On the convergence of averaging Hermite interpolators

We investigate two sequences of polynomial operators, H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x), of degrees 2n − 2 and 2n − 3, respectively, defined by interpolatory conditions similar to those of the classical Hermite-Féjer interpolators H_{2n − 1}(f, x). If H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x) are based on the zeros of the jacobi polynomials P_n^(α,β)(x), their convergence behaviour is similar to that of H_{2n − 1}(f;, x). If they are based on the zeros of (1 − x²)T_{n − 2}(x), their convergence behaviour is better, in some sense, than that of H_{2n − 1}(f, x).     

Oscillations of first-order neutral delay differential equations

Consider the neutral delay differential equation (*) (d/dt)[y(t) + py(t − τ)] + qy(t − σ) = 0, t t₀, where τ, q, and σ are positive constants, while p ε (−∞, −1) (0, + ∞). (For the case p ε [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p < − 1. Then every nonoscillatory solution y(t) of Eq. (*) tends to ± ∞ as t → ∞. Theorem 2. Assume p < − 1, τ > σ, and q(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Theorems 3. Assume p > 0. Then every nonoscillatory solution y(t) of Eq. (*) tends to zero as t → ∞. Theorem 4. Assume p > 0. Then a necessary condition for all solutions of Eq. (*) to oscillate is that σ > τ. Theorem 5. Assume p > 0, σ > τ, andq(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Extensions of these results to equations with variable coefficients are also obtained.     

On improved estimators of the generalized variance

Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p W_p(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ I_k) with ξ(p × k) unknown. Denote the columns of X by X₍₁₎ ,…, X_(k) and set ψ⁽⁰⁾(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ⁽ⁱ⁾(X, X) = min[ψ⁽ⁱ⁻¹⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |] and Ψ⁽ⁱ⁾(S, X) = min[ψ⁽⁰⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!}| S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ⁽⁰⁾(S, X) is dominated by each of Ψ⁽ⁱ⁾(S, X), i = 1,…,k and for every i, ψ⁽ⁱ⁾(S, X) is better than ψ⁽ⁱ⁻¹⁾(S, X). In particular, ψ^(k)(S, X) = min[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X₍₁₎X′₍₁₎|,…,| {(n − p + k + 2)!/(n + k + 2)!} | S + X₍₁₎X′₍₁₎ + + X_(k)X′_(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.     

Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Q_n(x)}^∞_n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Q_n(x)}^∞_n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: L_N[y](x)=∑^N_i=1 ℓ_i(x) y⁽ⁱ⁾(x)=λ_ny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Q_n(x)}^∞_n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.     

Jackson's theorem for compact connected Lie groups

We prove that for f ε E = C(G) or L^p(G), 1 p < ∞, where G is any compact connected Lie group, and for n 1, there is a trigonometric polynomial t_n on G of degree n so that f − t_nE C_rω_r(n⁻¹,f). Here ω_r(t, f) denotes the rth modulus of continuity of f. Using this and sharp estimates of the Lebesgue constants recently obtained by Giulini and Travaglini, we obtain “best possible” criteria for the norm convergence of the Fourier series of f.     

On Hilbert''s Integral Inequality

In this paper, we generalize Hilbert's integral inequality and its equivalent form by introducing three parameterst,a, andb.Iff, g L²[0, ∞), then[formula]where π is the best value. The inequality (1) is well known as Hilbert's integral inequality, and its equivalent form is[formula]where π²is also the best value (cf. [[1], Chap. 9]). Recently, Hu Ke made the following improvement of (1) by introducing a real functionc(x),[formula]wherek(x) = 2/π∫^∞₀(c(t²x)/(1 + t²)) dt − c(x), 1 − c(x) + c(y) ≥ 0, andf, g ≥ 0 (cf. [[2]]). In this paper, some generalizations of (1) and (2) are given in the following theorems, which are other than those in [ [2]].     

Comonotone Jackson's Inequality

Let 2s points y_i=−πy_2s<…<y₁<π be given. Using these points, we define the points y_i for all integer indices i by the equality y_i=y_i+2s+2π. We shall write fΔ⁽¹⁾(Y) if f is a 2π-periodic continuous function and f does not decrease on [y_i, y_i−1], if i is odd; and f does not increase on [y_i, y_i−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved. 1. If fΔ⁽¹⁾(Y), then for each nonnegative integer n there is a trigonometric polynomial τ_n(x) of order n such that τ_nΔ⁽¹⁾(Y), and |f(x)−π_n(x)|c(s) ω(f; 1/(n+1)), x , where ω(f; t) is the modulus of continuity of f, c(s)=const. Depending only on s.     

Second-order linearity of the general signed-rank statistic

Let X₁,…, X_n be i.i.d. random variables symmetric about zero. Let R_i(t) be the rank of |X_i − tn^−1/2| among |X₁ − tn^−1/2|,…, |X_n − tn^−1/2| and T_n(t) = Σ_{i = 1}ⁿφ((n + 1)⁻¹R_i(t))sign(X_i − tn^−1/2). We show that there exists a sequence of random variables V_n such that sup_{0 ≤ t ≤ 1} |T_n(t) − T_n(0) − tV_n| → 0 in probability, as n → ∞. V_n is asymptotically normal.     

An extremal property of Hermite polynomials

Let H_n be the nth Hermite polynomial, i.e., the nth orthogonal on polynomial with respect to the weight w(x)=exp(−x²). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f||H_n| at the zeros of H_n+1, then for k=1,…,n we have f^(k)H_n^(k), where · is the norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.     

