Periodic-type functionals and their extensions

Let Ω denote a simply connected domain in the complex plane and let $\text{K[Ω]}$ be the collection of all entire functions of exponential type whose Laplace transforms are analytic on Ω′, the complement of Ω with respect to the sphere. Define a sequence of functionals $\text{{L}{}_{\mathrm{n}}\text{}}\text{on}\text{K[Ω]}$ by $\text{L}{}_{\mathrm{n}}\text{(f) =}\text{1}\text{2πi}\text{∝}{}_{\mathrm{\Gamma }}\text{g}{}_{\mathrm{n}}\text{(ζ) F(ζ) dζ}$, where F denotes the Laplace transform of f, Γ ? Ω is a simple closed contour chosen so that F is analytic outside and on Ω, and g_n is analytic on Ω. The specific functionals considered by this paper are patterned after the Lidstone functions, L_2n(f) = f⁽²ⁿ⁾(0) and L_{2n + 1}(f) = f⁽²ⁿ⁾(1), in that their sequence of generating functions {g_n} are “periodic.” Set g_{pn + k}(ζ) = h_k(ζ) ζ^pn, where p is a positive integer and each h_k (k = 0, 1,…, p ? 1) is analytic on Ω. We find necessary and sufficient conditions for $\text{f ∈ k[Ω]}\text{with}\text{L}{}_{\mathrm{n}}\text{(f) = 0 (n = 0, 1,…)}$. DeMar previously was able to find necessary conditions [7]. Next, we generalize {L_n} in several ways and find corresponding necessary and sufficient conditions.     

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 a conjecture by D. Ž. D̄okovic

It is shown that if $\text{A?Ω}{}_{\mathrm{n}}\text{?{J}{}_{\mathrm{n}}\text{}}$ satisfies

$\text{nkσ}{}_{\mathrm{k}}\text{(A)?(n?k+1)}{}^2\text{σ}{}_{\mathrm{k?1}}\text{(A)}$

$\text{(k=1,2,…,n)}$

, where σ_k(A) denotes the sum of all kth order subpermanent of A, then Per[λJ_n+(1?λ)A] is strictly decreasing in the interval 0<λ<1.     

On exponential bases for the Sobolev spaces over an interval

We suppose that K is a countable index set and that $\text{Λ = {λ}{}_{\mathrm{k}}\text{¦ k ? K}}$ is a sequence of distinct complex numbers such that forms a Riesz (strong) basis for L²[a, b], a < b. Let Σ = {σ₁, σ₂,…, σ_m} consist of m complex numbers not in Λ. Then, with p(λ) = Π_{k = 1}^m (λ ? σ_k), forms a Riesz (strong) bas Sobolev space H^m[a, b]. If we take σ₁, σ₂,…, σ_m to be complex numbers already in Λ, then, defining p(λ) as before, forms a Riesz (strong) basis for the space H^?m[a, b]. We also discuss the extension of these results to “generalized exponentials” tⁿe^λ_kt.     

Construction of a recurrence relation for modified moments

In this paper we are constructing a recurrence relation of the form

$\text{∑}\text{i=0}\text{r}\text{ω}{}_{\mathrm{i}}\text{(k)m}{}_{\mathrm{k+i}}{}^{\mathrm{\{\lambda \}}}\text{[f] = ω(k)}$

for integrals (called modified moments)

$\text{m}{}_{\mathrm{k}}{}^{\mathrm{\{\lambda \}}}\text{[f]df=}\text{∫}\text{?1}\text{1}\text{f(x)C}{}_{\mathrm{k}}{}^{\mathrm{(\lambda )}}\text{(x)dx (k = 0,1,…)}$

in which C_k^(λ) is the k-th Gegenbauer polynomial of order $\text{λ(λ > ?}\text{1}\text{2}\text{)}$, and f is a function satisfying the differential equation

$\text{∑}\text{i=0}\text{n}\text{P}{}_{\mathrm{i}}\text{(x)f}{}^{\mathrm{(i)}}\text{(x) = p(x) (?1?x?1)}$

of order n, where p₀, p₁, …, p_n ? 0 are polynomials, and m_k^〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense.     

Limiting distributions of two random sequences

For fixed p (0 ≤ p ≤ 1), let {L₀, R₀} = {0, 1} and X₁ be a uniform random variable over {L₀, R₀}. With probability p let {L₁, R₁} = {L₀, X₁} or = {X₁, R₀} according as $\text{X}{}_1\text{≥}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}$; with probability 1 ? p let {L₁, R₁} = {X₁, R₀} or = {L₀, X₁} according as $\text{X}{}_1\text{≥}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}$, and let X₂ be a uniform random variable over {L₁, R₁}. For n ≥ 2, with probability p let {L_n, R_n} = {L_{n ? 1}, X_n} or = {X_n, R_{n ? 1}} according as $\text{X}{}_{\mathrm{n}}\text{≥}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}$, with probability 1 ? p let {L_n, R_n} = {X_n, R_{n ? 1}} or = {L_{n ? 1}, X_n} according as $\text{X}{}_{\mathrm{n}}\text{≥}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}$, and let X_{n + 1} be a uniform random variable over {L_n, R_n}. By this iterated procedure, a random sequence {X_n}_{n ≥ 1} is constructed, and it is easy to see that X_n converges to a random variable Y_p (say) almost surely as n → ∞. Then what is the distribution of Y_p? It is shown that the Beta, (2, 2) distribution is the distribution of Y₁; that is, the probability density function of Y₁ is g(y) = 6y(1 ? y) I_0,1(y). It is also shown that the distribution of Y₀ is not a known distribution but has some interesting properties (convexity and differentiability).     

