A Decay Result for Certain Windows Generating Orthogonal Gabor Bases

We consider tight Gabor frames (h,a=1,b=1) at critical density with h of the form Z ⁻¹(Zg/|Zg|). Here Z is the standard Zak transform and g is an even, real, well-behaved window such that Zg has exactly one zero, at , in [0,1)². We show that h and its Fourier transform have maximal decay as allowed by the Balian-Low theorem. Our result illustrates a theorem of Benedetto, Czaja, Gadziński, and Powell, case p=q=2, on sharpness of the Balian-Low theorem.      

The structure of the algebraic eigenspace to the spectral radius of eventually compact, nonnegative integral operators

Let K be an eventually compact linear integral operator on L^p(Ω, μ), 1 p < ∞, with nonnegative kernel k(x, y), where the underlying measure μ is totally σ-finite on the domain set Ω when P = 1. This work extends the previous analysis of the author who characterized the distinguished eigenvalues of K and K^*, and the support sets for the eigenfunctions and generalized eigenfunctions belonging to the spectral radius of K or K^*. The characterizations of the support sets for the algebraic eigenspaces of K or K^* are phrased in terms of significant k-components which are maximal irreducible subsets of Ω and which yield a positive spectral radius for the integral operator defined by the restriction of k(x, y) to the Cartesian product of such sets. In this paper, we show that a basis for the functions, constituting the algebraic eigenspaces of K and K^* belonging to the spectral radius of K, can be chosen to consist of elements which are positive on their sets of support, except possibly on sets of measure less than some arbitrarily specified positive number. In addition, we present necessary and sufficient conditions, in terms of the significant k-components, for both K and K^* to possess a positive eigenfunction (a.e. μ) corresponding to the spectral radius, as well as necessary and sufficient conditions for the sequence γⁿKⁿg _p to converge whenever g 0, where − _p denotes the norm in L^p(Ω, μ), and γ₁ the smallest (in modulus) characteristic value of K. This analysis is made possible by introducing the concepts of chains, lengths of chains, height, and depth of a significant k-component as was done by U. Rothblum [Lin. Alg. Appl. 12 (1975), 281–292] for the matrix setting.     

Markov-type Inequalities for Surface Gradients of Multivariate Polynomials

Let K ^d be a compact set with a smooth boundary and consider a polynomial p of total degree n such that p_C(K)1. Then we show that D_Tp(x)=o(n²) for any x Bd K and T a tangential direction at x. Moreover, the o(n²) term is given in terms of the modulus of smoothness of Bd K.     

Canonical correlation analysis based on information theory

In this article, we propose a new canonical correlation method based on information theory. This method examines potential nonlinear relationships between p×1 vector Y-set and q×1 vector X-set. It finds canonical coefficient vectors a and b by maximizing a more general measure, the mutual information, between a^TX and b^TY. We use a permutation test to determine the pairs of the new canonical correlation variates, which requires no specific distributions for X and Y as long as one can estimate the densities of a^TX and b^TY nonparametrically. Examples illustrating the new method are presented.     

The n-width of the unit ball of H

Let E be a compact subset of the open unit disc Δ and let H^q be the Hardy space of analytic functions f on Δ for which stf¦^q has a harmonic majorant. We determine the value of the Kolmogorov, Gel'fand, and linear n-widths in L^p(E, μ) of the restriction to E of the unit ball of H^q when p q or when 1 q < p < ∞ and E is “small”.     

Approximation ofk-Monotone Functions

It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inL_p, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inL_p, 1<p<∞, iso(n^−k/p).     

Ridge Functions and Orthonormal Ridgelets

Orthonormal ridgelets provide an orthonormal basis for L²(R²) built from special angularly-integrated ridge functions. In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x₁ cos θ+x₂ sin θ). We derive a formula for the ridgelet coefficients of a ridge function in terms of the 1-D wavelet coefficients of the ridge profile r(t). The formula shows that the ridgelet coefficients of a ridge function are heavily concentrated in ridge parameter space near the underlying scale, direction, and location of the ridge function. It also shows that the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1-D wavelet coefficients of the 1-D ridge profile r(t). In short, the full ridgelet expansion of a ridge function is in a certain sense equally as sparse as the 1-D wavelet expansion of the ridge profile. It follows that partial ridgelet expansions can give good approximations to objects which are countable superpositions of well-behaved ridge functions. We study the nonlinear approximation operator which “kills” coefficients below certain thresholds (depending on angular- and ridge-scale); we show that for approximating objects which are countable superpositions of ridge functions with 1-D ridge profiles in the Besov space B^1/p_p, p(R), 0<p<1, the thresholded ridgelet approximation achieves optimal rates of N-term approximation. This implies that appropriate thresholding in the ridgelet basis is equally as good, for certain purposes, as an ideally-adapted N-term nonlinear ridge approximation, based on perfect choice of N-directions.     

