Weighted tight frames of exponentials on a finite interval

Given a finite intervalI?R, a characterization is given for those discrete sets of real numbers Λ and associated sequences {c _λ}_λ∈Λ, withc _λ>0, having the properties that every functionf∈L ²(I) can be expanded inL ²(I) as the unconditionally convergent series $$f = \sum\limits_{\lambda \in \Lambda } {\hat f} (\lambda )c_\lambda e^{2\pi i\lambda x} $$ and that the range of the mappingL ²(I)→L _μ ² :f→f has finite codimension inL _μ ² , iff denotes the Fourier transform off and μ is the measure μ = ∑_λ∈Λ c _λ δ_λ.     

Factorization of matrix polynomials with multiple roots

The concept of a multiple root of matrix polynomial L(λ) is introduced, and associated spectral properties of L(λ) are investigated. A statement concerning factorization of L(λ) is presented. Applications are made to factorizations of the matrix polynomial L^α(λ), for any positive integer α.     

Factorization of a convolution-type operator

Three convolution-type equations are considered in the space of entire functions with topology ofd uniform convergence: $$\begin{gathered} M{_{\mu}{_1}} [f] \equiv \smallint _C f(z + t)d\mu _1 = 0, \hfill \\ M{_\mu{_1}} [f] \equiv \smallint _C f(z + t)d\mu _2 = 0, \hfill \\ M_\mu [f] \equiv \smallint _C f(z + t)d\mu = 0 \hfill \\ \end{gathered}$$ with respective characteristic functions L₁(λ), L₂(λ), L(λ)=L₁(λ)· L₂(λ), suppμ ?c, suppμ ₁ ?c, suppμ ₂ ?c. The necessary and sufficient conditions are found that every solutionf(z) of the equation M_μ[f[ can be written as a sumf ₁(z) +f ₂(z), wheref ₁(z) is the solution of the equation \(M{_\mu{_1}} [f] = 0\) ,f ₂(z) is the solution of the equation \(M{_\mu{_2}} [f] = 0\) .     

A pointwise abelian Ergodic theorem for Lp semigroups, 1 ⩽ p < ∞

Let (X, ∑, μ) be a σ-finite measure space and L_p(μ) = L_p(X, ∑, μ), 1 ? p ? ∞, the usual Banach spaces of complex-valued functions. Let {T_t: t ? 0} be a strongly continuous semigroup of positive L_p(μ) operators for some 1 ? p < ∞. Denote by R_λ the resolvent of {T_t}. We show that f?L_p(μ) implies λR_λf(x) → f(x) a.e. as λ → ∞.     

On spectral characterizations and embeddings of graphs

The least eigenvalue of the 0-1 adjacency matrix of a graph is denoted λ G. In this paper all graphs with λ(G) greater than ?2 are characterized. Such a graph is a generalized line graph of the form L(T;1,0,…,0), L(T), L(H), where T is a tree and H is unicyclic with an odd cycle, or is one of 573 graphs that arise from the root system E₈. If G is regular with λ(G)>?2, then Gis a clique or an odd circuit. These characterizations are used for embedding problems; λ_R(H) = sup{λ(G)z.sfnc;HinG; Gregular}. H is an odd circuit, a path, or a complete graph iff λ_R(H)> ?2. For any other line graph H, λ_R(H) = ?2. A similar result holds for complete multipartite graphs.     

Linear maps on matrices: the invariance of symmetric polynomials

Let L be a linear map on the space M_n of all n by n complex matrices. Let h(x₁,…,x_n) be a symmetric polynomial. If X is a matrix in M_n with eigenvalues λ₁,…,λ_n, denote h(λ₁,…,λ_n) by h(X). For a large class of polynomials h, we determine the structure of the linear maps L for which h(X)=h(L(X)), for all X in M_n.     

The higher rank numerical range of nonnegative matrices

In this article the rank-k numerical range ∧ _k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧ _k (A), we examine their location on the complex plane. Further, an application of this theory to ∧ _k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C _L .     

Quadratic matrix polynomials with a parameter

In this paper we examine matrix polynomials of the form L(λ) = Aλ² + εBλ + C in which ε is a parameter and A, B, C are positive definite. This arises in a natural way in the study of damped vibrating systems. The main results are concerned with the generic case in which det L(λ) has at least 2n − 1 distinct zeros for all ε ϵ [0, ∞). The values of ε at which there is a multiple zero of det L(λ) are of major interest in this analysis. The dependence of first degree factors of L(λ) on ε is also discussed.     

Uniform boundedness inL p (p=p(x)) of some families of convolution operators

Suppose that a measurable 2π-periodic essentially bounded function (the kernel) κ_λ =κ_λ(x) is given for any realλ≥1. We consider the following linear convolution operator inL ^p: $$\kappa _\lambda = \kappa _\lambda f = (\kappa _\lambda f)(x) = \int_{ - \pi }^\pi {f(t)} k_\lambda (t - x) dt.$$ Uniform boundedness of the family of operators {Κ_λ}_λ≥1 is studied. Conditions on the variable exponentp=p(x) and on the kernel κ_λ that ensure the uniform boundedness of the operator family {Κ_λ}_λ≥1 inL ^p are obtained. The condition on the exponentp=p(x) is given in its final form.     

