values, we show that vector-valued analytic functions in tubes TC=Rn+iC obtain vector-valued tempered distributions as boundary values. In a certain vector-valued case, we study the structure of this boundary value, which is shown to be the Fourier transform of the distributional derivative of a vector-valued continuous function of polynomial growth. A set of vector-valued functions used to show the structure of the boundary value is shown to have a one–one and onto relationship with a set of vector-valued distributions, which generalize the Schwartz space DL2′(Rn); the tempered distribution Fourier transform defines the relationship between these two sets. By combining the previously stated results, we obtain a Cauchy integral representation of the vector-valued analytic functions in terms of the boundary value.

Multiple vector-valued, mixed norm estimates for Littlewood-Paley square functions

We prove that for any $L^Q$-valued Schwartz function $f$ defined on $\mathbb{R}^d$, one has the multiple vector-valued, mixed norm estimate $$ \| f \|_{L^P(L^Q)} \lesssim \| S f \|_{L^P(L^Q)} $$ valid for every $d$-tuple $P$ and every $n$-tuple $Q$ satisfying $0 < P, Q < \infty$ componentwise. Here $S:= S_{d_1}\otimes ... \otimes S_{d_N}$ is a tensor product of several Littlewood-Paley square functions $S_{d_j}$ defined on arbitrary Euclidean spaces $\mathbb{R}^{d_j}$ for $1\leq j\leq N$, with the property that $d_1 + ... + d_N = d$. This answers a question that came up implicitly in our recent works and completes in a natural way classical results of the Littlewood-Paley theory. The proof is based on the \emph{helicoidal method} introduced by the authors.

A pointwise estimate for pseudo-differential operators

Let [Formula: see text] be a pseudo-differential operator defined by the symbol [Formula: see text] with [Formula: see text] [Formula: see text]. It is shown that if [Formula: see text], then the operator satisfies the following pointwise estimate: [Formula: see text] for all [Formula: see text] and all Schwartz function [Formula: see text] Here, [Formula: see text] denotes the John-Nirenberg sharp function of [Formula: see text] and [Formula: see text] stands for the Hardy–Littlewood maximal operator.

Preservation of classes of entire functions defined in terms of growth restrictions along the real axis under perturbations of their zero sets

Four special subsets of the Schwartz algebra are defined (this algebra consists of all entire functions of exponential type and of polynomial growth on the real axis). Perturbations of the zero sets for functions belonging to each of these subsets are studied. It is shown that the boundedness of the real part of the perturbing sequence is a sufficient and, generally speaking, unimprovable condition for preservation the subset from which the function in question is taken. An application of these results to spectral synthesis problems for differentiation-invariant subspaces of the Schwartz class on an interval of the real line is considered.

Two-point boundary value problem for systems of pseudo-differential equations under boundary conditions containing pseudo-differential operators

This paper deals with a two-point boundary value problem for pseudodifferential equations and for systems of second order pseudodifferential equations under boundary conditions containing pseudodifferential operators. The need to consider pseudodifferential operators is caused by two reasons, first, such equations appear more and more often in applied problems, and second, by considering such equations, it is possible to achieve the well-posedness of the boundary value problem in the Schwartz space S and in its dual space.First, we consider a scalar pseudodifferential equation with a symbol belonging to the space $C_{-\infty}^{\infty}$, consists of infinitely differentiable functions of polinomial growth. For this equation it is found of the boundary condition under which a specific type the boundary value problem is well-posed in the space S. In addition, an example of a differential-difference equation and a specific boundary condition with a convolution-type pseudo-differential operator under which this boundary value problem is well-posed in the space S are given.Then we consider a system of two pseudodifferential equations with symbols from the space $C _ {-\infty} ^ {\infty}$. For this system, we prove the existence of a well posed boundary value problem in the space S. The Fourier transform and the reduction of the system to a triangular form are used in the proof. In this case, we also give an example of a system and a specific boundary condition under which this boundary value problem is correct in the space S.Thus, the work proves that for any pseudo-differential equation, as well as for a system of two pseudo-differential equations, there is always a correct boundary value problem in the $S$ space, while the boundary conditions contain pseudo-differential operators. The algorithm for constructing correct boundary conditions is also indicated. They are pseudo-differential operators whose symbols depend on the symbols of pseudo-differential equations.