Remarks on some zero-sum problems

Let Z denote the ring of integers and for a prime p and positive integers r and d, let f_r(P, d) denote the smallest positive integer such that given any sequence of f_r(p, d) elements in (Z/pZ(^d, there exists a subsequence of (rp) elements whose sum is zero in (Z/pZ(^d. That f₁(p, 1) = 2p − 1, is a classical result due to Erdős, Ginzburg and Ziv. Whereas the determination of the exact value of f₁(p, 2) has resisted the attacks of many well known mathematicians, we shall see that exact values of f_r(p, 1) for r ≥ 1 can be easily obtained from the above mentioned theorem of Erdős, Ginzburg and Ziv and those of f_r(p, 2) for r ≥ 2 can be established by the existing techniques developed by Alon, Dubiner and Rónyai in connection with obtaining good upper bounds for f₁(p, 2). We shall also take this opportunity to describe some of the early results in the introduction.     

Codes over p-adic Numbers and Finite Rings Invariant under the Full Affine Group

Codes over p-adic numbers and over integers modulo p^d of block length p^m invariant under the full affine group AGL_m(F_p) are described.     

Efficient p-adic Cell Decompositions for Univariate Polynomials

Cell decompositions are constructed for polynomials f(x)Z_p[x] of degree n, such that n<p, using O(n²) cells. When f is square-free this yields a polynomial-time algorithm for counting and approximating roots in Z_p. These results extend to give a polynomial-time algorithm in the bit model for fZ[x].     

Inequalities for predictive ratios and posterior variances in natural exponential families

The predictive ratio is considered as a measure of spread for the predictive distribution. It is shown that, in the exponential families, ordering according to the predictive ratio is equivalent to ordering according to the posterior covariance matrix of the parameters. This result generalizes an inequality due to Chaloner and Duncan who consider the predictive ratio for a beta-binomial distribution and compare it with a predictive ratio for the binomial distribution with a degenerate prior. The predictive ratio at x₁ and x₂ is defined to be p_g(x₁)p_g(x₂)/[p_g( )]² = h_g(x₁, x₂), where p_g(x₁) = ∫ ƒ(x₁θ) g(θ) dθ is the predictive distribution of x₁ with respect to the prior g. We prove that h_g(x₁, x₂) ≥ h_g^*(x₁, x₂) for all x₁ and x₂ if ƒ(xθ) is in the natural exponential family and Cov_gx(θ) ≥ Cov_g^*x(θ) in the Loewner sense, for all x on a straight line from x₁ to x₂. We then restrict the class of prior distributions to the conjugate class and ask whether the posterior covariance inequality obtains if g and g^* differ in that the “sample size”     

On equivalence of moduli of smoothness of polynomials in ,

It is well known that for functions , 1p∞. For general functions fL_p, it does not hold for 0<p<1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p=∞. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the L_p space for 0<p∞. We then show its analogues for the Ditzian–Totik modulus of smoothness and the weighted Ditzian–Totik modulus of smoothness for polynomials with .     

Average Width and Optimal Interpolation of the Sobolev-Wiener Class Wpq(R) in the Metric Lq(R)

In this paper, we determine the exact value of average n − K width _n(W^r_pq(R), L_q(R)) of Sobolev-Wiener class W^r_pq(R) in the metric L_q(R) for 1 > q ≥ p > ∞ and get the value of _n(W^r_p(R), L_qp(R)) for the dual case. We also solve the optimal interpolation problems of W^r_pq(R) in the metric L_q(R) and W^r_p(R) in the metric L_qp(R) for 1 < q ≤ p < ∞.     

