William Frederick Trench

• PhD
• Professor at Trinity University

Following Andrew [1], we say that an n–vector x = [x 1 x 2 · · · x n ] T is symmetric if x j = x n−j+1 , 1 ≤ j ≤ n, or skew–symmetric if x j = −x n−j+1 , 1 ≤ j ≤ n. (Some authors call such vectors reciprocal and anti-reciprocal .) The following theorems are special cases of results stated explicitly by Cantoni and Butler [2], but already implicit i...

It is shown that a formula of Gohberg and Semencul for the inverse of a Toeplitz matrix is equivalent to an earlier formula of the author, and that a similar formula of Heinig follows from a formula of the author for the inverse of a Hankel matrix. This manuscript was submitted for publication in Linear Algebra and Its Ap-plications in 1989 or 1990...

Let P ∈ ℂ mxm and Q ∈ ℂ n×n be invertible matrices partitioned as P = [P 0 P 1 · · · P k−1 ] and Q = [Q 0 Q 1 · · · Q k−1 ], with P ℓ ∈ ℂ m×mℓ and Q ℓ ∈ ℂ n×nℓ , 0 ≤ ℓ ≤ k − 1. Partition P ⁻¹ and Q ⁻¹ as where P̂ ℓ ∈ ℂ mℓ ×m , Q̂ ℓ ∈ ℂ nℓ×n , P̂ℓP m = δ ℓm I mℓ , and Q̂ ℓ Q m = δ ℓm I nℓ , 0 ≤ ℓ, m ≤ k − 1. Let Z k = {0, 1, . . . , k − 1}. We stud...

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We consider the following inverse problems for the class of unilevel block α-circulants , where k > 1, C0, C1, …, , α ∈ {1,2, … ,k −1}, gcd(α,k) = 1, and ∥ ⋅ ∥ is the Frobenius norm. Problem 1 Find necessary and sufficient conditions on and for the existence of such that CZ = W, and find all such C if the conditions are satisfied. Problem 2 For arb...

Suppose that –∞a b a ≤ u 1n ≤ u 2n ≤ · · · ≤ unn ≤ b, and a ≤ v 1n ≤v 2n ≤ · · · ≤ v nn ≤ b for n ≥ 1. We simplify and strengthen Weyl's definition of equal distribution of {{uin} n i = 1}∞n = 1 and {{v in} ni = 1}∞n = 1 by showing that the following statements are equivalent: (i) limn→∞ 1/n Σni = 1 (F(uin) – F (v in )) = 0 for all F ∈ C [a, b], (i...

Let and , where , , , , …, are distinct complex numbers, and . We say that is -commutative if . We characterize the class of -commutative matrices and extend results obtained previously for the case where and , , with , . Our results are independent of , , …, , so long as they are distinct; i.e., if for some choice of , (all distinct), then for arb...

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May be copied, modified, redistributed, translated, and built upon subject to the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License

We consider the asymptotic behavior of solutions of a linear differential system x′=A(t)x, where A is continuous on an interval ([a,∞). We are interested in the situation where the system may not have a desirable asymptotic property such as stability, strict stability, uniform stability, or linear asymptotic equilibrium, but its solutions can be wr...

We say that a matrix R∈Cn×n is k-involutory if its minimal polynomial is xk-1 for some k⩾2, so Rk-1=R-1 and the eigenvalues of R are 1, ζ, ζ2,…,ζk-1, where ζ=e2πi/k. Let α,μ∈{0,1,…,k-1}. If R∈Cm×m,A∈Cm×n,S∈Cn×n and R and S are k-involutory, we say that A is (R,S,α,μ)-symmetric if RAS-α=ζμA. We show that an (R,S,α,μ)-symmetric matrix A can be useful...

Let Cn×n(I) denote the set of continuous n×n matrices on an interval I. We say that R∈Cn×n(I) is a nontrivial k-involution if R=P(⊕ℓ=0k-1ζℓIdℓ)P-1 where ζ=e-2πi/k, d0+d1+⋯+dk-1=n, and P′=P⊕ℓ=0k-1Uℓ with Uℓ∈Cdℓ×dℓ(I). We say that A∈Cn×n(I) is R-symmetric if R(t)A(t)R-1(t)=A(t), t∈I, and we show that if A is R-symmetric then solving x′=A(t)x or x′=A(...

