Schwartz Space Research Articles

The Generalization of the Decomposition of Functions by Energy Operators

This work starts with the introduction of a family of differential energy operators. Energy operators ($Psi_R^+$, $Psi_R^-$) were defined together with a method to decompose the wave equation in a previous work. Here the energy operators are defined following the order of their derivatives ($Psi_k^+$, $Psi_k^-$, k = {0,1,2,..}). The main part of the work is to demonstrate that for any smooth real-valued function f in the Schwartz space ($S^-(R)$), the successive derivatives of the n-th power of f (n in Z and n not equal to 0) can be decomposed using only $Psi_k^+$ (Lemma) or with $Psi_k^+$, $Psi_k^-$ (k in Z) (Theorem) in a unique way (with more restrictive conditions). Some properties of the Kernel and the Image of the energy operators are given along with the development. Finally, the paper ends with the application to the energy function.

Arithmetic properties of similitude theta lifts from orthogonal to symplectic groups

Adapting the work of Kudla and Millson we obtain a lifting of cuspidal cohomology classes for the symmetric space associated to GO(V) for an indefinite rational quadratic space V of even dimension to holomorphic Siegel modular forms on GSpn(A). For n = 2 we prove the Thom Lemma for hyperbolic 3-space, which together with results of Kudla and Millson imply an interpretation of the Fourier coefficients of the theta lift as period integrals of the cohomology class over certain cycles, and relates those over infinite geodesics to L-values of cusp forms for GL2 over imaginary quadratic fields. This allows us to prove, for almost all primes p, the p-integrality of the lift for a particular choice of Schwartz function. We further calculate the Hecke eigenvalues (including for some “bad” places) for this choice in the case of V of signature (3,1).

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

AbstractThe aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$\end{document} under the heat kernel transform on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n,$\end{document} using direct sum and direct integral of Bergmann spaces and certain unitary representations of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n$\end{document} which can be realized on the Hilbert space of Hilbert‐Schmidt operators on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^2(\mathbb {R}^n).$\end{document} We also show that the image of Sobolev space of negative order \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$\end{document} is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n$\end{document} under the heat kernel transform.

Smooth transfer of Kloosterman integrals (the Archimedean case)

We establish the existence of a transfer, which is compatible with Kloosterman integrals, between Schwartz functions on ${\rm GL}_n(\Bbb{R})$ and Schwartz functions on the variety of non-degenerate Hermitian forms. Namely, we consider an integral of a Schwartz function on ${\rm GL}_n(\Bbb{R})$ along the orbits of the two sided action of the groups of upper and lower unipotent matrices twisted by a non-degenerate character. This gives a smooth function on the torus. We prove that the space of all functions obtained in such a way coincides with the space that is constructed analogously when ${\rm GL}_n(\Bbb{R})$ is replaced with the variety of non-degenerate hermitian forms. We also obtain similar results for $\frak{gl}_n(\Bbb{R})$. The non-Archimedean case was done by H. Jacquet ({\it Duke Math. J.}, 2003) and our proof is based on the ideas of this work. However we have to face additional difficulties that appear only in the Archimedean case. Those results are crucial for the comparison of the Kuznetsov trace formula and the relative trace formula of ${\rm GL}_n$ with respect to the maximal unipotent subgroup and the unitary group, as done by H. Jacquet, and by B. Feigon, E. Lapid, and O. Offen.

On the inverse Bessel diamond kernel of Marcel Riesz

In this paper, we define the Bessel diamond kernel of Marcel Riesz and the Bessel diamond Marcel Riesz operator of order (α, β) on the function f by where α, β ∈ ℂ, the symbol * designates the convolution, and f∈𝒮, 𝒮 is the Schwartz space of functions. In this paper, we aim to study the convolution of and obtain the operator E (α, β)=[U (α, β)]−1 such that if U (α, β)(f)=ϕ, then E (α, β)ϕ=f.

De Rham theorem for Schwartz functions on Nash manifolds

In [2] the authors proved the de Rham theorem for Schwartz functions on affine Nash manifolds. Here we simplify the proof and generalize their result to the case of non-affine Nash manifolds.

