The minimal generating space of a polynomial

Given a polynomial $\text{P(X}{}_1\text{,…,X}{}_{\mathrm{N}}\text{)∈}\text{R}\text{[}\text{X}\text{]}$, we calculate a subspace G_p of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property $\text{P∈}\text{R}\text{[}\text{G}{}_{\mathrm{p}}\text{]}$ (the algebra generated by G_p, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X₁,…,X_n) and Q(X₁,…,X_n) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as $\text{P}{}^1\text{(Y}{}_1\text{,…,Y}{}_{\mathrm{k}}\text{)}$ and $\text{Q}{}^1\text{(Y}{}_{\mathrm{<mtext>k+</mtext><mtext>1</mtext>}}\text{,…,Y}{}_{\mathrm{n}}\text{)}$ respectively, where $\text{Y}\text{=T}\text{X}$.     

On Guerra's broken replica-symmetry bound

Consider a random Hamiltonian $\text{H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{)}$ for $\text{σ}\text{→}\text{∈Σ}{}_{\mathrm{N}}\text{={0,1}}{}^{\mathrm{<mtext>N</mtext>}}\text{.}$ We assume that the family $\text{(H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{))}$ is jointly Gaussian centered and that for $\text{σ}\text{→}{}^1\text{,}\text{σ}\text{→}{}^2\text{∈Σ}{}_{\mathrm{N}}\text{,}$$\text{N}{}^{\mathrm{?1}}\text{EH}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}{}^1\text{)H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}{}^2\text{)}$ =ξ(N^?1∑_i?Nσ¹_iσ²_i) for a certain function ξ on $\text{R}$. F. Guerra proved the remarkable fact that the free energy of the system with Hamiltonian $\text{H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{)+h∑}{}_{\mathrm{i?N}}\text{σ}{}_{\mathrm{i}}$ is bounded below by the free energy of the Parisi solution provided that ξ is convex on $\text{R}$. We prove that this fact remains (asymptotically) true when the function ξ is only assumed to be convex on $\text{R}{}^{\mathrm{+}}$. This covers in particular the case of the p-spin interaction model for any p. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On the Smale theorem in weak shape theory

We modify various lemmas from Dydak's paper on the Vietoris-Smale theorem to obtain more general results. We consider closed subsets X and Y of paracompact spaces M, N respectively, and a map F:M→N such that F∥X:X→Y is closed and surjective and X=F^-1(Y) to obtain the following result.(Theorem). If $\text{F}{}^{-1}\text{(y)ϵAC}{}^{\mathrm{n}}{}_{\mathrm{M}}\text{(}\text{K}\text{)}$ for each y ϵ Y and N is LCⁿ⁺¹, then the morphism in-pro-$\text{π}{}_{\mathrm{k}}\text{[F]:π}{}_{\mathrm{k}}\text{U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{M}}\text{(X,x)→π}{}_{\mathrm{k}}\text{U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{N}}\text{(Y,y)}$ induced by the morphism $\text{[F]:U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{M}}\text{(X,x)→U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{N}}\text{(Y,y)}$ is an isomorphism of in-pro-Grp for k⩽n and an epimorphism for k=n+1.     

Hermite series estimates of a probability density and its derivatives

The following estimate of the pth derivative of a probability density function is examined: $\text{Σ}{}_{\mathrm{k\; =\; 0}}{}^{\mathrm{N}}\text{a}\text{?}{}_{\mathrm{k}}\text{h}{}_{\mathrm{k}}\text{(x)}$, where h_k is the kth Hermite function and $\text{a}\text{?}{}_{\mathrm{k}}\text{= (}\text{(?1)}{}^{\mathrm{p}}\text{n}\text{)}$Σ_{i = 1}ⁿh_k^(p)(X_i) is calculated from a sequence X₁,…, X_n of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n^?α) and O(n^?α log n), respectively, where $\text{α =}\text{2(r ? p)}\text{(2r + 1)}$. Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n^?β) and $\text{O(n}{}^{\mathrm{<mtext>?\beta </mtext><mtext>2</mtext>}}\text{log}\text{n)}$, respectively, where $\text{β =}\text{(2(r ? p) ? 1)}\text{(2r + 1)}$.     

