Functional series representable as a sum of two universal series

Series with respect to systems Φ{φ_n(x)}_n=1^∞ of measurable and almost everywhere finite functions are discussed. A necessary and sufficient condition for representing any series with respect to a system Φ as a sum of two universal series is formulated. A consequence of the condition is that any series with respect to an arbitrary complete and orthonormal system Φ is a sum of two universal series.     

On universal formal power series

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C^∗. The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.     

Maximal smoothness of the anti-analytic part of a trigonometric null series

We proved recently (C. R. Acad. Sci. Paris, Ser. I 336 (2003) 475–478) that the anti-analytic part of a trigonometric series, converging to zero almost everywhere, may belong to L² on the circle. Here we prove that it can even be C^∞, and we characterize precisely the possible degree of smoothness in terms of the rate of decrease of the Fourier coefficients. This sharp condition might be viewed as a ‘new quasi-analyticity’. To cite this article: G. Kozma, A. Olevski??, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A null series with small anti-analytic part

We show that it is possible for an L² function on the circle, which is a sum of an almost everywhere convergent series of exponentials with positive frequencies, to not belong to the Hardy space H². A consequence in the uniqueness theory is obtained. To cite this article: G. Kozma, A. Olevskiǐ, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Power series solutions to Volterra integral equations

Power series type solutions are given for a wide class of linear and q-dimensional nonlinear Volterra equations on R^p. The basic assumption on the kernel K(x, y) is that K(x, xt) has a power series in x. For example, this holds for any analytic kernel.The kernel may be strongly singular, provided certain constants are finite. One and only one such power series solution exists. Its coefficients are given by a simple iterative formula. In many cases this may be solved explicitly. In particular an explicit formula for the resolvent is given.     

Functional time series prediction via conditional mode estimation

This Note focuses on an estimator of the conditional mode of a scalar response Y given a functional random variable X. We start by building a kernel estimator of the conditional density of Y given X; the conditional mode is defined as the value which maximizes this conditional density. We establish the almost complete convergence for this estimate under α-mixing assumption. To cite this article: F. Ferraty et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. To cite this article: F.M. Bleher, T. Chinburg, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Factorizing multivariate time series operators

Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a₁B + … + a_pB^p) X(t) = [Π_i=1^p (I ? α_iB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a₁, a₂,…, a_p. In fact the number of possibilities is limited upwards by $\text{(np)!}\text{(n!)}{}^{\mathrm{p}}$, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.     

BMO estimates for lacunary series

We proveBMO andL ^p norm inequalities inR ⁿ for lacunary Walsh and generalized trigonometric series.     

On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series

For a rational formal power series in N non-commuting indeterminates, with matrix coefficients, we establish the formula which relates the intersection of the null spaces of coefficients to the intersection of the null spaces of values of this series at N-tuples of $n\times n$ matrices, for n large enough. As an application, we formulate the criteria of observability, controllability, and minimality for a recognizable formal power series representation in terms of matrix substitutions. To cite this article: D. Alpay, D.S. Kalyuzhny??-Verbovetzki??, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Braids on surfaces and finite type invariants

We prove that there is no functorial universal finite type invariant for braids in Σ×I if the genus of Σ is positive. To cite this article: P. Bellingeri, L. Funar, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On series containing products of Legendre polynomials

Explicit formulas are established for infinite sums of products of three or four Legendre polynomials of nth order with coefficients 2n + 1; the series depends only the arguments of the polynomials and contains no other variables. We show that, for the product of three polynomials, the sum is inverse to the root of the product of four sine functions and, in the case of four polynomials, this expression additionally contains the elliptic integral K(k) as a multiplier. Analogs and particular cases are considered which allow one to compare the relationships proved in this note with results proved in various domains of mathematical physics and classical functional analysis.     

Théorèmes ergodiques aléatoires et suites de poids aléatoires régularisantsRandom ergodic theorems and regularizing random weights

In this work, we study the convergence of pointwise ergodic averages for random subsequences, in a universal framework. We give also results on the convergence of averages which are modulated by random weights. These results are obtained as a consequence of the study of the regularity of random trigonometric polynomials. The methods used in this work involve mainly Gaussian tools, random trigonometric polynomials and spectral theory. To cite this article: S. Durand, D. Schneider, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 375–378.     

Asymptotic distributions of functions of the eigenvalues of sample covariance matrix and canonical correlation matrix in multivariate time series

Let S = (1/n) Σ_t=1ⁿ X(t) X(t)′, where X(1), …, X(n) are p × 1 random vectors with mean zero. When X(t) (t = 1, …, n) are independently and identically distributed (i.i.d.) as multivariate normal with mean vector 0 and covariance matrix Σ, many authors have investigated the asymptotic expansions for the distributions of various functions of the eigenvalues of S. In this paper, we will extend the above results to the case when {X(t)} is a Gaussian stationary process. Also we shall derive the asymptotic expansions for certain functions of the sample canonical correlations in multivariate time series. Applications of some of the results in signal processing are also discussed.     

The topology of corank 1 multi-singularities of stable smooth mappings of equidimensional manifolds

We study conditions for the coexistence of singularities of a stable smooth mapping of a closed manifold into a manifold of the same dimension n. Assuming that this mapping has only singularities of corank 1, we find universal linear relations between the Euler characteristics of the manifolds of multi-singularities in the image of the considered mapping. To cite this article: V.D. Sedykh, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Régulateurs supérieurs,périodes et valeurs spéciales de la fonction L de degré 4 de GSp(4)

We present some results in direction of Beilinson's conjecture for the degree four L-function of an automorphic representation π of the symplectic group GSp(4). Our main result relates some motivic cohomology classes over powers of the universal abelian scheme over the Shimura variety of GSp(4) to the product of an archimedean integral, an occult period defined by Harris, a Deligne period and the special value of the L-function predicted by Beilinson's conjecture. We assume that π is stable, of multiplicity one, and has a Bessel model of a specific type. To cite this article: F. Lemma, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Absolute convergence of double series of Fourier-Haar coefficients for functions of bounded p-variation

We consider functions of two variables of bounded p-variation of the Hardy type on the unit square. For these functions we obtain a sufficient condition for the absolute convergence of series of positive powers of Fourier coefficients with power-type weights with respect to the double Haar system. This condition implies those for the absolute convergence of series of Fourier-Haar coefficients of one-variable functions which have a bounded Wiener p-variation or belong to the class Lip ??. We show that the obtained results are unimprovable. We also formulate N-dimensional analogs of the main result and its corollaries.     

Stabilisation d'ondes électromagnétiques dans un domaine extérieur 2D

We consider the stabilization of Maxwell equations in two dimensional on the exterior of an arbitrary bounded obstacle. Using some results concerning of wave equations with Neumann boundary condition in the exterior domain. We deduce under the ‘exterior geometric control condition’ the behavior of the solution for large time. To cite this article: A. Moulahi, C. R. Acad. Sci. Paris, Ser. I 342 (2006).