On a theorem of Titchmarch-Kodaira-Weidmann concerning absolutely continuous operators, I

For the Favard class F_r in the space C_2π of continuous 2π-periodic functions we solve the following problem. Given x∈$\text{R}$ and knots x₀< x₁ < ··· < x_v?1., x_u? 2π we determine weights x_ki(0 ?k · n, 0 ? j < r) such that

$\text{sup}\text{f∈F}{}_{\mathrm{r}}\text{f(x)?}\text{∑}\text{k=0}\text{n?1}\text{∑}\text{j=0}\text{r?1}\text{α}{}_{\mathrm{kj}}\text{f}{}^{\mathrm{(i)}}\text{(X}{}_{\mathrm{k}}\text{)}$

is minimal. The optimal weights are unique (except for a trivial case) and we obtain them from a system of periodic polynomial splines u_kj(0 ? k < n, 0 ?j< r): α_kj = u_kj(x). These splines induce an interpolation operator whose degree of approximation with respect to the class F_r is minimal if the knots are equidistant. Finally, we describe an efficient numerical procedure which shows how to compute the interpolation spline in the equidistant case.     

Engel-Type Conditions Involving Two Generalized Skew Derivations in Prime Rings

Let ? be a prime ring of characteristic different from 2, 𝒬_r the right Martindale quotient ring of ?, 𝒞 the extended centroid of ?, F, G two generalized skew derivations of ?, and k ≥ 1 be a fixed integer. If [F(r), r]_kr ? r[G(r), r]_k = 0 for all r ∈ ?, then there exist a ∈ 𝒬_r and λ ∈ 𝒞 such that F(x) = xa and G(x) = (a + λ)x, for all x ∈ ?.     

Lacunary Quadrature Formulae and Interpolation Singularity

Birkholl quadrature formulae (q.f.), which have algebraic degree of precision (ADP) greater than the number of values used, are studied. In particular, we construct a class of quadrature rules of ADP = 2n + 2r + 1 which are based on the information {ƒ^(j)(−1), ƒ^(j)(−1), j = 0, ..., r − 1 ; ƒ(x_i), ƒ^(2m)(x_i), i = 1, ..., n}, where m is a positive integer and r = m, or r = m − 1. It is shown that the corresponding Birkhoff interpolation problems of the same type are not regular at the quadrature nodes. This means that the constructed quadrature formulae are not of interpolatory type. Finally, for each In, we prove the existence of a quadrature formula based on the information {ƒ(x_i), ƒ^(2m)(x_i), i = 1, ..., 2m}, which has algebraic degree of precision 4m + 1.     

Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach

The problem of determining the pair w:={F(x,t);T₀(t)} of source terms in the parabolic equation u_t=(k(x)u_x)_x+F(x,t) and Robin boundary condition −k(l)u_x(l,t)=v[u(l,t)−T₀(t)] from the measured final data μ_T(x)=u(x,T) is formulated. It is proved that both components of the Fréchet gradient of the cost functional can be found via the same solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is derived. The obtained results permit one to prove existence of a quasi-solution of the considered inverse problem, as well as to construct a monotone iteration scheme based on a gradient method.     

Perfect spline approximation

Our study of perfect spline approximation reveals: (i) it is closely related to ΣΔ modulation used in one-bit quantization of bandlimited signals. In fact, they share the same recursive formulae, although in different contexts; (ii) the best rate of approximation by perfect splines of order r with equidistant knots of mesh size h is h^r−1. This rate is optimal in the sense that a function can be approximated with a better rate if and only if it is a polynomial of degree <r.The uniqueness of best approximation is studied, too. Along the way, we also give a result on an extremal problem, that is, among all perfect splines with integer knots on , (multiples of) Euler splines have the smallest possible norms.     

Extreme convex set functions with finite carrier: General theory

Let Ω={1,…,n} and P={X:SΩ}. A mapping e : P→R⁺ is a convex set function if e()=0 and e(S) + e(T)e(S∩T) + e(S T) for all S. TεP. The set of convex set functions for fixed Ω is a convex cone and the paper is dealing with the extreme points of the base of this cone. To this end a representation theorem is proved: every e ε ¹ can be written as e(·)=max(m¹(·)−α₁…m^t(·)−α_t), where m¹,…,m^t are measures on P and α₁,…,α_t are nonnegative reals. Given additional requirements, the representation is unique and called “canonical”. Fix H {1,…,r},|H| 2. There is a certain subsystem of sets SεP such that m^τ(S)−α_τ=e(S) (τε H}, that is, the subsystem of sets S such that m^τ(S)−α_τ(τεH) is a maximal term in the representation of e by m¹,…,m^τ and α₁,…α_t.e is called nondegenerate is these subsystems determine the measures m¹,…,m^τ uniquely and it turns out that nondegeneracy and extremality are equivalent for e ε ¹. Moreover, it is seen that nondegeneracy is closely related to a generalized version of the problem “represent a given integer λ o by means of integer weights g,…,g_r 0 via σ^r=₁ag=λ such that the integer coefficients a satisfy 0ak (=1,…,r), where k are prescribed integer bounds. Find r such representations with the additional property that the coefficients form a nonsingular matrix.” A solution to the generalized version of this number theoretical problem is given and, finally, a few examples are discussed.     

Approaching an extinction point in one-dimensional semilinear heat equations with strong absorption

This paper deals with the Cauchy problem u_t − u_xx + u^p = 0; − ∞ < x < + ∞, t>0, u(x, 0) = u₀(x); − ∞ < x < + ∞, where 0 < p < 1 and u₀(x) is continuous, nonnegative, and bounded. In this case, solutions are known to vanish in a finite time T, and interfaces separating the regions where u(x, t) > 0 and u(x, t) = 0 appear when t is close to T. We describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached. We also give conditions under which some of these behaviours actually occur.     