Admissibility and complete class results for the multinomial estimation problem with entropy and squared error loss

Let X ≡ (X₁, …, X_t) have a multinomial distribution based on N trials with unknown vector of cell probabilities p ≡ (p₁, …, p_t). This paper derives admissibility and complete class results for the problem of simultaneously estimating p under entropy loss (EL) and squared error loss (SEL). Let and f(x¦p) denote the (t − 1)-dimensional simplex, the support of X and the probability mass function of X, respectively. First it is shown that δ is Bayes w.r.t. EL for prior P if and only if δ is Bayes w.r.t. SEL for P. The admissible rules under EL are proved to be Bayes, a result known for the case of SEL. Let Q denote the class of subsets of of the form T = _j=1^kF_j where k ≥ 1 and each F_j is a facet of which satisfies: F a facet of such that F naF_jF ncT. The minimal complete class of rules w.r.t. EL when N ≥ t − 1 is characterized as the class of Bayes rules with respect to priors P which satisfy P( ⁰) = 1, ξ(x) ≡ ∫ f(x¦p) P(dp) > 0 for all x in {x : sup ^₀ f(x¦p) > 0} for some ⁰ in Q containing all the vertices of . As an application, the maximum likelihood estimator is proved to be admissible w.r.t. EL when the estimation problem has parameter space Θ = but it is shown to be inadmissible for the problem with parameter space Θ = ( minus its vertices). This is a severe form of “tyranny of boundary.” Finally it is shown that when N ≥ t − 1 any estimator δ which satisfies δ(x) > 0 x is admissible under EL if and only if it is admissible under SEL. Examples are given of nonpositive estimators which are admissible under SEL but not under EL and vice versa.     

Common Principal Components for Dependent Random Vectors

Let the kp-variate random vector X be partitioned into k subvectors X_i of dimension p each, and let the covariance matrix Ψ of X be partitioned analogously into submatrices Ψ_ij. The common principal component (CPC) model for dependent random vectors assumes the existence of an orthogonal p by p matrix β such that β^tΨ_ijβ is diagonal for all (i, j). After a formal definition of the model, normal theory maximum likelihood estimators are obtained. The asymptotic theory for the estimated orthogonal matrix is derived by a new technique of choosing proper subsets of functionally independent parameters.     

Weak Limits for Multivariate Random Sums

Let {Xi, i1} be a sequence of i.i.d. random vectors inRd, and letνp, 0<p<1, be a positive, integer valued random variable, independent ofXis. Theν-stable distributions are the weak limits of properly normalized random sums ∑νpi=1 Xiasνp ∞ andpνp ν. We study the properties ofν-stable laws through their representation via stable laws. In particular, we estimate their tail probabilities and provide conditions for finiteness of their moments.     

Mean Location and Sample Mean Location on Manifolds: Asymptotics, Tests, Confidence Regions

In a previous investigation we studied some asymptotic properties of the sample mean location on submanifolds of Euclidean space. The sample mean location generalizes least squares statistics to smooth compact submanifolds of Euclidean space. In this paper these properties are put into use. Tests for hypotheses about mean location are constructed and confidence regions for mean location are indicated. We study the asymptotic distribution of the test statistic. The problem of comparing mean locations for two samples is analyzed. Special attention is paid to observations on Stiefel manifolds including the orthogonal groupO(p) and spheresS^k−1, and special orthogonal groupsSO(p). The results also are illustrated with our experience with simulations.     

The complexity of minimum difference cover

The complexity of searching minimum difference covers, both in Z⁺ and in Z_n, is studied. We prove that these two optimization problems are NP-hard. To obtain this result, we characterize those sets—called extrema—having themselves plus zero as minimum difference cover. Such a combinatorial characterization enables us to show that testing whether sets are not extrema, both in Z⁺ and in Z_n, is NP-complete. However, for these two decision problems we exhibit pseudo-polynomial time algorithms.     

Generalization of Kummer's criterion for divisibility of class numbers

We give a necessary and sufficient condition for the relative class number of an imaginary field contained in Q(e^2πi/p?) to be divisible by p. We also give a sufficient condition for the class number of a real field contained in Q(e^2πi/p?) not to be divisible by p.     

Dimension in diffeology

We define the dimension function for diffeological spaces, a simple but new invariant. We show then how it can be applied to prove that, for two different integers m and n the quotient spaces R^m/O(m) and Rⁿ/O(n) are not diffeomorphic, and not diffeomorphic to the half-line [0, ∞[ R.