In a previous paper we characterized unilevel block α-circulants , Am∈Cd1×d2, 0⩽m⩽n-1, in terms of the discrete Fourier transform FA={F0,F1,…,Fn-1} of A={A0,A1…,An-1}, defined by . We showed that most theoretical and computational problems concerning A can be conveniently studied in terms of corresponding problems concerning the Fourier coefficient...

Let , and .All operations in indices are modulo k.It is well known that if d1=d2=1 then , where .However, to our knowledge it has not been emphasized that FA plays a fundamental role in connection with all the matrices , with d1,d2 arbitrary.We begin by adapting a theorem of Ablow and Jenner with d1=d2=1 to the case where d1 and d2 are arbitrary.We...

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We say that a matrix R∈Cn×n is k-involutory if its minimal polynomial is xk-1 for some k⩾2, so Rk-1=R-1 and the eigenvalues of R are 1, ζ,ζ2,…,ζk-1, where ζ=e2πi/k. Let μ∈{0,1,…,k-1}. If R∈Cm×m, A∈Cm×n,S∈Cn×n and R and S are k-involutory, we say that A is (R,S,μ)-symmetric if RAS-1=ζμA. If R,A∈Cn×n, we say that A is (R,μ)-symmetric if RAR-1=ζμA. We...

We consider the asymptotic relationship as n → ∞ between the eigenvalues λ1 n {less-than or slanted equal to} ⋯ {less-than or slanted equal to} λn n and μ1 n {less-than or slanted equal to} ⋯ {less-than or slanted equal to} μn n of the Sturm-Liouville problems defined for n {greater than or slanted equal to} 2 k + 1 by{A formula is presented} and{A...

Let n=(n1,…,nk) be a multiindex and κ(n̲)=∏j=1knj. We say that n→∞ if ni→∞, 1⩽i⩽k. If r=(r1,…,rk) and s=(s1,…,sk), let ∣r−s∣=(∣r1−s1∣,…,∣s1−sk∣). We say that a multilevel Toeplitz matrix of the form Tn̲=[t|r̲-s̲|]r̲,s̲=1̲∞̲ is totally symmetric. Let Qk be the k-fold Cartesian product of Q=[−π,π] with itself, and let {tr̲}r̲=-∞̲∞̲ be the Fourier coe...

Let n=n1n2⋯nk where k>1 and n1,…,nk are integers >1. For 1⩽i⩽k, let pi=∏j=1i-1nj and qi=∏j=i+1knj, and suppose that Ui∈Cni×ni is a nontrivial involution; i.e., Ui=Ui-1≠±Ini. Let Ri=Ipi⊗Ui⊗Iqi, 1⩽i⩽k, and denote R=(R1,…,Rk). If μ∈{ 0, 1,l…,2k−1}, let μ=∑i=1kℓiμ2i-1 be its binary expansion. We say that A∈Cn×n is (R, μ)-symmetric if RiARi=(-1)ℓiμA, 1⩽...

We define a class of formal expansions a Sigma(l)(infinity)=-infinity alpha(l)z(l) of a rational function with at least one non-zero pole. To distinct formal expansions Sigma(l)(infinity)=-infinity alpha(l)z(l) and Sigma(l)(infinity)=-infinity beta(l)z(l) in this class we associate structured arrays A = (alpha(ij))(i,j=1)(infinity) and B = (b(ij))(...

Let $R\in \mathbb{C}^{m\times m}$ and $S\in \mathbb{C}^{n\times n}$ be nontrivial involutions; i.e., $R=R^{-1}\ne\pm I_m$ and $S=S^{-1}\ne\pm I_n$. We say that $A\in \mathbb{C}^{m\times n}$ is (R,S)-symmetric ((R,S)-skew symmetric) if RAS=A (RAS=-A).We give an explicit representation of an arbitrary (R,S)-symmetric matrix A in terms of matrices P a...

Let R∈Cm×m and S∈Cn×n be nontrivial involutions; thus R=R−1≠±I and S=S−1≠±I. We say that A∈Cm×n is (R,S)-symmetric ((R,S)-skew symmetric) if RAS=A (RAS=−A). Let S be the class of m×n (R,S)-symmetric matrices or the class of m×n (R,S)-skew symmetric matrices. Let Z∈Cn×q and W∈Cm×q. We study the following problems:(i)Give necessary and sufficient con...

