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.     

Matrices dont deux lignes quelconque coïncident dans un nombre donne de positions communes

Le nombre maximal de lignes de matrices seront désignées par:

1. (a) R(k, λ) si chaque ligne est une permutation de nombres 1, 2,…, k et si chaque deux lignes différentes coïncide selon λ positions;

2. (b) S⁰(k, λ) si le nombre de colonnes est k et si chaque deux lignes différentes coïncide selon λ positions et si, en plus, il existe une colonne avec les éléments y₁, y₂, y₃, ou y₁ = y₂ ≠ y₃;

3. (c) T⁰(k, λ) si c'est une (0, 1)-matrice et si chaque ligne contient k unités et si chaque deux lignes différentes contient les unités selon λ positions et si, en plus, il existe une colonne avec les éléments 1, 1, 0.

La fonction T⁰(k, λ) était introduite par Chvátal et dans les articles de Deza, Mullin, van Lint, Vanstone, on montrait que T⁰(k, λ) max(λ + 2, (k − λ)² + k − λ + 1). La fonction S⁰(k, λ) est introduite ici et dans le Théorème 1 elle est étudiée analogiquement; dans les remarques 4, 5, 6, 7 on donne les généralisations de problèmes concernant T⁰(k, λ), S⁰(k, λ), dans la remarque 9 on généralise le problème concernant R(k, λ). La fonction R(k, λ) était introduite et étudiée par Bolton. Ci-après, on montre que R(k, λ) S⁰(k, λ) T⁰(k, λ) d'où découle en particulier: R(k, λ) λ + 2 pour λ k + 1 − (k + 2)^1/2; R(k, λ) = 0(k²) pour k − λ = 0(k); R(k, λ) (k − 1)² − (k + 2) pour k 1191.     

A Two-Parameter Family of Orthogonal Polynomials with Respect to a Jacobi-Type Weight on the Unit Circle

A two-parameter family of polynomials is introduced by a recursion formula. The polynomials are orthogonal on the unit circle with respect to the weight ω_{α, β}(θ) = |(1 − z)^α(1 + z)^β|², α, β > − , z = e^iθ. Explicit representation, norm estimates, shift identities, and explicit connection to Jacobi polynomials on the real interval [−1, 1] is presented.     

On the Behaviour of Zeros of Jacobi Polynomials

Denote by x_n,k(α,β) and x_n,k(λ)=x_n,k(λ−1/2,λ−1/2) the zeros, in decreasing order, of the Jacobi polynomial P^(α,β)_n(x) and of the ultraspherical (Gegenbauer) polynomial C^λ_n(x), respectively. The monotonicity of x_n,k(α,β) as functions of α and β, α,β>−1, is investigated. Necessary conditions such that the zeros of P^(a,b)_n(x) are smaller (greater) than the zeros of P^(α,β)_n(x) are provided. A. Markov proved that x_n,k(a,b)<x_n,k(α,β) (x_n,k(a,b)>x_n,k(α,β)) for every n and each k, 1kn if a>α and b<β (a<α and b>β). We prove the converse statement of Markov's theorem. The question of how large the function f_n(λ) could be such that the products f_n(λ)x_n,k(λ), k=1,…,[n/2] are increasing functions of λ, for λ>−1/2, is also discussed. Elbert and Siafarikas proved that f_n(λ)=(λ+(2n²+1)/(4n+2))^1/2 obeys this property. We establish the sharpness of their result.     

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.     

Where does smoothness count the most for Fredholm equations of the second kind with noisy information?

We study the complexity of Fredholm problems (I−T_k)u=f of the second kind on I^d=[0,1]^d, where T_k is an integral operator with kernel k. Previous work on the complexity of this problem has assumed either that we had complete information about k or that k and f had the same smoothness. In addition, most of this work has assumed that the information about k and f was exact. In this paper, we assume that k and f have different smoothness; more precisely, we assume that fW^r,p(I^d) with r>d/p and that kW^s,∞(I^2d) with s>0. In addition, we assume that our information about k and f is contaminated by noise. We find that the nth minimal error is Θ(n^−μ+δ), where μ=min{r/d,s/(2d)} and δ is a bound on the noise. We prove that a noisy modified finite element method has nearly minimal error. This algorithm can be efficiently implemented using multigrid techniques. We thus find tight bounds on the -complexity for this problem. These bounds depend on the cost c(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation is proportional to δ^−t, then the -complexity is roughly (1/)^t+1/μ.     

Reflection sieve obstructions in finite groups of type dn

In [4] we constructed certain homology representations of a finite group G of type A_n, B_n or C_n, and showed that these representations can be used to sift out the reflection compound characters of G. In the present note, we show that for a group G of type D_n, each reflection compound character π^(k), 2 k n − 2, determines a unique “obstruction” character θ^(k), which occurs with positive multiplicity in every homology representation containing π^(k).     

