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## ‣ Sets of injectivity for weighted twisted spherical means and support theorems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In this article, we show that the spheres $S_R(o)=\{z\in\mathbb C^n: |z|=R\}$
are sets of injectivity for the weighted twisted spherical means (WTSM) for a
suitable class of functions on $\mathbb C^n$. The weights here are spherical
harmonics on $S^{2n-1}.$ In general, the question of set of injectivity for the
twisted spherical means (TSM) with real analytic weight is still open. We would
like to refer to \cite{NRR}, for some results on the sets of injectivity for
the spherical means with real analytic weights in the Euclidean setup.; Comment: Published: J. Fourier Anal. Appl., 18(2012), no. 3, 592-608. arXiv
admin note: text overlap with arXiv:1103.4571

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## ‣ Note on a result of Kerman and Weit

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/09/2011
Português

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A result in \cite{Ker-Weit} states that a real valued continuous function $f$
on the circle and its nonnegative integral powers can generate a dense
translation invariant subspace in the space of all continuous functions on the
circle if $f$ has a unique maximum or a unique minimum. In this note we
endeavour to show that this is quite a general phenomenon in harmonic analysis.; Comment: 5 pages

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## ‣ A spectral gap property for subgroups of finite covolume in Lie groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/03/2009
Português

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Let G be a real Lie group and H a lattice or, more generally, a closed
subgroup of finite covolume in G. We show that the unitary representation
lambda_{G/H} of G on L^2(G/H) has a spectral gap, that is, the restriction of
lambda_{G/H} to the orthogonal of the constants in L^2(G/H) does not have
almost invariant vectors. This answers a question of G. Margulis. We give an
application to the spectral geometry of locally symmetric Riemannian spaces of
infinite volume.; Comment: 9 pages, no figure

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## ‣ Nilpotent Gelfand pairs and spherical transforms of Schwartz functions III. Isomorphisms between Schwartz spaces under Vinberg's condition

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/10/2012
Português

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#Mathematics - Functional Analysis#Mathematics - Commutative Algebra#Mathematics - Representation Theory#Primary: 13A50, 43A32, Secondary: 43A85, 43A90

Let (N,K) be a nilpotent Gelfand pair, i.e., N is a nilpotent Lie group, K a
compact group of automorphisms of N, and the algebra D(N)^K of left-invariant
and K-invariant differential operators on N is commutative. In these
hypotheses, N is necessarily of step at most two. We say that (N,K) satisfies
Vinberg's condition if K acts irreducibly on $n/[n,n]$, where n= Lie(N).
Fixing a system D of d formally self-adjoint generators of D(N)^K, the
Gelfand spectrum of the commutative convolution algebra L^1(N)^K can be
canonically identified with a closed subset S_D of R^d. We prove that, on a
nilpotent Gelfand pair satisfying Vinberg's condition, the spherical transform
establishes an isomorphism from the space of $K$-invariant Schwartz functions
on N and the space of restrictions to S_D of Schwartz functions in R^d.; Comment: 51 pages

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## ‣ The domain of the Fourier integral

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/02/2011
Português

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We consider the problem of determining the Fourier integral in the Hilbert
space of square integrable functions. Fourier integral is the scalar product of
two functions belonging to the Hilbert space of square integrable functions and
the Hilbert space of almost periodic functions. Scalar product for different
Hilbert spaces defined at the intersection of these spaces, which contains only
one zero element. Therefore, the Fourier integral is not defined in the Hilbert
space of square integrable functions with nonzero norm.; Comment: PDF, 4 p

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## ‣ Compact symmetric spaces, triangular factorization, and Cayley coordinates

