Let d denote the dimension of the vector space consisting of all solutions of the equation − (p(t)y′)′ + q(t)y = 0, a ≤ t < ∞; that lie in the function space L2[a, ∞). By means of certain bounds on the solutions of this equation, sufficiency criteria are obtained for the cases d = 0 and d = 2.

This paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expression

on [0,∞), together with a real homogeneous boundary condition at t = 0.

In this paper a spectral theory for completely coupled linear operator systems is developed. These systems take the form

where Ak, Bk are n × n matrices with operator entries. Λ is an n × n matrix with complex scalar entries and xk is an n × 1 column vector. The main result is a Parseval equality and expansion theorem.

This paper is concerned with integral inequalities of the form

where p, q are real-valued coefficients, with p and w non-negative, on the compact interval [a, b] and D is a linear manifold of functions so chosen that all three integrals are absolutely convergent.

It has been known for some time that certain least-squares problems are “ill-conditioned”, and that it is therefore difficult to compute an accurate solution. The degree of ill-conditioning depends on the basis chosen for the subspace in which it is desired to find an approximation. This paper characterizes the degree of ill-conditioning, for a general inner-product space, in terms of the basis.

The results are applied to least-squares polynomial approximation. It is shown, for example, that the powers {1, z, z2,…} are a universally bad choice of basis. In this case, the condition numbers of the associated matrices of the normal equations grow at least as fast as 4n, where n is the degree of the approximating polynomial.

Analogous results are given for the problem of finite interpolation, which is closely related to the least-squares problem.

Applications of the results are given to two algorithms—the Method of Moments for solving linear equations and Krylov's Method for computing the characteristic polynomial of a matrix.

This paper obtains, under certain general conditions on the coefficient q, a best-possible upper bound on the real parameter λ for the differential equation

to have a non-trivial solution in the integrable-square space L2 (a, ∞).

We consider the Friedrichs extension A of a minimal Sturm-Liouville operator L0 and show that A admits a Schrödinger factorization, i.e. that one can find first order differential operators Bk with where the μk are suitable numbers which optimally chosen are just the lower eigenvalues of A (if any exist). With the help of this theorem we derive for the special case L0u = −u″ + q(x)u with q(x) → 0 (|x| → ∞) the inequality

σd(A) being the discrete spectrum of A. This inequality is seen to be sharp to some extent.

The Epstein operator is defined by

where x = (x1, …, xn) ∈ Rn, y ∈ R,

and H, K, L, M are real constants such that c2(y) > 0. The operator arises in the study of acoustic wave propagation in plane-stratified fluids with sound speed c(y) at depth y. In this paper it is shown that A defines a selfadjoint operator in the Hilbert space ℋ = L2(Rn + 1c−2(y) dx dy) where dx = dx1 … dxn. The spectral family of A is constructed, the spectrum is shown to be continuous and an eigenfunction expansion for A is given in terms of families of improper eigenfunctions.

We study the initial value problem for the nonlinear Volterra integrodifferential equation

where μ > 0 is a small parameter, a is a given real kernel, and F, g are given real functions; (+) models the elongation ratio of a homogeneous filament of a certain polyethylene which is stretched on the time interval (— ∞ 0], then released and allowed to undergo elastic recovery for t > 0. Under assumptions which include physically interesting cases of the given functions a, F, g, we discuss qualitative properties of the solution of (+) and of the corresponding reduced problem when μ = 0, and the relation between them as μ → 0+, both for t near zero (where a boundary layer occurs) and for large t. In particular, we show that in general the filament does not recover its original length, and that the Newtonian term —μy′ in (+) has little effect on the ultimate recovery but significant effect during the early part of the recovery.

This paper considers semilinear elliptic boundary value problems of the form

where the partial derivative ∂f/∂u is bounded above by the least eigenvalue of the linear elliptic operator L. Existence and uniqueness of solutions is proved by using monotone operator theory and sub and supersolution techniques.

Conditions are obtained which ensure that the maximal operator generated by the formally self-adjoint second-order differential expression

in L2(Rn), n ≥ 1 has the Dirichlet and separation properties.

In this paper the boundary behaviour of the solution of Dirichlet's problem for the equation Δu = f in domains with C2,α-boundary is studied. Without using Schauder's a priori estimates, we prove the following theorem, which is indispensable in Schauder's method to solve Dirichlet's problem for a general linear elliptic equation.

Theorem: Let (f, g)∈ C0,α (Ḡ)×C2,α (∂G) and u ∈ C0.0(Ḡ) ∩C2,α(G) be a solution of the problem Δu = f, u|∂G = g. Then u ∈ C2,α(Ḡ).