Let R∈Cn×n be a nontrivial unitary involution; i.e., R=R∗=R−1≠±I. We say that A∈Cn×n is R-symmetric (R-skew symmetric) if RAR=A (RAR=−A). Let S be one of the following subsets of Cn×n: (i) hermitian matrices; (ii) hermitian R-symmetric matrices; (iii) hermitian R-skew symmetric matrices. Given Z, W∈Cn×m, we characterize the matrices A in S that min...

Let {aℓ(f)}ℓ=−∞∞ be the Fourier coefficients of f∈L[−π,π] and consider the Toeplitz matrices {Tn(f)}, where Tn(f)=(ai−j(f))i,j=1n; thus, {Tn(f)} is generated by the symbol f. There are many results on asymptotic properties of {Tn(f)} as n→∞. If r is a positive integer and Tn(r)(f)=(ar(i−j))i,j=1n, then {Tn(r)(f)}={Tn(fr)} where fr is easily obtaine...

Let be a nontrivial involution; i.e., R=R−1≠±I. We say that is R-symmetric (R-skew symmetric) if RAR=A (RAR=−A). Let be one of the following subsets of : (i) R-symmetric matrices; (ii) Hermitian R-symmetric matrices; (iii) R-skew symmetric matrices; (iv) Hermitian R-skew symmetric matrices. Let with rank(Z)=m and Λ=diag(λ1,…,λm).The inverse eigenpr...

Let be a nontrivial involution; i.e., R=R−1≠±I. We say that is R-symmetric (R-skew symmetric) if RAR=A (RAR=−A).There are positive integers r and s with r+s=n and matrices and such that , , RP=P, and RQ=−Q. We give an explicit representation of an arbitrary R-symmetric matrix A in terms of P and Q, and show that solving Az=w and the eigenvalue prob...

Abstract

We strengthen a theorem of Kuijlaars and Serra Capizzano on the distribution of zeros of a sequence of orthogonal polynomials {pn}n=1∞ for which the coefficients in the three term recurrence relation are clustered at finite points. The proof uses a matrix argument motivated by a theorem of Tyrtyshnikov.

Let U={{u(in)}(i=1)d(n)}(nn"0) and V={{v(in)}(i=1)d(n)}(nn"0), where u 1nu 2n⋯u dn,n, v 1nv 2n⋯v dn,n, nn 0, and lim n→∞d n=∞. Let F be a set of continuous real-valued functions on R. Then U and V are equally distributed with respect to F if∑(i=1)d(n)(F(u(in))-F(v(in)))=o(d(n)), F∈F,or absolutely equally distributed with respect to F if∑(i=1)d(n)|F...

If −∞<α<β<∞ let mid(α,x,β)=α if x<α, x if α⩽x⩽β, β if x>β. Let An=Bn+Pn where Bn and Pn are n×n Hermitian matrices. We show that if ∥Pn∥F2=o(n) then, for any [α,β], (A) ∑i=1n|F(mid(α,λi(An),β))−F(mid(α,λi(Bn),β))|=o(n) if F∈C[α,β]. (Eigenvalues numbered in nondecreasing order.) We consider the special case where {Pn} are real Hankel matrices. We al...

Let y1 and y2 be principal and nonprincipal solutions of the non- oscillatory differential equation (r(t)y0)0 + f(t)y = 0. In an earlier paper we showed that if R 1(f g)y1y2 dt converges (perhaps conditionally), and a related improper integral converges absolutely and sufficently rapidly, then the differential equation (r(t)x0)0 + g(t)x = 0 has sol...

We consider the asymptotic spectral distribution of Hermitian Toeplitz matrices {Tn}1n=1 formally generated by a rational function h(z) = (f(z)f�(1/z))/(g(z)g�(1/z)), where the numerator and denominator have no common zeros, deg(f) < deg(g), and the zeros of g are in the open punc- tured disk 0 < |z| < 1. From Szego's theorem, the eigenvalues of {T...