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Let U/K represent a connected, compact symmetric space, where theta is an
involution of U that fixes K, phi: U/K to U is the geodesic Cartan embedding,
and G is the complexification of U. We investigate the intersection of phi(U/K)
with the Bruhat decomposition of G corresponding to a theta-stable triangular,
or LDU, factorization of the Lie algebra of G. When g in phi(U/K) is generic,
the corresponding factorization g=ld(g)u is unique, where l in N^-, d(g) in H,
and u in N^+. In this paper we present an explicit formula for d in Cayley
coordinates, compute it in several types of symmetric spaces, and use it to
identify representatives of the connected components of the generic part of
phi(U/K). This formula calculates a moment map for a torus action on the
highest dimensional symplectic leaves of the Evens-Lu Poisson structure on U/K.; Comment: 19 pages: Main proof entirely rewritten, sections reorganized,
exposition made more precise and concise. To appear in Pacific Journal of
Mathematics

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## ‣ Complete monotonicity for inverse powers of some combinatorially defined polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Combinatorics#Mathematical Physics#Mathematics - Classical Analysis and ODEs#Mathematics - Representation Theory#05C31 (Primary), 05A15, 05A20, 05B35, 05C05, 05C50, 05E99, 15A15,
15B33, 15B57, 17C99, 26A48, 26B25, 26C05, 32A99, 43A85, 44A10, 60C05, 82B20
(Secondary)

We prove the complete monotonicity on $(0,\infty)^n$ for suitable inverse
powers of the spanning-tree polynomials of graphs and, more generally, of the
basis generating polynomials of certain classes of matroids. This generalizes a
result of Szego and answers, among other things, a long-standing question of
Lewy and Askey concerning the positivity of Taylor coefficients for certain
rational functions. Our proofs are based on two_ab initio_ methods for proving
that $P^{-\beta}$ is completely monotone on a convex cone $C$: the
determinantal method and the quadratic-form method. These methods are closely
connected with harmonic analysis on Euclidean Jordan algebras (or equivalently
on symmetric cones). We furthermore have a variety of constructions that, given
such polynomials, can create other ones with the same property: among these are
algebraic analogues of the matroid operations of deletion, contraction, direct
sum, parallel connection, series connection and 2-sum. The complete
monotonicity of $P^{-\beta}$ for some $\beta > 0$ can be viewed as a strong
quantitative version of the half-plane property (Hurwitz stability) for $P$,
and is also related to the Rayleigh property for matroids.; Comment: LaTeX2e, 70 pages (v2) or 82 pages (v3). Version 2 (accepted for
publication in Acta Mathematica) is significantly reorganized at the
suggestion of a referee; also...

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## ‣ Quantitative property A, Poincare inequalities, L^p-compression and L^p-distortion for metric measure spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We introduce a quantitative version of Property A in order to estimate the
L^p-compressions of a metric measure space X. We obtain various estimates for
spaces with sub-exponential volume growth. This quantitative property A also
appears to be useful to yield upper bounds on the L^p-distortion of finite
metric spaces. Namely, we obtain new optimal results for finite subsets of
homogeneous Riemannian manifolds. We also introduce a general form of Poincare
inequalities that provide constraints on compressions, and lower bounds on
distortion. These inequalities are used to prove the optimality of some of our
results.; Comment: 26 pages

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## ‣ When is a Riesz distribution a complex measure?

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Classical Analysis and ODEs#Mathematics - Representation Theory#Mathematics - Statistics Theory#43A85 (Primary) 17A15, 17C99, 28C10, 44A10, 46F10, 47G10, 60E05,
62H05 (Secondary)

Let R_\alpha be the Riesz distribution on a simple Euclidean Jordan algebra,
parametrized by the complex number \alpha. I give an elementary proof of the
necessary and sufficient condition for R_\alpha to be a locally finite complex
measure (= complex Radon measure).; Comment: LaTeX2e, 15 pages. Version 2 contains some small changes suggested by
a referee

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## ‣ Ramanujan's Master Theorem for Riemannian symmetric spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Representation Theory#Mathematics - Functional Analysis#43A85 (Primary) 22E45, 33E20 (Secondary)