Semiclassical Limit for Generalized KdV Equations Before the Gradient Catastrophe

We study the semiclassical limit of the (generalised) KdV equation, for initial data with Sobolev regularity, before the time of the gradient catastrophe of the limit conservation law. In particular, we show that in the semiclassical limit the solution of the KdV equation: i) converges in $H^s$ to the solution of the Hopf equation, provided the initial data belongs to $H^s$, ii) admits an asymptotic expansion in powers of the semiclassical parameter, if the initial data belongs to the Schwartz class. The result is also generalized to KdV equations with higher order linearities.

On the inverse diamond kernel of Marcel Riesz

In this paper, we define the diamond Marcel Riesz operator of order (α, β) on the function f by M (f) = Kα,β ∗ f, where Kα,β is diamond kernel of Marcel Riesz, α, β ∈ C, the symbol ∗ designates the convolution, and f ∈ S, S is the Schwartz space of functions. Our purpose of this paper is to obtain the operator N (α,β) = [ M (α,β) ]−1 such that if M (α,β)(f) = φ, then N (α,β)φ = f. Our results generalize formulae appearing in A. Kananthai [On the convolutions of the diamond kernel of Marcel Riesz, Applied Mathematics and Computation, 114(2000), 95 − 101]. Mathematics Subject Classification: 46F10, 46F12

IRREGULAR GABOR FRAMES

In the article we investigate irregular Gabor frames. It is shown that for any Schwartz function ψ the family of its time–frequency shifts constitutes a weighted Gabor frame if only the lattice of sampling time– frequnecy–values is dense enough. The proof uses classical analysis tool only and bases on the localization of the kernel of the Gabor transform, further, the method allows to find the density bound explicitly. A numerical example is presented.

Fourier transform of Schwartz functions on the Heisenberg group

Let $H_1$ be the 3-dimensional Heisenberg group. We prove that a modified version of the spherical transform is an isomorphism between the space $\mathcal S_m(H_1)$ of Schwartz functions of type $m$ and the space $\mathcal S(\varSigma _m)$ consisting of r

Clark-Ocone formula by the S-transform on the Poisson white noise space

Given ' a square-integrable Poisson white noise functionals we show that the Segal-Bargmann (S-) transform t 㜀! S'(Pt(g)) is absolutely continuous with d dt S '(P t(g)) = S @ � E(@t'jFt ) � (g) for almost all t 2 R, where Pt(g) = 1(1 , t) �g for g in the Schwartz space S on R, and @t means the Poisson white noise derivative. After integration with respect to t and applying the inverse S-transform, this identity recovers the Clark-Ocone formula for '.

A regularized Siegel–Weil formula onU(1,1)and application to theta lifting fromU(1)toU(1,1)

In this paper, we obtain a second term identity of the regularized Siegel–Weil formula for the unitary dual pair ( U ( 1 , 1 ) , U ( 1 , 1 ) ) , and we use this to derive a Rallis inner product formula for theta lifting from U ( 1 ) to U ( 1 , 1 ) . By choosing a good Schwartz function in the inner product formula, we obtain a special value formula for some Hecke L-function at s = 1 in terms of the norm of a theta function on U ( 1 , 1 ) .

Tight wave packet frames for [formula omitted] and [formula omitted

In this paper we establish a characterization of tight wave packet frames for L2(R) and we also prove that it is possible to construct frames in H2(R) which are given by dilation, translation and modulation of a single function ψ, where ψ, as well as ψˆ, belongs to the Schwartz class.

Characterizations of function spaces by the discrete Radon transform

Let be the lattice in and the set of all discrete hyperplanes in . Similarly, as in the Euclidean case, for a function f on , the discrete Radon transform R f is defined by the integral of f over discrete hyperplanes, and R maps functions on to functions on . In this paper, we determine the Radon transform images of the Schwartz space , the space of compactly supported functions on , and a discrete Hardy space .