On existence of singular solutions

In the paper sufficient conditions are given under which the differential equation y⁽ⁿ⁾=f(t,y,…,y⁽ⁿ⁻²⁾)g(y⁽ⁿ⁻¹⁾) has a singular solution y :[T,τ)→R, τ<∞ fulfilling

     

On a Conjecture Concerning Strong Unicity Constants

Let fC[−1, 1] be real-valued. We consider the sequence of strong unicity constants (γ_n(f))_n induced by the polynomials of best uniform approximation of f. It is proved that lim inf_n→∞ γ_n(f)=0, whenever f is not a polynomial.     

On an infinite series of Abel occurring in the theory of interpolation

The purpose of this paper is to show that for a certain class of functions f which are analytic in the complex plane possibly minus (−∞, −1], the Abel series f(0) + Σ_{n = 1}^∞ f⁽ⁿ⁾(nβ) z(z − nβ)^{n − 1}/n! is convergent for all β>0. Its sum is an entire function of exponential type and can be evaluated in terms of f. Furthermore, it is shown that the Abel series of f for small β>0 approximates f uniformly in half-planes of the form Re(z) − 1 + δ, δ>0. At the end of the paper some special cases are discussed.     

Coboundaries in

Let T be an ergodic automorphism of a probability space, f a bounded measurable function, . It is shown that the property that the probabilities μ(|S_n(f)|>n) are of order n^−p roughly corresponds to the existence of an approximation in L^∞ of f by functions (coboundaries) g−g○T, gL^p. Similarly, the probabilities μ(|S_n(f)|>n) are exponentially small iff f can be approximated by coboundaries g−g○T where g have finite exponential moments.

Résumé

Soit T un automorphisme ergodique d'un espace probabilisé, f une fonction bornée mesurable et . Une correspondance est établie entre l'existence de l'estimation des probabilités μ(|S_n(f)|>n) d'ordre n^−p et l'existence de l'approximation dans L^∞ de la fonction f par des cobords g−g○T où g est “presque” dans L^p. De manière similaire, les probabilités μ(|S_n(f)|>n) sont d'ordre e^−cn, pour un certain c>0, n=1,2… , si et seulement si f admet une approximation dans L^∞ par des cobords g−g○T avec g ayant des moments exponentiels.     

Second Jackson Inequality in a Sign-Preserving Approximation of Periodic Functions

We consider a 2-periodic function f continuous on and changing its sign at 2s points y _i [–, ). For this function, we prove the existence of a trigonometric polynomial T _n of degree n that changes its sign at the same points y _i and is such that the deviation |f(x) – T _n(x)| satisfies the second Jackson inequality.     

Stability conditions for systems of linear nonautonomous delay differential equations

Systems of linear nonautonomous delay differential equations are considered which are of the form y′_i(t) = ∑_{k = 1}ⁿ ∝₀^T b_ik(t, s) y_k(t − s) dη_ik(s) − c_i(t) y_i(t), where I = 1,…, n. Sufficient conditions are derived for both the asymptotic stability and the instability of the zero solution. The main result is found by a monotone technique using elementary methods only. Moreover, additional criteria are obtained by using the method of Lyapunov functionals.     

A continuous function space with a Faber basis

Let be compact with #S=∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (P_n)_n=0^∞ with deg P_n=n such that every fC(S) has a unique representation f=∑_i=0^∞ α_iP_i and call such a basis Faber basis. In the special case of , 0<q<1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with A(S_q)≠U(S_q)=C(S_q), where A(S_q) is the so-called Wiener algebra and U(S_q) is the set of all fC(S_q) which are uniquely represented by its Fourier series.     

On almost sure convergence of Cesaro averages of subsequences of vector-valued functions

A remarkable theorem proved by Komlòs [4] states that if {f_n} is a bounded sequence in L¹(R), then there exists a subsequence {f_nk} and f L¹(R) such that f_nk (as well as any further subsequence) converges Cesaro to f almost everywhere. A similar theorem due to Révész [6] states that if {f_n} is a bounded sequence in L²(R), then there is a subsequence {f_nk} and f L²(R) such that Σ_k=1^∞ a_k(f_nk − f) converges a.e. whenever Σ_k=1^∞ | a_k |² < ∞. In this paper, we generalize these two theorems to functions with values in a Hilbert space (Theorems 3.1 and 3.3).     

On Approximate ℓ 1 Systems in Banach Spaces

Let X be a real Banach space and let (f(n)) be a positive nondecreasing sequence. We consider systems of unit vectors (x_i)^∞_i=1 in X which satisfy ∑_iA±x_i|A|−f(|A|), for all finite A and for all choices of signs. We identify the spaces which contain such systems for bounded (f(n)) and for all unbounded (f(n)). For arbitrary unbounded (f(n)), we give examples of systems for which [x_i] is H.I., and we exhibit systems in all isomorphs of ℓ₁ which are not equivalent to the unit vector basis of ℓ₁. We also prove that certain lacunary Haar systems in L₁ are quasi-greedy basic sequences.     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.     

The asymptotic behavior of nonoscillatory solutions of nonlinear neutral type difference equations

In this paper, the authors study the asymptotic behavior of solutions of second-order neutral type difference equations of the form

Δ²(y_n+py_n−k)+f(n,y_n−ℓ)=0,n