We consider the functional difference system ( A ) j x i ( n )= f i ( n ; X ), 1 h i h k , where X =( x 1 ,…, x k ) and f 1 (·; X ),…, f k (·; X ) are real-valued functionals of X , which may depend quite arbitrarily on values of X ( l ) for multiple values of l ] Z . We give sufficient conditions for ( A ) to have solutions that approach specified...

Let {x1n} and {x2n} be recessive and dominant solutions of the nonoscillatory difference equation Δ(rn−1Δxn−1)+pnxn=0. In an earlier paper we showed that if ∑∞fnx1nx2n converges (perhaps conditionally) and a related infinite series converges absolutely and sufficiently rapidly, then the difference equation Δ(rn−1Δyn−1)+pnyn=fnyn is also nonoscillat...

If ζ is a nonzero complex number and P is a monic polynomial with real coefficients, let Kn(ζ;P)=(P(|r−s|)ρ|r−s|ei(r−s)φ)r,s=1n. We call the class of matrices Tn=∑jcjKn(ζj;Pj) (cj real, finite sum) generalized Kac–Murdock–Szegömatrices. If |ζj|<1 for all j, the family {Tn} has a generating function in C[−π,π], and Szegö's distribution theorem impli...

We consider functional perturbations of the nonoscillatory equation Δ(rn-1Δxn-1) + pnxn = 0. Let Sk be the set of all real sequences of the form Y = {y n}n=kinfin;. For each n > k, let f n(Y) denote a real-valued functional of Y ∈ Sk. We give sufficient conditions on {fn(Y)}n=k+1∞ for the equation Δ(rn-1Δy n-1) + pnyn = fn(Y), n> k, to have a solut...

We consider generalizations of the Kac-Murdock-Szegö matrices [cf. M. Kac, W. L. Murdock and G. Szegö, J. Rat. Mech. Analysis 2, 767-800 (1953; Zbl 0051.30302)] of the forms L n =(ρ |r-s| c min(r,s) ) r,s=1 n and U n =(ρ |r-s| c max(r,s) ) r,s=1 n , where ρ and c 1 ,c 2 ,⋯,c n are real numbers. We obtain explicit expressions for the determinants an...

If Tn=(tr−s)r,s=0n is a real symmetric Toeplitz (RST) matrix then Rn has a basis consisting of ⌈n/2⌉ eigenvectors x satisfying (A) Jx=x and ⌊n/2⌋ eigenvectors y satisfying (B) Jy=−y, where J is the flip matrix. We say that an eigenvalue λ of Tn is even if a λ-eigenvector of Tn satisfies (A), or odd if a λ-eigenvector of Tn satisfies (B). We call th...

In this article we revisit the classical subject of infinite products. For standard definitions and theorems on this subject see [1] or almost any textbook on complex analysis. We will restate parts of this material required to set the stage for our results, as follows. The infinite product P = ∞ (1+a n) of complex numbers is said to converge if th...

We consider symmetric Toeplitz matrices T-n = (t(/r - s/))(r,s=1)(n) with t(r) = alpha rho(r) + beta/rho(r), where alpha and beta are real and 0 < rho < 1. We give formulas for det(T-n) and T-n(-1), and show that if alpha - beta = 1 and beta not equal 0 then T-n has eigenvalues lambda(1n) < lambda(2n) < ... < lambda(nn) such that lim(n-->infinity)...

The standard definition of convergence of an infinite product of scalars is extended to the infinite product P = Pi(n=1)(infinity)B(n) of k x k matrices; that is, P is convergent according to the definition given here if and only if there is an integer N such that B-n is invertible for n greater than or equal to N and P = lim(n-->infinity) Pi(m=N)(...

We give sufficient conditions for a k×k linear system of difference equations to have linear asymptotic equilibrium if , where for some and are sequences of scalars. The conditions involve convergence (perhaps conditional) of certain iterated sums involving these sequences.

Following an idea originated by Conti for continuous matrix functions, an equivalence relation called summable similarity is defined on pairs of n × n matrix functions A and B. Special cases of the results show that if A and B are summably similar and the system (A) Δxm = Amxm, m = 0, 1, 2, ... is uniformly, exponentially, or strictly stable, or ha...

H. J. Landau has recently given a nonconstructive proof of an existence theorem for the inverse eigenvalue problem for real symmetric Toeplitz (RST) matrices. This paper presents a procedure for the numerical solution of this problem. The procedure is based on an implementation of Newton's method that exploits Landau's theorem and other special spe...