Trace cocharacters and the Kronecker products of Schur functions

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.     

Fourier Analysis of Semistable Distributions

We consider the generalized convolution powers G _α ^*u (x) of an arbitrary semistable distribution function G _α (x) of exponent α∈(0,2), and prove that for all j, k∈{0,1,2,…} and u>0 the derivatives G _α ^(k,j)(x;u)=∂ ^k+j G _α ^*u (x)/∂ x ^k ∂ u ^j , x∈ℝ, are of bounded variation on the whole real line ℝ. The proof, along with an integral recursion in j, is new even in the special case of stable laws, and the result provides a framework for possible asymptotic expansions in merge theorems from the domain of geometric partial attraction of semistable laws. An erratum to this article can be found at     

On the Interpolation by Discrete Splines with Equidistant Nodes

In this paper we consider equidistant discrete splines S(j), j , which may grow as O(|j|^s) as |j|→∞. Such splines are relevant for the purposes of digital signal processing. We give the definition of the discrete B-splines and describe their properties. Discrete splines are defined as linear combinations of shifts of the B-splines. We present a solution to the problem of discrete spline cardinal interpolation of the sequences of power growth and prove that the solution is unique within the class of discrete splines of a given order.     

Analysis for the identification of an unknown diffusion coefficient via semigroup approach

This paper presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u_x) in the inhomogenenous quasi‐linear parabolic equation u_t(x, t)=(k(u_x)u_x(x, t))_x +F(u), with the Dirichlet boundary conditions u(0, t)=ψ₀, u(1, t)=ψ₁ and source function F(u). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]:??→C¹[0, T], Ψ[·]:??→C¹[0, T] via semigroup theory. Copyright © 2009 John Wiley & Sons, Ltd.     

Automatic Smoothing Spline Projection Pursuit

Abstract

A highly flexible nonparametric regression model for predicting a response y given covariates {x_k}^d _k=1 is the projection pursuit regression (PPR) model ? = h(x) = β₀ + Σ_jβ_jf_j(α^T _jx) where the f_j , are general smooth functions with mean 0 and norm 1, and Σ^d _k=1α² _kj=1. The standard PPR algorithm of Friedman and Stuetzle (1981) estimates the smooth functions f_j using the supersmoother nonparametric scatterplot smoother. Friedman's algorithm constructs a model with M _max linear combinations, then prunes back to a simpler model of size M ≤ M _max, where M and M _max are specified by the user. This article discusses an alternative algorithm in which the smooth functions are estimated using smoothing splines. The direction coefficients α_j, the amount of smoothing in each direction, and the number of terms M and M _max are determined to optimize a single generalized cross-validation measure.     

О наилучшем выборе уз лов при интерполиров ании функций эрмитовыми с плайнами

The problem of optimal choice of knots is considered for the functions belonging to the classW ^2m+1 V, concerning interpolation by means of Hermite splines. The problem of asymptotically best choice of the knots for interpolation of a fixed functionf(x) (f^(2m+2)(x)>0, 0x1) by Hermite splines is also treated.     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

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.     

Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations

In this paper we study the existence of periodic solutions of the fourth-order equations u^iv − pu″ − a(x)u + b(x)u³ = 0 and u^iv − pu″ + a(x)u − b(x)u³ = 0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P₁) and (P₂) for these equations are considered respectively with the boundary conditions u(0) = u(L) = u″(0) = u″(L) = 0. Existence of nontrivial solutions for (P₁) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P₂) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation u^iv + pu″ + a(x)u − b(x)u² − c(x)u³ = 0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used.     

On Equations Which Describe Pseudospherical Surfaces

Using the notion of a differential equation which describes an η-pseudospherical surface (η-p.s.s.), we give a characterization of the equations of type u_xt = F(u, u_x,…, ?^ku / ?x^k), k ≥ 2, with this property. We obtain a systematic procedure to determine a linear problem for which a given equation is the integrability condition. The equations of type u_xt = F(u, u_x) were characterized by Rabelo and Tenenblat in another paper. The theory is applied to several equations, some of which were not known to describe η-p.s.s.     

Compactly supported wavelet bases for Sobolev spaces

In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and in satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψ_jk:=2^j/2ψ(2^j·−k) ( ) form a Riesz basis for . If, in addition, φ lies in the Sobolev space , then the derivatives 2^j/2ψ^(m)(2^j·−k) ( ) also form a Riesz basis for . Consequently, is a stable wavelet basis for the Sobolev space . The pair of φ and are not required to be biorthogonal or semi-orthogonal. In particular, φ and can be a pair of B-splines. The added flexibility on φ and allows us to construct wavelets with relatively small supports.     

Analysis of a semigroup approach in the inverse problem of identifying an unknown coefficient

This article presents a semigroup approach to the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u_x) in the quasi‐linear parabolic equation u_t(x, t)=(k(u_x)u_x(x, t))_x+F(x, t), with Dirichlet boundary conditions u(0, t)=ψ₀, u(1, t)=ψ₁ and source function F(x, t). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]: ?? → C¹[0, T], Ψ[·]: ?? → C¹[0, T] via semigroup theory. Copyright © 2008 John Wiley & Sons, Ltd.     

An inverse polynomial method for the identification of the leading coefficient in the Sturm–Liouville operator from boundary measurements

An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u^′(x))^′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^′(x)≠0,u^″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^′(x₀)=0,x₀(0,1);u^′(x)≠0,x≠x₀;u^″(x)≠0,x[0,1]) and severely ill-conditioned (u^′(x₀)=u^″(x₀)=0,x₀(0,1);u^′(x)≠0,u^″(x)≠0,x≠x₀). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.