Systems of functions which are complete in measure

In this note it is proved that if a complete orthonormal system {? _n} in L₂[0, 1] contains a subsystem {? _nk} of a lacunary order p>2, then for some bounded measurable function h(x) the system {h(x)? _n(x)}_n≠_nk is complete in L₂[0, 1].     

A sufficient condition for mean convergence of orthogonal series for Freud weights

Let S_n[f](x) be the n-th partial sum of the orthonormal polynomial series expansion for f corresponding to a Freud weight W(x)=e^−Q(x). We give a sufficient condition for the inequality      

A sufficient condition for mean convergence of orthogonal series for Freud weights

Let S_n[f](x) be the n-th partial sum of the orthonormal polynomial series expansion for f corresponding to a Freud weight W(x)=e^−Q(x). We give a sufficient condition for the inequality to hold. This sufficient condition is much easier to apply than that of Jha and Lubinsky [2].     

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

More mutually orthogonal latin squares

Wilson's construction for mutually orthogonal Latin squares is generalized. This generalized construction is used to improve known bounds on the function n_r (the largest order for which there do not exist r MOLS). In particular we find n₇?780, n₈?4738, n₉?5842, n₁₀?7222, n₁₁?7478, n₁₂?9286, n₁₃?9476, n₁₅?10632.     

Orthogonality criteria for compactly supported refinable functions and refinable function vectors

A refinable function φ(x):ℝⁿ→ℝ or, more generally, a refinable function vector Φ(x)=[φ₁(x),...,φ_r(x)]^T is an L¹ solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φ_j(x−α):α∈ℤⁿ, 1≤j≤r form an orthogonal set of functions in L²(ℝⁿ). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multiwavelet bases of L²(ℝⁿ). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.     

On Existence for Polynomial-Like Iterative Equations

Many known results on the iterative equation α_i=1 ⁿ λ_i?ⁱ(x) = F(x) require a condition that λ₁ > 0 for technical reasons. A problem on the existence of solutions of this iterative equation with the natural restriction λ_n ≠ 0 is raised. In this paper we study an auxiliary functional equation for its invertible solutions. Then we apply our results on the auxiliary equation to solve the problem in some cases.     

Functional series representable as a sum of two universal series

Series with respect to systems Φ{φ_n(x)}_n=1^∞ of measurable and almost everywhere finite functions are discussed. A necessary and sufficient condition for representing any series with respect to a system Φ as a sum of two universal series is formulated. A consequence of the condition is that any series with respect to an arbitrary complete and orthonormal system Φ is a sum of two universal series.     

Limit Theory for Moderate Deviations from a Unit Root Under Innovations with a Possibly Infinite Variance

An asymptotic theory was given by Phillips and Magdalinos (J Econom 136(1):115–130, 2007) for autoregressive time series Y _t ?=?ρY _t?1?+?u _t , t?=?1,...,n, with ρ?=?ρ _n ?=?1?+?c/k _n , under (2?+?δ)-order moment condition for the innovations u _t , where δ?>?0 when c?<?0 and δ?=?0 when c?>?0, {u _t } is a sequence of independent and identically distributed random variables, and (k _n ) _n?∈?? is a deterministic sequence increasing to infinity at a rate slower than n. In the present paper, we established similar results when the truncated second moment of the innovations $l(x)=\textsf{E} [u_1^2I\{|u_1|\le x\}]$ is a slowly varying function at ∞, which may tend to infinity as x?→?∞. More interestingly, we proposed a new pivotal for the coefficient ρ in case c?<?0, and formally proved that it has an asymptotically standard normal distribution and is nuisance parameter free. Our numerical simulation results show that the distribution of this pivotal approximates the standard normal distribution well under normal innovations.     

On Haar series of <Emphasis Type="Italic">A</Emphasis>–integrable functions

In this paper we obtain a necessary and sufficient condition on the sequence of natural numbers {q _n } such that the almost everywhere convergence of the cubic partial sums S _qn (x) of the multiple Haar series Σ_n a _nχ_n(x) and the condition lim inf \(\lambda \cdot mes\left\{ {x:\begin{array}{*{20}{c}} {\sup } \\ n \end{array}\left| {S{}_{qn}\left( x \right)} \right| \succ \lambda } \right\} = 0\), imply that the coefficients a _n can be uniquely determined by the sum of the series. Also, we have obtained a necessary and sufficient condition for the series \(\sum\limits_{n = 1}^\infty {{\varepsilon _n}{a_n}} {\chi _n}\left( x \right)\) with an arbitrary bounded sequence {ε _n} to be a Fourier-Haar series of an A-integrable function.     

Computations of coefficients in the polynomials of Pade´approximations by solving systems of linear equations

We explore reliability, stability and accuracy of determining the polynomials which define the Pade´approximation to a given function h(x) by solving a system of linear equations to get the coefficients in the denominator polynomial B_n(x). The coefficients in the numerator polynomial A_m(x) follow directly from those for B_n(x). Our approach is in the main heuristic. For the numerics we use the models e^?x1n(1 +x), (1 +x)^{± 1/2} and the exponential integral, each with m=n. The system of equations, with matrix of Toeplitz type, was solved by Gaussian elimination (Crout algorithm) with equilibration and partial pivoting. For each model, the maximum number of incorrect figures in the coefficients is of the order n at least, thus indicating that the matrix becomes ill conditioned as n increases. Let δ_n(x)andω_n(x) be the errors in A_n(x) and B_n(x) respectively, due to errors in the coefficients of B_n(x). For x fixed, δ_n(x) and ω_n(x) and the corresponding relative errors increase as n increases. However, for a considerable range on n, the relative errors in A_n(x)B_n(x) are virtually nil. This has the following theoretical explanation. Now B_n(x)h(x) ?A_m(m) = 0 (x^{m+n+ 1}). It can be shown that ω_n(x)h(x) ? δ_m(x) = 0(x^{m+ 1}). In this sense both A_m(x)B_n(x)andδ_m(x)ω_n(x) are approximations to h(x). Thus if the difference of these two approximations and ω_n(x)B_n(x), the relative error in B_n(x), are sufficiently small, then the relative error in A_m(x)/B_n(x) is of no consequence.     

