arXiv subjects

Philippe Jaming

Publications and source records attributed to Philippe Jaming.

68 records · Page 4Linked to original sources

arXiv2008

On the extremal rays of the cone of positive, positive definite functions

The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on $\R^d$. Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there many other extremals than the gaussians, thus disproving a conjecture of G. Choquet and that no reasonable conjecture can be made on the full set of extremals. The last feature of this article is to show that many characterizations of positive definite functions available in the literature are actually particular cases of the Choquet integral representations we obtain.

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arXiv2007

Distances sets that are a shift of the integers and Fourier basis for planar convex sets

The aim of this paper is to prove that if a planar set $A$ has a difference set $Δ(A)$ satisfying $Δ(A)\subset \Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if $A$ is a set of exponentials mutually orthogonal with respect to any symmetric convex set $K$ in the plane with a smooth boundary and everywhere non-vanishing curvature, then $ # (A \cap {[-q,q]}^2) \leq C(K) q$ where $C(K)$ is a constant depending only on $K$. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from \cite{IKP01} and \cite{IKT01} that if $K$ is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then $L^2(K)$ does not possess an orthogonal basis of exponentials.

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arXiv2007

An efficient algorithm for positive realizations

We observe that successive applications of known results from the theory of positive systems lead to an {\it efficient general algorithm} for positive realizations of transfer functions. We give two examples to illustrate the algorithm, one of which complements an earlier result of \cite{large}. Finally, we improve a lower-bound of \cite{mn2} to indicate that the algorithm is indeed efficient in general.

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arXiv2006

Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups

In this paper, we characterize the class of distributions on an homogeneous Lie group $\fN$ that can be extended via Poisson integration to a solvable one-dimensional extension $\fS$ of $\fN$. To do so, we introducte the $ß'$-convolution on $\fN$ and show that the set of distributions that are $ß'$-convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of $L^1$-functions. Moreover, we show that the $ß'$-convolution of such a distribution with the Poisson kernel is harmonic and has the expected boundary behaviour. Finally, we show that such distributions satisfy some global weak-$L^1$ estimates.

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arXiv2006

Nazarov's uncertainty principles in higher dimension

In this paper we prove that there exists a constant $C$ such that, if $S,Σ$ are subsets of $\R^d$ of finite measure, then for every function $f\in L^2(\R^d)$, $$\int_{\R^d}|f(x)|^2 dx \leq C e^{C \min(|S||Σ|, |S|^{1/d}w(Σ), w(S)|Σ|^{1/d})} (\int_{\R^d\setminus S}|f(x)|^2 dx + \int_{\R^d\setminusΣ}|\hat{f}(x)|^2 dx) $$ where $\hat{f}$ is the Fourier transform of $f$ and $w(Σ)$ is the mean width of $Σ$. This extends to dimension $d\geq 1$ a result of Nazarov \cite{pp.Na} in dimension $d=1$.

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arXiv2006

Uncertainty principles for orthonormal bases

In this survey, we present various forms of the uncertainty principle (Hardy, Heisenberg, Benedicks). We further give a new interpretation of the uncertainty principles as a statement about the time-frequency localization of elements of an orthonormal basis, which improves previous unpublished results of H. Shapiro. Finally, we show that Benedicks' result implies that solutions of the Shrödinger equation have some (appearently unnoticed) energy dissipation property.

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arXiv2006

Zero-free regions of radar ambiguity functions and moments

In this article, we give an estimate of the zero-free region around the origin of the ambiguity function of a signal $u$ in terms of the moments of $u$. This is done by proving an uncertainty relation between the first zero of the Fourier transform of a non-negative function and the moments of the function. As a corollary, we also give an estimate of how much a function needs to be translated to obtaina function that is orthogonal to the original function.