On the probability that k positive integers are relatively prime II

Given a set S of positive integers let $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t)}$ denote the number of k-tuples 〈m₁, …, m_k〉 for which $\text{m}{}_{\mathrm{i}}\text{∈ S ? [1, t]}$ and (m₁, …, m_k) = 1. Also let $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n)}$ denote the probability that k integers, chosen at random from $\text{S ? [1, n]}$, are relatively prime. It is shown that if P = {p₁, …, p_r} is a finite set of primes and S = {m : (m, p₁ … p_r) = 1}, then $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t) = (td(S))}{}^{\mathrm{k}}\text{Π}{}_{\mathrm{\nu ?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) + O(t}{}^{\mathrm{k?1}}\text{)}$ if k ≥ 3 and $\text{Z}{}_2{}^{\mathrm{S}}\text{(t) = (td(S))}{}^2\text{Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^2\text{) + O(t}\text{log}\text{t)}$ where d(S) denotes the natural density of S. From this result it follows immediately that $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) = (ζ(k))}{}^{\mathrm{?1}}\text{Π}{}_{\mathrm{p\in P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$ as n → ∞. This result generalizes an earlier result of the author's where $\text{P = ?}$ and S is then the whole set of positive integers. It is also shown that if S = {p₁^x₁ … p_r^x_r : x_i = 0, 1, 2,…}, then $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → 0}$ as n → ∞.     

Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

An elementary proof is given of the author's transformation formula for the Lambert series $\text{G}{}_{\mathrm{p}}\text{(x) =}\text{Σ}{}_{\mathrm{n?1}}{}^{\mathrm{<mtext>\infty </mtext>}}\text{n}{}^{\mathrm{?p}}\text{x}{}^{\mathrm{n}}\text{(1?x}{}^{\mathrm{n}}\text{)}$ relating G_p(e^2πiτ) to G_p(e^2πiAτ), where p > 1 is an odd integer and $\text{Aτ =}\text{(aτ + b)}\text{(cτ + d)}$ is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function $\text{log}\text{η(τ) =}\text{πiτ}\text{12}\text{? G}{}_1\text{(e}{}^{\mathrm{2\pi i\tau }}\text{)}$, and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.     

Cesàro asymptotics for the orders of SLk(Zn) and GLk(Zn) as n→∞

Given an integer k>0, our main result states that the sequence of orders of the groups $\text{SL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$ (respectively, of the groups $\text{GL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$) is Cesàro equivalent as n→∞ to the sequence C₁(k)n^k2?1 (respectively, C₂(k)n^k2), where the coefficients C₁(k) and C₂(k) depend only on k; we give explicit formulas for C₁(k) and C₂(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function ?(n) is Cesàro equivalent to $\text{n}\text{6}\text{π}{}^2$. We present some experimental facts related to the main result. To cite this article: A.G. Gorinov, S.V. Shadchin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

The asymptotic eigenvalue-distribution for a certain class of random matrices

For an n × n Hermitean matrix A with eigenvalues λ₁, …, λ_n the eigenvalue-distribution is defined by $\text{G(x, A) :=}\text{1}\text{n}$ · number {λ_i: λ_i ? x} for all real x. Let A_n for n = 1, 2, … be an n × n matrix, whose entries a_ik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, $\text{R}$, p) with the same distribution F_a. Suppose that all moments $\text{E}$ | a | ^k, k = 1, 2, … are finite, $\text{E}$a=0 and $\text{E}$ | a | ². Let

$\text{M(A)=}\text{∑}\text{σ=1}\text{s}\text{θ}{}_{\mathrm{\sigma }}\text{P}{}_{\mathrm{\sigma }}\text{(A,A}{}^1\text{)}$

with complex numbers θ_σ and finite products P_σ of factors A and $\text{A}{}^1$ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G₀(x) = G₁x) + G₂(x), where G₁ is a step function and G₂ is absolutely continuous, such that with probability $\text{1 G(x, M(}\text{A}{}_{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{))}$ converges to G₀(x) as n → ∞ for all continuity points x of G₀. The density g of G₂ vanishes outside a finite interval. There are only finitely many jumps of G₁. Both, G₁ and G₂, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples $\text{A}{}_{\mathrm{r}}\text{A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{+ A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{A}{}^{\mathrm{1r}}\text{± A}{}^{\mathrm{1r}}\text{A}{}^{\mathrm{r}}$, r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form

$\text{lim sup}\text{n→∞}\text{∑}\text{i=1}{}^{\mathrm{n}}\text{|γ}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{|}{}^2\text{?6A}{}_{\mathrm{n}}\text{6}{}^2\text{? 0.8228…}$

of Schur's inequality for the eigenvalues λ_i⁽ⁿ⁾ of A_n holds. Consequently random matrices do not tend to be normal matrices for large n.     