. An iterative method based on displacement structure is proposed for computing eigenvalues and eigenvectors of a class of Hermitian Toeplitz--like matrices which includes matrices of the form T T where T is arbitrary Toeplitz matrix, Toeplitz--block matrices and block--Toeplitz matrices. The method obtains a specific individual eigenvalue (i.e., t...

The standard definition of convergence of an infinite product of scalars is extended to the infinite product P = ∞ m=1 Bm of k × k matrices; that is, P is convergent ac-cording to the definition given here if and only if there is an integer N such that Bm is invertible for m ≥ N and P = limn→∞ n m=N (I + Am) is invertible. Sufficient condi-tions fo...

Sufficient conditions are given for the n × n system y'=(A+P(t))y to have a solution ŷ such that as t → ∞, where λ is an eigenvalue of the constant matrix A and v is an associated eigenvector. The integrability conditions on P allow conditional convergence and the o(1) terms are specified precisely.

Sufficient conditions are given for the Poincaré recurrence system y(m+1) = (A + P(m)) y(m) to have a solution ŷ such that , where λ is an eigenvalue of the constant matrix A and v is an associated eigenvector. The summability conditions on P permit conditional convergence and the o(1) terms are specified precisely.

Necessary conditions are given for the Hermitian Toeplitz matrix T(n) = (t(r-s))r,s=1(n) to have a repeated eigenvalue lambda with multiplicity M > 1 and for an eigenpolynomial of T(n) associated with lambda to have a given number of zeros off the unit circle Absolute value of z = 1. It is assumed that t(r) 1/2pi integral-pi/-pi f(theta)e(-irtheta)...

It is shown that for every n the even and odd spectra of real symmetric Toeplitz matrices Tn=(tr−s)nr,s=1 are interlaced if where f is monotonic and nonconstant.

It is known that if P1 j|pj| < 1 then the Emden-Fowler difference equation (A) �2yn 1 = pny n ( > 0) has a positive solution {yn}, defined for n sufficiently large, such that limn!1 yn = c > 0, while if P1 j|pj| < 1 then (A) has a positive solu- tion {yn}, defined for n sufficiently large, such that limn!1�yn = c > 0. Here it is shown that these co...

It is shown that if the zeros lambda1, lambda2, . . . , lambda(n) of the polynomial q(lambda) = lambda(n) + a1lambda(n-1) + . . . + a(n) are distinct and r is an integer in {1, 2, . . . , n} such that \lambda(s)\ not-equal \lambda(r)\ if s not-equal r, then the Poincare difference equation y(n + m) + (a1 + p1(m))y(n + m - 1) + . . . + (a(n) = p(n)(...

It is pointed out that the author's O(n 2) algorithm for computing individual eigenvalues of an arbitrary n × n Hermitian Toeplitz matrix T n reduces to an O(rn) algorithm if T n is banded with bandwidth r.

It is known that if P1 j|pj| < 1, then the difference equation has solutions {y1n} and {y2n} such that limn!1 y1n = limn!1 y2n/n = 1. Here it is shown that this conclusion holds if the series P1 jpj converges (perhaps condi- tionally) and satisfies a second condition which is weaker than absolute convergence. Estimates of {yin} and {�yin} (i = 1,2)...

Conditions are given for a nonlinear perturbation of a linear differential equation to have a solution on a given semiinfinite interval which behaves asymptotically like a specified solution of the unperturbed equation. Unlike most previous results on this question, our integrability conditions allow conditional convergence of some of the improper...

It is shown that the equation DELTA-u + p(Absolute value of x)u(gamma) = 0 has positive radially symmetric solutions on a given exterior domain E(a) = {x is-an-element-of R(n)\Absolute value of x > a greater-than-or-equal-to 0} which behave asymptotically (as Absolute value of x --> infinity) like constant multiples of the radial solutions v1 = 1 a...

An iterative procedure is proposed for computing the eigenvalues and eigenvectors of a class of specially structured Hermitian Toeplitz matrices which in-cludes Hermitian Toeplitz and Toeplitz–plus–Hankel matrices. The computational cost per eigenvalue–eigenvector for a matrix of order n is 0(n 2) in serial mode. Results of numerical experiments on...