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arXiv2006

Uncertainty principles for orthonormal sequences

The aim of this paper is to provide complementary quantitative extensions of two results of H.S. Shapiro on the time-frequency concentration of orthonormal sequences in $L^2 (\R)$. More precisely, Shapiro proved that if the elements of an orthonormal sequence and their Fourier transforms are all pointwise bounded by a fixed function in $L^2(\R)$ then the sequence is finite. In a related result, Shapiro also proved that if the elements of an orthonormal sequence and their Fourier transforms have uniformly bounded means and dispersions then the sequence is finite. This paper gives quantitative bounds on the size of the finite orthonormal sequences in Shapiro's uncertainty principles. The bounds are obtained by using prolate spheroïdal wave functions and combinatorial estimates on the number of elements in a spherical code. Extensions for Riesz bases and different measures of time-frequency concentration are also given.

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arXiv2005

Discrete radar ambiguity problems

In this paper, we pursue the study of the radar ambiguity problem started in \cite{Ja,GJP}. More precisely, for a given function $u$ we ask for all functions $v$ (called \emph{ambiguity partners}) such that the ambiguity functions of $u$ and $v$ have same modulus. In some cases, $v$ may be given by some elementary transformation of $u$ and is then called a \emph{trivial partner} of $u$ otherwise we call it a \emph{strange partner}. Our focus here is on two discrete versions of the problem. For the first one, we restrict the problem to functions $u$ of the Hermite class, $u=P(x)e^{-x^2/2}$, thus reducing it to an algebraic problem on polynomials. Up to some mild restriction satisfied by quasi-all and almost-all polynomials, we show that such a function has only trivial partners. The second discretization, restricting the problem to pulse type signals, reduces to a combinatorial problem on matrices of a special form. We then exploit this to obtain new examples of functions that have only trivial partners. In particular, we show that most pulse type signals have only trivial partners. Finally, we clarify the notion of \emph{trivial partner}, showing that most previous counterexamples are still trivial in some restricted sense.

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arXiv2004

Maximum boundary regularity of bounded Hua-harmonic functions on tube domains

In this paper we prove that bounded Hua-harmonic functions on tube domains that satisfy some boundary regularity condition are necessarily pluriharmonic. In doing so, we show that a similar theorem is true on one-dimensional extensions of the Heisenberg group or equivalently on the Siegel upper half-plane.

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arXiv2003

Reconstruction of functions from their triple correlations

Suppose that A is a subset of an abelian group G. To know the 3-deck of A is to know the number of occurences in A of translates of each possible multiset {0,a,b}. The concept of the 3-deck is naturally extended to integrable functions on G. In this paper we study when the 3-deck of a function determines the function up to translations. The method is to look at the Fourier Transform of the function. Our emphasis is on the real line and the cyclic groups.

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arXiv2001

Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions $f$ on $\R^d$ which may be written as $P(x)\exp (Ax,x)$, with $A$ a real symmetric definite positive matrix, are characterized by integrability conditions on the product $f(x)\hat{f}(y)$. We also give the best constant in uncertainty principles of Gelf'and Shilov type. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.

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arXiv1999

Harmonic functions on the real hyperbolic ball I : Boundary values and atomic decomposition of Hardy spaces

We study harmonic functions for the Laplace-Beltrami operator on the real hyperbolic ball. We obtain necessary and sufficient conditions for this functions and their normal derivatives to have a boundary distribution.In doing so, we put forward different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball. We then study Hardy spaces of hyperbolic harmonic extensions of distributions belonging to the Hardy spaces of the sphere. In particular, we obtain an atomic decomposition of these spaces.

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arXiv1999

Harmonic functions on the real hyperbolic ball II Hardy and Lipschitz spaces

In this paper, we pursue the study of harmonic functions on the real hyperbolic ball started by the second named author. Our focus here is on the theory of Hardy, Hardy-Sobolev and Lipschitz spaces of these functions. We prove here that these spaces admit Fefferman-Stein like characterizations in terms of maximal and square functionals. We further prove that the hyperbolic harmonic extension of Lipschitz functions on the boundary extend into Lipschitz functions on the whole ball.

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