The probability that k positive integers are relatively r-prime

If r, k are positive integers, then $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ denotes the number of k-tuples of positive integers (x₁, x₂, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r = 1. An explicit formula for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ is derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}\text{n}{}^{\mathrm{k}}\text{=}\text{1}\text{ζ(rk)}$.If S = {p₁, p₂, …, p_a} is a finite set of primes, then 〈S〉 = {p₁^a₁p₂^a₂…p_s^a_s; p_i ∈ S and a_i ≥ 0 for all i} and $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ denotes the number of k-tuples (x₁, x₃, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r ∈ 〈S〉. Asymptotic formulas for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ are derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}\text{n}{}^{\mathrm{k}}\text{=}\text{(p}{}_1\text{… p}{}_{\mathrm{a}}\text{)}{}^{\mathrm{rk}}\text{ζ(rk)(p}{}_1{}^{\mathrm{rk}}\text{? 1) … (p}{}_{\mathrm{s}}{}^{\mathrm{rk}}\text{? 1)}$.     

On the closability of some positive definite symmetric differential forms on c0∞(ω)

Let $\text{S}$ be a Dirichlet form in $\text{L}{}^2\text{(Ω; m)}$, where Ω is an open subset of $\text{R}$ⁿ, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let $\text{S}$_k be a Dirichlet form on some k-dimensional submanifold $\text{Ω}{}_{\mathrm{k}}$ of Ω. The paper is devoted to the study of the closability of the forms E with domain $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ and defined by: $\text{(?,g)=}$$\text{E}$$\text{(?, g)+}\text{∑}\text{i}\text{p}\text{=1}$$\text{E}$k_i where 1 ? k_p < ? < n, and where , g^k_i denote restrictions of ?, g in $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ to . Conditions are given for E to be closable if, for each i = 1,…, p, one has k_i = n ? i. Other conditions are given for E to be nonclosable if, for some i, k_i < n ? i.     

Best pseudo-isolated Gerschgorin disks for Eigenvalues

Let A be an arbitrary n×n matrix, partitioned so that if A=[A_ij], then all submatrices A_ii are square. If x is a positive vector, it is well-known that $\text{G(}\text{x}\text{) =∪}{}^{\mathrm{N}}{}_{\mathrm{i=1}}\text{G}{}_{\mathrm{i}}\text{(}\text{x}\text{)}$, where

$\text{G}{}_{\mathrm{i}}\text{(}\text{x}\text{) =}\text{z}\text{6(zI ? A}{}_{\mathrm{ii}}\text{)}{}^{\mathrm{?1}}\text{6}{}^{\mathrm{?1}}\text{?}\text{1}\text{x}{}_{\mathrm{i}}\text{∑}\text{j = 1}\text{j ≠ i}\text{N}\text{`6A}{}_{\mathrm{ij}}\text{6x}{}_{\mathrm{j}}$

, contains all the eigenvalues of A. The purpose of this paper is to give a new definition of the concept of an isolated subregion of G(x). An algorithm is given for obtaining the best such isolated subregion in a certain sense, and examples are given to show that tighter bounds for some eigenvalues of A may be obtained than with previous algorithms. For ease of computation, each subregion G_i(x) is replaced by the union of circular disks centered at the eigenvalues of A_ii.     