Explicit formulas are given for the weighting coefficients inthe linear minimum variance predictor of a wide sense stationary autoregressive-moving average time series k steps ahead, given n + 1 successive observations of a realization of the process. The formulas involve determinants whose entries are the values of certain polynomials related to t...

Let Tn = (ti j)ni,j=1 (n � 3) be a real symmetric Toeplitz matrix such that Tn 1 and Tn 2 have no eigenvalues in common. We consider the evolution of the spectrum of Tn as the parameter t = tn 1 varies over (1 ,1). It is shown that the eigenvalues of Tn associated with symmetric (reciprocal) eigenvectors are strictly increasing functions of t, whil...

This paper offers an alternative to the standard method of applying the Schauder-Tychonoff theorem to establish the existence of solutions to mtxed initial and final value problems for a system x'=Fx(f > to) of functional differential equations. The main result reduces the application of the Schauder-Tychonoff theorem to merely verifying that the f...

It is shown that a formula of Heinig for the inverse of a Toeplitz matrix follows from a formula of the author for the inverse of a Hankel matrix.

Recently the asymptotic properties of solutions of systems of functional differential equations have begun to be studied. (See, e.g., (1)-(10).) Here we give sufficient conditions for rather general functional differential systems to have solutions that approach given constant vectors as t ! 1. This question has been thoroughly investigated for ord...

Sufficient conditions are given for a nonlinear system of differential equations with deviating arguments to have solutions which approach finite limits as t→∞. No specific assumptions other than continuity are imposed on the deviating arguments. The nonlinearities may be superlinear, sublinear, or singular in form, or a mixture of these. Some of t...

A numerical method is proposed for finding all eigenvalues of symmetric Toeplitz matrices $T_n = ( t_{j - i} )_{ij = 1}^n $, where the $\{ t_j \}$ are the coefficients in a Laurent expansion of a rational function. Matrices of this kind occur, for example, as covariance matrices of ARMA processes. The technique rests on a representation of the char...

A method is proposed for solving linear algebraic systems with Toeplitz matrices generated by $T ( z ) = C( z )\Phi ( z )$, where $C( z )$ is a Laurent polynomial and $\Phi ( z )$ is a formal Laurent series, and a convenient method is available for solving systems with Toeplitz matrices generated by $\Phi ( z )$. Special cases of the method provide...

The problem of asymptotic behavior of solutions of an nth order linear differential equation is reconsidered, and a result obtained by Hartman and Wintner under integral smallness conditions on the perturbing terms which require absolute integrability is shown to hold under weaker integrability conditions requiring only ordinary (perhaps conditiona...

An adhoc procedure is given for obtaining first integrals of second order differential equations in which the non-linear term is a power of the dependent variable, as in the Emden-Fowler equation. The main theorem is a considerable extension of previous results along these lines. A corollary implies several known examples.

Formulas are given for the characterstic polynomials {pn (λ)} and eigenvectors of the family {Tn } of real synometric Toeptitz matrices generated by a retional function R(z) with real coefficients such that R(z)=R(1/z). The formulas are in terms of the zeros of a fixed polynomial P(w;λ) with coefficients which are simple functions of λ and the coef...

An algorithm is presented which reduces the problem of solving a Toeplitz system (1) TX=Y to simple recursive computations and solving a related Toeplitz system which is of lower order if T is nearly triangular. The method does not require that T or any of its principal submatrices be nonsingular, but only that (1) have a solution X for the given Y...

We study the eigenvalue problem for a class $\mathcal{H}$ of band matrices which includes as a proper subclass all band matrices with Toeplitz inverses. Toeplitz matrices of this kind occur, for example, as autocorrelation matrices of purely autoregressive stationary time series. A formula is given for the characteristic polynomial $p_n ( \lambda )...

Sufficient conditions are given for a generalized Emden-Fowler equation with deviating argument to have nonoscillatory solutions with prescribed asymptotic behavior as t → ∞. The integrability condition on the nonlinear term requires only conditional convergence, supplemented by a condition on the order of convergence, which is automatically satisf...

It is shown that the equation (r(t)x ' ) ' +g(t)x=0 has solutions which behave asymptotically like those of a nonoscillatory equation (r(t)y ' ) ' +f(t)y=0, provided that a certain integral involving f-g converges (perhaps conditionally) and satisfies a second condition which has to do with its order of convergence. The result improves upon a theor...