An extremal problem on non-full colorable graphs

For a given graph G of order n, a k-L(2,1)-labelling is defined as a function f:V(G)→{0,1,2,…k} such that |f(u)-f(v)|?2 when d_G(u,v)=1 and |f(u)-f(v)|?1 when d_G(u,v)=2. The L(2,1)-labelling number of G, denoted by λ(G), is the smallest number k such that G has a k-L(2,1)-labelling. The hole index ρ(G) of G is the minimum number of integers not used in a λ(G)-L(2,1)-labelling of G. We say G is full-colorable if ρ(G)=0; otherwise, it will be called non-full colorable. In this paper, we consider the graphs with λ(G)=2m and ρ(G)=m, where m is a positive integer. Our main work generalized a result by Fishburn and Roberts [No-hole L(2,1)-colorings, Discrete Appl. Math. 130 (2003) 513-519].     

On the absolute continuity of Schrödinger operators with spherically symmetric,long-range potentials,II

Let H = ?Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V(r) = V_S(r) + V_L. Let λ = lim sup_r→∞V_L(r) < ∞ (we allow λ = ? ∞) and set λ⁺ = max(λ, 0). Assume that for some r₀, V_L(r) ?C^2k(r₀, ∞) and that there exists δ > 0 such that $\text{(}\text{d}\text{dr}\text{)}{}^{\mathrm{j}}\text{V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r) · (λ}{}^{\mathrm{+}}\text{? V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r) + 1)}{}^{\mathrm{?1}}\text{= O(r}{}^{\mathrm{?j\delta }}\text{), j = 1,…, 2k,}\text{as}\text{r → ∞}$. Assume further that $\text{∝}{}_1{}^{\mathrm{\infty }}\text{(}\text{dr}\text{¦ V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r)¦}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{) = ∞}$ and that 2kδ > 1. It is shown that: (a) The restriction of H to C^∞(Rⁿ) is essentially self-adjoint, (b) The essential spectrum of H contains the closure of (λ, ∞). (c) The part of H over (λ, ∞) is absolutely continuous.     

Friedrichs extensions of Schrödinger operators with singular potentials

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d²/dx²+Bx²+Ax⁻²+λx^−α (B>0, A?0) in L₂(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L₂(0,∞), namely C₀^∞(0,∞) and D(T_2,F)∩D(M_λ,α), where the latter is a subspace of the Sobolev space W_2,2(0,∞). Adjoints of these differential operators on C₀^∞(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) in L₂(0,∞) lead to the other adjoints. Further density properties C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T_2,F)∩D(M_λ,α).     

The linearity coefficient of metric projections onto a Chebyshev subspace

The linearity coefficient λ(Y) of a metric projection P _Y onto a subspace Y in a Banach space X is determined. This coefficient turns out to be related to the Lipschitz norm of the operator P _Y . It is proved that, for any Chebyshev subspace Y in the space C or L ₁, either λ(Y) = 1 (which corresponds to the linearity of P _Y ) or λ(Y) ≤ 1/2.     

On singular integrals depending on three parameters

In this paper we will prove the pointwise convergence of L(f; x, y, λ) to f(x₀, y₀), as (x, y, λ) tends to (x₀, y₀, λ₀) in the space L_2π, by the three parameter family of singular operators. In contrast to previous works, the kernel function is radial.     

On Solvability of Boundary Value Problems for Kinetic Operator-Differential Equations

We study solvability of boundary value problems for the so-called kinetic operator-differential equations of the form B(t)u _t ?L(t)u = f, where L(t) and B(t) are families of linear operators defined in a complex Hilbert space E. We do not assume that the operator B is invertible and that the spectrum of the pencil L ?λ B is included into one of the half-planes Re λ < a or Re λ > a \({(a\in {\mathbb{R}})}\) . Under certain conditions on the above operators, we prove several existence and uniqueness theorems and study smoothness questions in weighted Sobolev spaces for solutions.     

Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,…,λ} such that |f(u)−f(v)|≥p if u and v are adjacent, and |f(u)−f(v)|≥q if u and v are at distance 2 apart. The minimum value of λ for which G has L(p,q)-labeling is denoted by λ_p,q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks.In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p−1)+q(bc(G)−2), where bc(G) is the biclique number of G. Since λ_p,q(G)≥p+q(bc(G)−2) for any bipartite graph G, the upper bound is at most p−1 far from optimal.     