Rearranged series by Haar system

For the orthonormal Haar system {X _n} the paper proves that for each 0 < ? < 1 there exist a measurable set E ? [0, 1] with measure | E | > 1 ? ? and a series of the form Σ _n=1 ^∞ a _n X _n with a _i ↘ 0, such that for every function f ∈ L ¹(0, 1) one can find a function \(\tilde f\) ∈ L ¹(0, 1) coinciding with f on E, and a series of the form

$\sum\limits_{i = 1}^\infty {\delta _i a_i \chi _i } where \delta _i = 0 or 1$

, that would converge to \(\tilde f\) in L ¹(0, 1).

     

Packing and covering constants for certain families of trees. I

If ? denotes a family of rooted trees, let p_k(n) and c_k(n) denote the average value of the k-packing and k-covering numbers of trees in ? that have n nodes. We assume, among other things, that the generating function y of trees in ? satisfies a relation of the type y = x?(y) for some power series ?. We show that the limits of p_k(n)/n and c_k(n)/n as n → ∞ exist and we describe how to evaluate these limits.     

Orthogonal polynomials on the disc

We consider the space P_n of orthogonal polynomials of degree n on the unit disc for a general radially symmetric weight function. We show that there exists a single orthogonal polynomial whose rotations through the angles , j=0,1,…,n forms an orthonormal basis for P_n, and compute all such polynomials explicitly. This generalises the orthonormal basis of Logan and Shepp for the Legendre polynomials on the disc.Furthermore, such a polynomial reflects the rotational symmetry of the weight in a deeper way: its rotations under other subgroups of the group of rotations forms a tight frame for P_n, with a continuous version also holding. Along the way, we show that other frame decompositions with natural symmetries exist, and consider a number of structural properties of P_n including the form of the monomial orthogonal polynomials, and whether or not P_n contains ridge functions.     

Analytic theory of finite asymptotic expansions in the real domain. Part I: two-term expansions of differentiable functions

We establish a general analytic theory of asymptotic expansions of type 1 $$f(x) = a_1 \varphi _1 (x) + \cdots + a_n \varphi _n (x) + o(\varphi _n (x)) x \to x_0 ,$$ , where the given ordered n-tuple of real-valued functions (? ₁, ..., ? _n ) forms an asymptotic scale at x ₀ ?? . By analytic theory, as opposed to the set of algebraic rules for manipulating finite asymptotic expansions, we mean sufficient and/or necessary conditions of general practical usefulness in order that (*) hold true. Our theory is concerned with functions which are differentiable (n ? 1) or n times and the presented conditions involve integro-differential operators acting on f, ? ₁, ..., ? _n . We essentially use two approaches; one of them is based on canonical factorizations of nth-order disconjugate differential operators and gives conditions expressed as convergence of certain improper integrals, very useful for applications. The other approach starts from simple geometric considerations and gives conditions expressed as the existence of finite limits, as x ?? x ₀, of certain Wronskian determinants constructed with f, ? ₁, ..., ? _n . There is a link between the two approaches and it turns out that some of the integral conditions found via the factorizational approach have geometric meanings. Our theory extends to more general expansions the theory of real-power asymptotic expansions thoroughly investigated in previous papers. In the first part of our work we study the case of two comparison functions ? ₁, ? ₂ because the pertinent theory requires a very limited theoretical background and completely parallels the theory of polynomial expansions.     

Single wavelets in n-dimensions

Under very minimal regularity assumptions, it can be shown that 2ⁿ−1 functions are needed to generate an orthonormal wavelet basis for L²(ℝⁿ). In a recent paper by Dai et al. it is shown, by abstract means, that there exist subsets K of ℝⁿ such that the single function ψ, defined by , is an orthonormal wavelet for L²(ℝⁿ). Here we provide methods for construucting explicit examples of these sets. Moreover, we demonstrate that these wavelets do not behave like their one-dimensional couterparts.     

Local connectivity of a random graph

A graph is locally connected if for each vertex ν of degree ≧2, the subgraph induced by the vertices adjacent to ν is connected. In this paper we establish a sharp threshold function for local connectivity. Specifically, if the probability of an edge of a labeled graph of order n is p = ((3/2 +?_n) log n/n)^1/2 where ?_n = (log log n + log(3/8) + 2x)/(2 log n), then the limiting probability that a random graph is locally connected is exp(-exp(-x)).     

Reducibility of Punctual Hilbert Schemes of Cone Varieties

In this short note we show that for any pair of positive integers (d, n) with n > 2 and d > 1 or n = 2 and d > 4, there always exist projective varieties X ? ? ^N of dimension n and degree d and an integer s ₀ such that Hilb ^s (X) is reducible for all s ≥ s ₀. X will be a projective cone in ? ^N over an arbitrary projective variety Y ? ? ^N?1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points.     

Bijective maps preserving commutators on a solvable classical group

Let F be a field, T n the group consisting of all n × n invertible upper triangular matrices over F . In this article we classify bijective maps φ from T n to itself satisfying φ[x, y] = [φ(x), φ(y)]. We show that each such map differs only slightly from an automorphism of T n .