We consider Toeplitz matrices Tn = (ti−j)ni,j=0, where Σ∞−∞tjzj is a formal Laurent series of a rational function R(z). A criterion is given for Tn to be invertible, in terms of the nonvanishing of a determinant Dn involving the zeros of R(z), and of order and form independent of n; i.e., n enters into Dn as a parameter, and not so as to complicate...

A theorem of Wintner concerning sufficient conditions for a system y' = A(t)y to have linear asymptotic equilibrium is extended to a system x' = A(t)x + f(t, x). The integrability conditions imposed on f permit conditional convergence of some of the improper integrals that occur. The results improve on Wintner's even if f = 0.

Without Abstract

Summary Conti defined t8similarity of systems (1)x'=tA(t)xand (2)y'=B(t)y, and showed that it is an equivalence relation which preserves uniform and strict stability. Here the definition is weakened by imposing less stringent integrability conditions, some in terms of perhaps conditionally convergent improper integrals, on the matrix function relat...

Explicit formulas are given for the elements of $T_n^{ - 1} $ and the solution of $T_n X = Y$, where $T_n $ is an $( n + 1 ) \times ( n + 1 )$ Toeplitz band matrix with bandwidth $k\leqq n$. The formulas involve $k \times k$ determinants whose entries are powers of the zeros of a certain kth degree polynomial $P ( z )$ which is independent of n, or...

Conditions are given which imply that the functional differential equation (r(t)x ' (t)) ' +q(t)x(t)=f(t,x(g(t))) has a solution x ¯ which behaves for large t in a precisely defined way like a given solution y ¯ of the ordinary differential equation (r(t)y ' ) ' q(t)y=0· It is not assumed that g(t)-t is sign-constant, and f(t,u) need only be define...

Most known conditions implying that a nonlinear differential equation y ( n )+ f(t, y) = 0 has solutions such that lim t →∞ t − k y ( i ) = c ≠ 0 are local in that the solutions are guaranteed to exist only for sufficiently large t . This paper presents conditions ensuring that the solutions exist on a given interval and have the prescribed asympto...

A formula is given for the characteristic polynomial of an nth order Toeplitz band matrix, with bandwidth k < n, in terms of the zeros of a kth degree polynomial with coefficients independent of n. The complexity of the formula depends on the bandwidth k, and not on the order n. Also given is a formula for eigenvectors, in terms of the same zeros a...

Conditions are given for the nonlinear differential equation (1)L n y+f(t, y, ..., ...,y (n–1)=0to have solutions which exist on a given interval [t0, )and behave in some sense like specified solutions of the linear equation (2)L n z=0as t.The global nature of these results is unusual as compared to most theorems of this kind, which guarantee th...

Conditions are given which imply that a system X'(t) = F[ t, X(g(t))] has a solution with some components which approach given limits with specified orders of convergence as t → ∞, while the other components have specified orders of magnitude. The integral smallness conditions on F permit conditional convergence of some of the improper integrals th...

A sufficient condition is given for eventual disconjugacy of an nth-order linear differential equation. The condition is weaker than one given in an earlier paper with the same title, and the conclusion is sharper.

It is shown that solutions of a system $x' = f(t, x)$ approach constant vectors as $t \rightarrow \infty$, under assumptions which do not require that $\|f(t, x)\| \leqslant w(t, \|x\|)$, where $w$ is nondecreasing in $\|x\|$, and which permit some or all of the integral smallness conditions on $f$ to be stated in terms of ordinary-rather than abso...

In separate papers, D. L. Lovelady has related oscillation of solutions of certain linear differential equations of odd order $\geqslant3$ and even order $\geqslant4$ to oscillation of an associated second order equation. This paper presents a unified proof of Lovelady's results for equations of arbitrary order $\geqslant3$. The results are somewha...

Functions are exhibited which interpolate the magnitude of a solution $y$ of a linear, homogeneous, second-order differential equation at its critical points, $|y'|$ at the zeros of $y$, and $|\int^x_{x_0}y(t)h(t) dt|$ at the zeros of $y$. Except for a normalization condition, the interpolating functions are independent of the specific solution $y$...