Ramanujan's Master theorem states that, under suitable conditions, the Mellin
transform of a power series provides an interpolation formula for the
coefficients of this series. Based on the duality of Riemannian symmetric
spaces of compact and noncompact type inside a common complexification, we
prove an analogue of Ramanujan's Master Theorem for the spherical Fourier
transform of a spherical Fourier series. This extend the results proven by
Bertram for Riemannian symmetric spaces of rank-one.; Comment: As accepted by JFA

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## ‣ Invariant differential operators on nonreductive homogeneous spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/08/2000
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A systematic exposition is given of the theory of invariant differential
operators on a not necessarily reductive homogeneous space. This exposition is
modelled on Helgason's treatment of the general reductive case and the special
non-reductive case of the space of horocycles. As a final application the
differential operators on (not a priori reductive) isotropic pseudo-Riemannian
spaces are characterized.; Comment: 11 pages, electronic version of old 1981 report

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## ‣ Wavelets in Banach Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Functional Analysis#Mathematical Physics#Mathematics - Complex Variables#Mathematics - Representation Theory#Quantum Physics#43A85 (Primary)#32M99, 43A32, 46E10, 47A60, 47A67, 47C99, 81R30,
81S10 (Secondary)

We describe a construction of wavelets (coherent states) in Banach spaces
generated by ``admissible'' group representations. Our main targets are
applications in pure mathematics while connections with quantum mechanics are
mentioned. As an example we consider operator valued Segal-Bargmann type spaces
and the Weyl functional calculus.
Keywords: Wavelets, coherent states, Banach spaces, group representations,
covariant, contravariant (Wick) symbols, Heisenberg group, Segal-Bargmann
spaces, Weyl functional calculus (quantization), second quantization, bosonic
field.; Comment: 37 pages; LaTeX2e; no pictures; 27/07/99: many small corrections

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## ‣ Fourier restriction Theorem and characterization of weak $L^2$ eigenfunctions of the Laplace--Beltrami operator

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/04/2012
Português

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In this paper we prove the Fourier restriction theorem for $p=2$ on
Riemannian symmetric spaces of noncompact type with real rank one which extends
the earlier result proved in \cite[Theorem 1.1]{KRS}. This result depends on
the weak $L^2$ estimates of the Poisson transform of $L^2$ function. By using
this estimate of the Poisson transform we also characterizes all weak $L^2$
eigenfunction of the Laplace--Beltrami operator of Riemannian symmetric spaces
of noncompact type with real rank one and eigenvalue $-(\lambda^2+\rho^2)$ for
$\lambda\in\R\setminus\{0\}$.

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## ‣ Crested products of Markov chains

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/03/2009
Português

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In this work we define two kinds of crested product for reversible Markov
chains, which naturally appear as a generalization of the case of crossed and
nested product, as in association schemes theory, even if we do a construction
that seems to be more general and simple. Although the crossed and nested
product are inspired by the study of Gelfand pairs associated with the direct
and the wreath product of two groups, the crested products are a more general
construction, independent from the Gelfand pairs theory, for which a complete
spectral theory is developed. Moreover, the $k$-step transition probability is
given. It is remarkable that these Markov chains describe some classical models
(Ehrenfest diffusion model, Bernoulli--Laplace diffusion model, exclusion
model) and give some generalization of them. As a particular case of nested
product, one gets the classical Insect Markov chain on the ultrametric space.
Finally, in the context of the second crested product, we present a
generalization of this Markov chain to the case of many insects and give the
corresponding spectral decomposition.; Comment: Published in at http://dx.doi.org/10.1214/08-AAP546 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## ‣ On the Uniqueness of Solutions of the Schr\"odinger Equation on Riemannian Symmetric Spaces of the Noncompact Type

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Let X be a Riemannian symmetric space of the noncompact type. We prove that
the solution of the time-dependent Schr\"odinger equation on X with square
integrable initial condition f is identically zero at all times t whenever f
and the solution at a time t0 > 0 are simultaneously very rapidly decreasing.
The stated condition of rapid decrease is of Beurling type. Conditions
respectively of Gelfand-Shilov, Cowling-Price and Hardy type are deduced.; Comment: 20 pages, To appear in Annales de l Institut Fourier