Boundary effects on convergence rates for Tikhonov regularization

We consider the Tikhonov regularizer f_λ of a smooth function f ε H^2m[0, 1], defined as the solution (see [1]) to We prove that if f^(j)(0) = f^(j)(1) = 0, J = m, …, k < 2m − 1, then ¦f − f_λ¦_j² Rλ^{(2k − 2j + 3)/2m}, J = 0, …, m. A detailed analysis is given of the effect of the boundary on convergence rates.     

Error bounds for least squares approximation by polynomials

Let f ε Cⁿ⁺¹[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials q_k associated with a distribution dα on [−1, 1]. It is shown that if q_n+1/q_n max(q_n+1(1)/q_n(1), −q_n+1(−1)/q_n(−1)), then f − H[f] f^{n + 1} · q_n+1/q_{n + 1}^{(n + 1)}, where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)^α (1 + t)^β dt, α, β > −1, the condition on q_n+1/q_n is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.     

On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle

Let u(r,θ) be biharmonic and bounded in the circular sector ¦θ¦ < π/4, 0 < r < ρ (ρ > 1) and vanish together with δu/δθ when ¦θ¦ = π/4. We consider the transform û(p,θ) = ∝₀¹r^{p − 1}u(r,θ)dr. We show that for any fixed θ₀ u(p,θ₀) is meromorphic with no real poles and cannot be entire unless u(r, θ₀) ≡ 0. It follows then from a theorem of Doetsch that u(r, θ₀) either vanishes identically or oscillates as r → 0.     

On αrγs(k)-perfect graphs

For some integer k0 and two graph parameters π and τ, a graph G is called πτ(k)-perfect, if π(H)−τ(H)k for every induced subgraph H of G. For r1 let α_r and γ_r denote the r-(distance)-independence and r-(distance)-domination number, respectively. In (J. Graph Theory 32 (1999) 303–310), I. Zverovich gave an ingenious complete characterization of α₁γ₁(k)-perfect graphs in terms of forbidden induced subgraphs. In this paper we study α_rγ_s(k)-perfect graphs for r,s1. We prove several properties of minimal α_rγ_s(k)-imperfect graphs. Generalizing Zverovich's main result in (J. Graph Theory 32 (1999) 303–310), we completely characterize α_2r−1γ_r(k)-perfect graphs for r1. Furthermore, we characterize claw-free α₂γ₂(k)-perfect graphs.     

L∞-upper bound of L2-projections onto splines at a geometric mesh

For an integer k 1 and a geometric mesh (qⁱ)_−∞^∞ with q ε (0, ∞), let M_i,k(x): = k[q^{i + k}](· − x)₊^{k − 1}, N_i,k(x): = (q^{i + k} − qⁱM_i,k(x)/k, and let A_k(q) be the Gram matrix (∝M_i,kN_j,k)_i,jεz. It is known that A_k(q)⁻¹_∞ is bounded independently of q. In this paper it is shown that A_k(q)⁻¹_∞ is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for A_k (q)⁻¹ are obtained: for all q ε (0, ∞).     

Representations for the first associated q-classical orthogonal polynomials

Let {p_k(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation D_q(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {p_k⁽¹⁾(x;q)} be associated polynomials of the polynomials {p_k(x; q)}. Explicit forms of the coefficients b_n,k and c_n,k in the expansions

are given in terms of basic hypergeometric functions. Here _k(x) equals x^k if σ⁺(0)=0, or (x;q)_k if σ⁺(1)=0, where σ⁺(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by p_n−1⁽¹⁾(x;q) in a special form, recurrence relations (in k) for b_n,k and c_n,k are obtained in terms of σ and τ.     

Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems. VI. Asymptotics in the two-cluster region

We study the asymptotic behavior of the ground-state wave function of multiparticle quantum systems without statistics in that region of configuration space where the particles break up into two well-defined clusters very far apart. One example of our results is the following: consider a system of N particles moving in three dimensions with rotationally invariant two-body potentials which are bounded and have compact support. Let D = C₁,C₂ be a partition into two clusters so that H(C₁) and H(C₂) have discrete ground states η₁ and η₂ of energy ε₁ and ε₂. Suppose that Σ = ε₁ + ε₂ = inf σ_ess(H) and that H has a discrete ground state of energy E. Let ζ₁and ζ₂ denote internal coordinates for the clusters C₁ and c₂ and let R be the difference of the centers of mass of the clusters. Let μ = M₁M₂/M₁ + M₂with M_i the mass of clusters C_i and define k by k²/2m = Σ-E. Then as R → a8 with ¦ζ_i¦ bounded, we prove that (ζ₁,ζ₂, R) = cη(ζ₁)η(ζ₂)e^−kRR⁻¹(1+O(e^−γR)) for some γ, c > 0. We prove weaker conclusions under weaker hypotheses, including results in the atomic case.     