Eigenvalues of p-summing and lp-type operators in Banach spaces

This paper is a study of the distribution of eigenvalues of various classes of operators. In Section 1 we prove that the eigenvalues (λ_n(T)) of a p-absolutely summing operator, p ? 2, satisfy

$\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{?π}{}_{\mathrm{p}}\text{(T).}$

This solves a problem of A. Pietsch. We give applications of this to integral operators in L_p-spaces, weakly singular operators, and matrix inequalities.In Section 2 we introduce the quasinormed ideal Π₂⁽ⁿ⁾, P = (p₁, …, p_n) and show that for T ∈ Π₂⁽ⁿ⁾, 2 = (2, …, 2) ∈ Nⁿ, the eigenvalues of T satisfy

$\text{∑}\text{i∈N}\text{|λ}{}_{\mathrm{i}}\text{(T)|}{}^{\mathrm{<mtext>2</mtext><mtext>n</mtext>}}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}\text{?π}{}^{\mathrm{n}}{}_2\text{(T).}$

More generally, we show that for T ∈ Π_p⁽ⁿ⁾, P = (p₁, …, p_n), pi ? 2, the eigenvalues are absolutely p-summable,

$\text{1}\text{p}\text{=}\text{∑}\text{i=1}\text{n}\text{1}\text{p}{}_{\mathrm{i}}\text{and}\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{?C}{}_{\mathrm{p}}\text{π}{}^{\mathrm{n}}{}_{\mathrm{P}}\text{(T).}$

We also consider the distribution of eigenvalues of p-nuclear operators on L_r-spaces.In Section 3 we prove the Banach space analog of the classical Weyl inequality, namely

$\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}\text{? C}{}_{\mathrm{p}}\text{∑}\text{n∈N}\text{α}{}_{\mathrm{n}}\text{(T)}{}^{\mathrm{p}}$

, 0 < p < ∞, where α_n denotes the Kolmogoroff, Gelfand of approximation numbers of the operator T. This solves a problem of Markus-Macaev.Finally we prove that Hilbert space is (isomorphically) the only Banach space X with the property that nuclear operators on X have absolutely summable eigenvalues. Using this result we show that if the nuclear operators on X are of type l₁ then X must be a Hilbert space.     

Asymptotic results for partitions (II) and a conjecture of Bateman and Erdös

Asymptotic results are obtained for p_A^(k)(n), the kth difference of the function p_A(n) which is the number of partitions of n into integers from A. Under certain restrictions on A it is shown that

$\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k+1)}}\text{(n)}\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k)}}\text{(n)}\text{= O(n}{}^{\mathrm{?1/2}}\text{) (n→ ∫)}$

thereby verifying for these A a conjecture of Bateman and Erdös.     

Multiplicativity of lp norms for matrices. II

For 1 ? p ? ∞, let

$\text{|A|}{}_{\mathrm{p}}\text{=}\text{Σ}\text{i=1}\text{m}\text{Σ}\text{j=1}\text{n}\text{, |α}{}_{\mathrm{ij}}\text{|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$

, be the l_p norm of an m × n complex A = (α_ij) ?C_{m × n}. The main purpose of this paper is to find, for any p, q ? 1, the best (smallest) possible constants τ(m, k, n, p, q) and σ(m, k, n, p, q) for which inequalities of the form

$\text{|AB|}{}_{\mathrm{p}}\text{? τ(m, k, n, p, q) |A|}{}_{\mathrm{p}}\text{|B|}{}_{\mathrm{q}}\text{, |AB|}{}_{\mathrm{p}}\text{? σ (m, k, n, p, q)|A|}{}_{\mathrm{q}}\text{|B|}{}_{\mathrm{p}}$

hold for all A?C_{m × k}, B?C_{k × n}. This leads to upper bounds for inner products on C^k and for ordinary l_p operator norms on C_{m × n}.     

Critical exponents for the Pucci's extremal operatorsLes exposants critiques pour l'opérateur extrémal de Pucci

In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation

(1)$\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}\text{(D}{}^2\text{u)+u}{}^{\mathrm{p}}\text{=0,u?0}\text{in}\text{R}{}^{\mathrm{N}}\text{.}$

Here N?3, p>1 and $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}$ denotes the Pucci's extremal operators with parameters 0<λ?Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents $\text{1<p}{}^{\mathrm{s}}{}_{\mathrm{+}}\text{<p}{}^1{}_{\mathrm{+}}\text{<p}{}^{\mathrm{p}}{}_{\mathrm{+}}$, that satisfy: (i) If $\text{1<p<p}{}^1{}_{\mathrm{+}}$ then there is no nontrivial solution of ($\text{1}$). (ii) If $\text{p=p}{}^1{}_{\mathrm{+}}$ then there is a unique fast decaying solution of ($\text{1}$). (iii) If $\text{p}{}^1\text{<p?p}{}^{\mathrm{p}}{}_{\mathrm{+}}$ then there is a unique pseudo-slow decaying solution to ($\text{1}$). (iv) If p^p₊<p then there is a unique slow decaying solution to ($\text{1}$). Similar results are obtained for the operator $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{?}}$. To cite this article: P.L. Felmer, A. Quaas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 909–914.     