Invariant subspaces of analytic multiparticle Hamiltonians

The quantum mechanics of n particles interacting through analytic two-body interactions can be formulated as a problem of functional analysis on a Hilbert space $\text{G}$ consisting of analytic functions. On $\text{G}$, there is an Hamiltonian H with resolvent R(λ). These quantities are associated with families of operators H(?) and R(λ, ?) on $\text{L}$, the case ? = 0 corresponding to standard quantum mechanics. The spectrum of H(?) consists of possible isolated points, plus a number of half-lines starting at the thresholds of scattering channels and making an angle 2? with the real axis.Assuming that the two-body interactions are in the Schmidt class on the two-particle space $\text{G}$, this paper studies the resolvent R(λ, ?) in the case ? ≠ 0. It is shown that a well known Fredholm equation for R(λ, ?) can be solved by the Neumann series whenever ¦λ¦ is sufficiently large and λ is not on a singular half-line. Owing to this, R(λ, ?) can be integrated around the various half-lines to yield bounded idempotent operators P_p(?) (p = 1, 2,…) on $\text{L}$. The range of P_p(?) is an invariant subspace of H(?). As ? varies, the family of operators P_p(?) generates a bounded idempotent operator P_p on a space $\text{G}$. The range of this is an invariant subspace of H. The relevance of this result to the problem of asymptotic completeness is indicated.     

A framework for analyzing nonlinear eigenproblems and parametrized linear systems

Associated with an n×n matrix polynomial of degree , are the eigenvalue problem P(λ)x=0 and the linear system problem P(ω)x=b, where in the latter case x is to be computed for many values of the parameter ω. Both problems can be solved by conversion to an equivalent problem L(λ)z=0 or L(ω)z=c that is linear in the parameter λ or ω. This linearization process has received much attention in recent years for the eigenvalue problem, but it is less well understood for the linear system problem. We develop a framework in which more general versions of both problems can be analyzed, based on one-sided factorizations connecting a general nonlinear matrix function N(λ) to a simpler function M(λ), typically a polynomial of degree 1 or 2. Our analysis relates the solutions of the original and lower degree problems and in the linear system case indicates how to choose the right-hand side c and recover the solution x from z. For the eigenvalue problem this framework includes many special cases studied in the literature, including the vector spaces of pencils L₁(P) and L₂(P) recently introduced by Mackey, Mackey, Mehl, and Mehrmann and a class of rational problems. We use the framework to investigate the conditioning and stability of the parametrized linear system P(ω)x=b and thereby study the effect of scaling, both of the original polynomial and of the pencil L. Our results identify situations in which scaling can potentially greatly improve the conditioning and stability and our numerical results show that dramatic improvements can be achieved in practice.     

Deflating quadratic matrix polynomials with structure preserving transformations

Given a pair of distinct eigenvalues (λ₁,λ₂) of an n×n quadratic matrix polynomial Q(λ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(λ) into a quadratic of the form having the same eigenvalue s as Q(λ), with Q_d(λ) an (n-1)×(n-1) quadratic matrix polynomial and q(λ) a scalar quadratic polynomial with roots λ₁ and λ₂. This block diagonalization cannot be achieved by a similarity transformation applied directly to Q(λ) unless the eigenvectors corresponding to λ₁ and λ₂ are parallel. We identify conditions under which we can construct a family of 2n×2n elementary similarity transformations that (a) are rank-two modifications of the identity matrix, (b) act on linearizations of Q(λ), (c) preserve the block structure of a large class of block symmetric linearizations of Q(λ), thereby defining new quadratic matrix polynomials Q₁(λ) that have the same eigenvalue s as Q(λ), (d) yield quadratics Q₁(λ) with the property that their eigenvectors associated with λ₁ and λ₂ are parallel and hence can subsequently be deflated by a similarity applied directly to Q₁(λ). This is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action.     

The resolvent of a dilation-analytic three-particle system

If the potential in a three-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has a dilation-analytic continuation H(φ). The continuous spectrum of H(φ) consists of half-lines Y(λ_p, φ) starting at the thresholds λ_p of scattering channels and making angles 2φ with the positive real axis. If the interaction is the sum of local two-body potentials in suitable $\text{L}$^p-spaces, each half-line Y(λ_p, φ) is associated with an operator P(λ_p, φ) that projects onto an invariant subspace of H(φ). Suppose Y(λ_p, φ) does not pass through any two- or three-particle eigenvalues λ ≠ λ_p when φ runs through some interval $\text{0 < α ? φ ? β <}\text{π}\text{2}$. For φ in [α, β], this paper shows that the resolvent R(λ, φ) has smoothness properties near Y(λ_p, φ) that are sufficient for P(λ_p, φ)[H(φ) ? λ_p] e^?2iφ to be spectral and to generate a strongly differentiable group. The projection, the group, and the spectral resolution operators are norm continuous in φ. These results are not affected by any spurious poles of the resolvent equation. At a spurious pole λ = λ_p + ze^2iφ, the resolvent R(λ_p + ze^2iφ,φ) is examined by a method that uses two resolvent equations in succession and shows that there is norm continuity in z, φ. The case of spurious poles on Y(λ_p, φ) is included.