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## ‣ A Lie-algebraic approach to the local index theorem on compact homogeneous spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Functional Analysis#Mathematics - Differential Geometry#Mathematics - Representation Theory#58J20, 35K08, 43A85 (Primary) 17B70, 58J35 (Secondary)

Using a K-theory point of view, Bott related the Atiyah-Singer index theorem
for elliptic operators on compact homogeneous spaces to the Weyl character
formula. This article explains how to prove the local index theorem for compact
homogenous spaces using Lie algebra methods. The method follows in outline the
proof of the local index theorem due to Berline and Vergne. But the use of
Kostant's cubic Dirac operator in place of the Riemannian Dirac operator leads
to substantial simplifications. An important role is also played by the quantum
Weil algebra of Alekseev and Meinrenken.; Comment: reduced in length; error in Prop. 5.4 [v2] corrected (now Prop. 4.4)

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## ‣ Coxeter system of lines are sets of injectivity for the twisted spherical means on $\mathbb C$

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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It is well known that a line in $\mathbb R^2$ is not a set of injectivity for
the spherical means for odd functions about that line. We prove that any line
passing through the origin is a set of injectivity for the twisted spherical
means (TSM) for functions $f\in L^2(\mathbb C),$ whose each of spectral
projection $ e^{\frac{1}{4}|z|^2}f\times\varphi_k$ is a polynomial. Then, we
prove that any Coxeter system of even number of lines is a set of injectivity
for the TSM for $L^q(\mathbb C),~1\leq q\leq2.$; Comment: This article is accepted in J. Funct. Anal. jointly with
arXiv:1204.2773

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## ‣ Two body problem on two point homogeneous spaces, invariant differential operators and the mass center concept

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematical Physics#Mathematics - Differential Geometry#70F05, 81R15, 43A85 (Primary), 22E70, 22F30, 70G65 (Secondary)

We consider the two body problem with central interaction on two point
homogeneous spaces from point of view of the invariant differential operators
theory. The representation of the two particle Hamiltonian in terms of the
radial differential operator and invariant operators on the symmetry group is
found. The connection of different mass center definitions for these spaces to
the obtained expression for Hamiltonian operator is studied.; Comment: 26 pages, LaTeX, no figures, text improved

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## ‣ Spectral multipliers for the Kohn Laplacian on forms on the sphere in $\mathbb{C}^n$

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Analysis of PDEs#Mathematics - Functional Analysis#42B15, 43A85 (Primary), 32V20 (Secondary)

The unit sphere $\mathbb{S}$ in $\mathbb{C}^n$ is equipped with the
tangential Cauchy-Riemann complex and the associated Laplacian $\Box_b$. We
prove a H\"ormander spectral multiplier theorem for $\Box_b$ with critical
index $n-1/2$, that is, half the topological dimension of $\mathbb{S}$. Our
proof is mainly based on representation theory and on a detailed analysis of
the spaces of differential forms on $\mathbb{S}$.; Comment: 28 pages

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## ‣ Eigenmodes of Lens and Prism Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Spectral Theory#Mathematics - Geometric Topology#51P05, 85A40, 43A85 (primary) 57M99 (secondary)

Cosmologists are taking a renewed interest in multiconnected spherical
3-manifolds (spherical spaceforms) as possible models for the physical
universe. To understand the formation of large scale structures in such a
universe, cosmologists express physical quantities, such as density
fluctuations in the primordial plasma, as linear combinations of the eigenmodes
of the Laplacian, which can then be integrated forward in time. This need for
explicit eigenmodes contrasts sharply with previous mathematical
investigations, which have focused on questions of isospectrality rather than
eigenmodes. The present article provides explicit orthonormal bases for the
eigenmodes of lens and prism spaces. As a corollary it extends known results on
spectra from homogeneous lens spaces L(p,1) [Ikeda 1995] to arbitrary lens
spaces L(p,q).; Comment: 19 pages, 3 figures

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