On the length of optimal TSP circuits in sets of bounded diameter

Let V be a set of n points in R^k. Let d(V) denote the diameter of V, and l(V) denote the length of the shortest circuit which passes through all the points of V. (Such a circuit is an “optimal TSP circuit”.) l^k(n) are the extremal values of l(V) defined by l^k(n)=max{l(V)|V∈V_n^k}, where V_n^k={V|V?R^k,|V|=n, d(V)=1}. A set V∈V_n^k is “longest” if l(V)=l^k(n). In this paper, first some geometrical properties of longest sets in R² are studied which are used to obtain l²(n) for small n′s, and then asymptotic bounds on l^k(n) are derived. Let δ(V) denote the minimal distance between a pair of points in V, and let: δ^k(n)=max{δ(V)|V∈V_n^k}. It is easily observed that $\text{δ}{}^{\mathrm{k}}\text{(n)=O(n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{)}$. Hence, $\text{c}{}_{\mathrm{k}}\text{=}\text{lim sup}{}_{\mathrm{n\to \infty }}\text{δ}{}^{\mathrm{k}}\text{(n)n}{}^{\mathrm{<mtext>1</mtext><mtext>k</mtext>}}$ exists. It is shown that for all $\text{n, c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{≤δ}{}^{\mathrm{k}}\text{(n)}$, and hence, for all $\text{n, l}{}^{\mathrm{k}}\text{(n)≥ c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>1?1</mtext><mtext>k</mtext>}}$. For k=2, this implies that $\text{l}{}^2\text{(n)≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, which generalizes an observation of Fejes-Toth that $\text{lim}{}_{\mathrm{n\to \infty }}\text{l}{}^2\text{(n)n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$. It is also shown that $\text{l}{}^{\mathrm{k}}\text{(n) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]nδ}{}^{\mathrm{k}}\text{(n) + o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{+ o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{)}$. The above upper bound is used to improve related results on longest sets in k-dimensional unit cubes obtained by Few (Mathematika2 (1955), 141–144) for almost all k′s. For k=2, Few's technique is used to show that $\text{l}{}^2\text{(n)≤(}\text{πn}\text{2}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ O(1)}$.     

Un analogue d'un théorème de Hardy pour la transformation de DunklAn analogue of Hardy's theorem for the Dunkl transform

In this Note we give a generalization of Hardy's theorem for the Dunkl transform $\text{F}{}_{\mathrm{D}}$ on $\text{R}{}^{\mathrm{d}}$. More precisely, for all a>0, b>0 and p,q∈[1,+∞], we determine the measurable functions f such that and , where $\text{L}{}_{\mathrm{k}}{}^{\mathrm{p}}\text{(}\text{R}{}^{\mathrm{d}}\text{)}$ are the L^p spaces associated with the Dunkl transform. To cite this article: L. Gallardo, K. Trimèche, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 849–854.     

Positive definite distributions on K\GK

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space $\text{C}$¹(G). We show that the corresponding property is no longer true for the space of double cosets $\text{K}\text{G}\text{K}$. If G is of real-rank 1, we construct liner functionals $\text{T}{}_{\mathrm{p}}\text{? (C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{))′}$ for each p, 0 < p ? 2, such that $\text{T}{}_{\mathrm{p}}\text{(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$ but T_p does not extend to a continuous functional on $\text{C}{}^{\mathrm{p}}\text{(K}\text{G}\text{K}\text{)}$. In particular, if p ? 1, T_v does not extend to a continuous functional on $\text{C}{}^1\text{(K}\text{G}\text{K}\text{)}$. We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that $\text{T(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$. The main tool used is a theorem of Trombi-Varadarajan.