The construction of stationary two-dimensional Markoff fields with an application to quantum field theory

It is shown that if φ(f)  ∝_R^dφ(y) f(y) dy is a Markoff random field and X_α are multiplicative functionals of φ (with E(X_α) = 1) which converge locally in L₁, then there exists a locally Markoff random field $\text{φ}{}_1$ such that $\text{E(exp(iφ}{}_1\text{(f))) =}\text{lim}{}_{\mathrm{\alpha }}\text{E(X}{}_{\mathrm{<mtext>\alpha </mtext>}}\text{exp}\text{(iφ(φ)))}$. We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take X_α proportional to exp(?λ∝_R² : P(φ(y)) : g_α(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of g_α → 1 and small λ, {X_α} does converge locally in L₁ and that the corresponding $\text{φ}{}_1$ is stationary.     

Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

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 → ∞.     

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 $\text{Y}\text{=T}\text{X}$.     

C0-semigroups on a locally convex space

A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C₀-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)^?1 exists and the family {(λ ? β)^k(λI ? A)^?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)^?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra $\text{B}{}_{\mathrm{г}}\text{(X)}$ which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has lim_{k → z} (I ? k^?1tA)^?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C₀-semigroup by an operator which is an element of $\text{B}{}_{\mathrm{г}}\text{(X)}$ is obtained.     

Sur la convergence uniforme presque complète dans l'estimation de la densité spectrale d'un processus à temps continu après échantillonnage du temps

Let $\text{X={X(t)}}{}_{\mathrm{<mtext>t\in </mtext><mtext>R</mtext>}}$ be a continuous-time strictly stationary and strongly mixing process. In this paper, we prove in the setting of spectral density estimation, at first, under some hard conditions on the spectral density φ_X (because of aliasing phenomenon), the uniformly complete convergence of the spectral density estimate from periodic sampling. Afterwards, to overcome aliasing, we consider the sampled process $\text{{X(t}{}_{\mathrm{n}}\text{)}}{}_{\mathrm{<mtext>n\in </mtext><mtext>Z</mtext>}}$, where {t_n} is a stationary point process independent from X. The uniform complete convergence of the spectral estimate based on the discrete time observations {X(t_k),t_k} is also obtained. The convergence rates are also established. To cite this article: M. Rachdi, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Bounds for the variance of certain stationary point processes

For the variance of stationary renewal and alternating renewal processes N_n(·) the paper establishes upper and lower bounds of the form

$\text{?B}{}_1\text{?varN}{}_8\text{(0,x–Aλx?B}{}_2\text{(0<x<∞)}$

, where λ=EN₈(0,1), with constants A, B₁ and B₂ that depend on the first three moments of the interval distributions for the processes concerned. These results are consistent with the value of the constant A for a general stationary point process suggested by Cox in 1963 [1].     

On the invariants of some Zi-extensions

Let k ? k₁ ? … ? K be a Z_i-extension. The relations of $\text{λ(}\text{K}\text{k}\text{)}$ and $\text{λ(}\text{KF}\text{F}\text{)}$ is studied, where $\text{F}\text{k}$ is a cyclic l-extension. If $\text{M}\text{k}$ is another Z_i-extension of k, it is shown that for i ? 0$\text{λ(}\text{Mk}{}_{\mathrm{i}}\text{k}{}_{\mathrm{i}}\text{) = rl}{}^{\mathrm{i}}\text{+ C}$, under minimal additional hypotheses. Finally if $\text{MK}\text{k}$ has a unique totally ramified prime, and X_K is cyclic, it is shown that MK can contain at most one Z_i-extension with non-zero μ invariant.     

Ω-Theorems for the real part of the Riemann zeta function

For s = σ + it, σ > 1, and integer k ? 1$\text{ζ}{}^{\mathrm{k}}\text{(s) = ∑}\text{d}{}_{\mathrm{k}}\text{(n)}\text{n}{}^3$. In a previous paper, Ω results for $\text{ζ(c + it),}\text{1}\text{2}\text{? c ? 1}$, where obtained in an elementary way by choosing N = N(k, c) so that the size of the single term d_k(N) gave Ω results slightly better than existing ones. Here the method will be shown to give the same results for Re ζ(c + it).     

On the calculation of generalized binomial coefficients

The generalized binomial coefficients (_κ^λ) are defined by

$\text{Cλ(I}{}_{\mathrm{m}}\text{+ R)}\text{Cλ(I}{}_{\mathrm{m}}\text{)}\text{=}\text{Σ}\text{k=0}\text{l}\text{Σ}\text{κ}\text{(}\text{λ}\text{κ}\text{)}\text{C}{}_{\mathrm{\kappa }}\text{(R)}\text{C}{}_{\mathrm{\kappa }}\text{(I}{}_{\mathrm{m}}\text{)}$

, where the C_k(R) are the zonal polynomials of the m × m matrix R. In this paper some simple expressions are derived which allow straightforward calculation of a large number of these coefficients.     

On the sign of some Dedekind sums

Rademacher asked the following: if $\text{h}{}_1\text{k}{}_1$ and $\text{h}{}_2\text{k}{}_2$ are adjacent terms in a Farey series and if the Dedekind sums s(h₁, k₁) and s(h₂, k₂) are both positive, then is s(h₁ + h₂, k₁ + k₂) ≥ 0? In this note, an infinite family of counter-examples is constructed.