Universal cycles and homological invariants of locally convex algebras

Using an appropriate notion of locally convex Kasparov modules, we show how to induce isomorphisms under a large class of functors on the category of locally convex algebras; examples are obtained from spectral triples. Our considerations are based on the action of algebraic K-theory on these functors, and involve compatibility properties of the induction process with this action, and with Kasparov-type products. This is based on an appropriate interpretation of the Connes–Skandalis connection formalism. As an application, we prove Bott periodicity and a Thom isomorphism for algebras of Schwartz functions. As a special case, this applies to the theories kk for locally convex algebras considered by Cuntz.

On a Shallow Water Equation Perturbed in Schwartz Class

We discuss the Camassa-Holm equation perturbed in Schwartz class around a suitable constant. This paper is concerned with the wave breaking mechanism for periodic case where two special classes of initial data were involved. The asymptotic behavior of solutions is also analyzed in the following sense: the corresponding solution to initial data with algebraic decay at infinity will retain this property at infinity in its lifespan.

NEW TOPOLOGICAL ℂ-ALGEBRAS WITH APPLICATIONS IN LINEAR SYSTEMS THEORY

Motivated by the Schwartz space of tempered distributions [Formula: see text] and the Kondratiev space of stochastic distributions [Formula: see text] we define a wide family of nuclear spaces which are increasing unions of (duals of) Hilbert spaces [Formula: see text], with decreasing norms ‖⋅‖p. The elements of these spaces are functions on a free commutative monoid. We characterize those rings in this family which satisfy an inequality of the form ‖f * g‖p≤ A(p - q)‖f‖q‖g‖pfor all p ≥ q + d, where*denotes the convolution in the monoid, A(p - q) is a strictly positive number and d is a fixed natural number (in this case we obtain commutative topological ℂ-algebras). Such an inequality holds in [Formula: see text], but not in [Formula: see text]. We give an example of such a ring which contains [Formula: see text]. We characterize invertible elements in these rings and present applications to linear system theory.

Condition de non-résonance pour l’oscillateur harmonique quantique perturbé

On R d , we consider the differential operator −�+jj xjj +R(x)+M. Here R is in the Schwartz class and M is a bounded operator of L (R). Under a localization hypothesis on the spectrum of −� + jj xjj + R(x), which is automatically satisfied if d = 1, one proves that there are arbitrarily small operator M, i.e. jj M jj � 1, such that for all r � 3, for small Cauchy data of order in high Sobolev norms, the nonlinear Schrodinger equation admits a solution which is bounded by 2 on a time existence interval of length � −r . The case R = 0 has been proved by Grebert-Imekraz-Paturel. Here we do not need any explicit spectral analysis (eigenvalues and eigenfunctions) of −� + jj xjj + R(x). The role of M is making the spectrum of −�+ jj xjj + R(x)+ M nonresonant. We also give a partial answer in the case M = 0 : there are some potentials R such that the spectrum of −@2 x + x2 + R(x) is nonresonant. For this, we use some Chelkak-Kargaev-Korotyaev's results about the inverse spectral problem of harmonic oscillator and construct explicit dual basis of the squared Hermite functions. Sur R d , nous considerons l'operateur differentiel −� + jj xjj + R(x) +

A Dimensionally Continued Poisson Summation Formula

We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, our methods work to establish the (Voronoi) summation formulae associated with functions satisfying (modular) transformations of the Jacobi imaginary type by means of a density argument (as opposed to the usual Mellin transform approach). In particular, we construct a family of generalized theta series from Jacobi theta functions from which these summation formulae can be obtained. This family contains several families of modular forms, but is significantly more general than any of them. Our result also relaxes several of the hypotheses in the standard statements of these summation formulae. The density result we prove for Gaussians in the Schwartz space may be of independent interest.

Invariant subspace problem for classical spaces of functions

We construct continuous linear operators without non-trivial invariant subspaces on several classical non-Banach spaces of analysis: the Schwartz space of rapidly decreasing functions, the space of smooth functions on a compact smooth manifold, the space of holomorphic functions on the unit disc or on an arbitrary polydisc, the space of entire functions on C d for d ⩾ 2 . As these are reflexive spaces, the result gives an analogous statement for the dual spaces, e.g. the space of tempered distributions. The construction works with Köthe sequence spaces and is based on methods developed by Read to construct his famous example on the space l 1 .