Asymptotic expansions in the conditional central limit theorem

Let X_n, n , be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r (O(lg lgn/n^{(r − 2)/2)}) for r ).     

A general framework of compactly supported splines and wavelets

Let {V_k} be a nested sequence of closed subspaces that constitute a multiresolution analysis of L²( ). We characterize the family Φ = {φ} where each φ generates this multiresolution analysis such that the two-scale relation of φ is governed by a finite sequence. In particular, we identify the ε Φ that has minimum support. We also characterize the collection Ψ of functions η such that each η generates the orthogonal complementary subspaces W_k of V_k, . In particular, the minimally supported ψ ε Ψ is determined. Hence, the “B-spline” and “B-wavelet” pair (, ψ) provides the most economical and computational efficient “spline” representations and “wavelet” decompositions of L² functions from the “spline” spaces V_k and “wavelet” spaces W_k, k . A very general duality principle, which yields the dual bases of both {(·−j):j and {η(·−j):j } for any η ε Ψ by essentially interchanging the pair of two-scale sequences with the pair of decomposition sequences, is also established. For many filtering applications, it is very important to select a multiresolution for which both and ψ have linear phases. Hence, “non-symmetric” and ψ, such as the compactly supported orthogonal ones introduced by Daubechies, are sometimes undesirable for these applications. Conditions on linear-phase φ and ψ are established in this paper. In particular, even-order polynomial B-splines and B-wavelets φ_m and ψ_m have linear phases, but the odd-order B-wavelet only has generalized linear phases.     

Sharp inequalities for convolution-type operators

Let μ be a probability measure on [− a, a], a > 0, and let x₀ε[− a, a], f ε Cⁿ([−2a, 2a]), n 0 even. Using moment methods we derive best upper bounds to ¦∫_−a^a ([f(x₀ + y) + f(x₀ − y)]/2) μ(dy) − f(x₀)¦, leading to sharp inequalities that are attainable and involve the second modulus of continuity of f⁽ⁿ⁾ or an upper bound of it.     

Summability of orthogonal expansions of several variables

Summability of spherical h-harmonic expansions with respect to the weight function ∏_j=1^d |x_j|^2κ_j (κ_j0) on the unit sphere S^d−1 is studied. The main result characterizes the critical index of summability of the Cesàro (C,δ) means of the h-harmonic expansion; it is proved that the (C,δ) means of any continuous function converge uniformly in the norm of C(S^d−1) if and only if δ>(d−2)/2+∑_j=1^d κ_j−min_1jd κ_j. Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and S^d−1, the (C,δ) means of the h-harmonic expansion of a continuous function f converges pointwisely to f if δ>(d−2)/2. Similar results are established for the orthogonal expansions with respect to the weight functions ∏_j=1^d |x_j|^2κ_j(1−|x|²)^μ−1/2 on the unit ball B^d and ∏_j=1^d x_j^κ_j−1/2(1−|x|₁)^μ−1/2 on the simplex T^d. As a related result, the Cesàro summability of the generalized Gegenbauer expansions associated to the weight function |t|^2μ(1−t²)^λ−1/2 on [−1,1] is studied, which is of interest in itself.     

On the degree of weak convergence of a sequence of finite measures to the unit measure under convexity

This is a study of the degree of weak convergence under convexity of a sequence of finite measures μ_j on ^k, k 1, to the unit measure δ_x0. LetQ denote a convex and compact subset of ^k, let ƒ ε C^m(Q), m 0, satisfy a convexity condition and let μ be a finite measure on Q. Using standard moment methods, upper bounds and best upper bounds are obtained for ¦∝_Qƒdμ − ƒ(x₀)¦. They sometimes lead to sharp inequalities which are attained for particular μ and ƒ. These estimates are better than the corresponding ones found in the literature.     

A Remez-Type Inequality for Non-dense Müntz Spaces with Explicit Bound

LetΛ :=(λ_k)^∞_k=0be a sequence of distinct nonnegative real numbers withλ₀ :=0 and ∑^∞_k=1 1/λ_k<∞. Let(0, 1) and(0, 1−) be fixed. An earlier work of the authors shows that [formula]is finite. In this paper an explicit upper bound forC(Λ, , ) is given. In the special caseλ_k :=k^α,α>1, our bounds are essentially sharp.