J. P. McClure

Infinite systems of differential equations. II

i = 1, 2, . . . . Here 5 is a nonnegative real number, Rs = {/ Ç R: t ^ 5}, and x{i) = {x\(t), x2(t), . . .} denotes a sequence-valued function. Conditions on the coefficient matrix A (t) = [a i ;(/)] and the nonlinear perturbation / ( / , x) = {/i(̂ > 2?)} were established which guarantee that for each initial value c = {ct} Ç Z , the system (1.1) has a strongly continuous /^valued solution x(t) (i.e., each xt(t) is continuous and ||x(0ll = 2^T=i| Xi(t)\ converges uniformly on compact subsets of Rs). A theorem was also given which yields the exponential stability for the nonlinear system (1.1). For more information we refer the reader to [6]. Our purpose here is to give conditions which ensure the existence of a strongly continuous solution x(t) to (1.1) which also converges to a limit as t —> co. It is easy to see that the continuous functions on Rs which have this property are precisely the restrictions of the continuous functions on the one point compactification [s, 00] of Rs. We say such functions are convergent at 00 , and call a solution to (1.1) with this property a convergent solution. Our interest in this problem was motivated by a theorem for an abstract linear nonhomogeneous equation given in [5, p. 153]. However, we found that the hypotheses of the abstract result were difficult to formulate in terms of the matrix entries. Moreover, examination of similar results for finite systems (see [1] and [3]) suggested that some of the hypotheses required for the abstract equation might be unnecessary in the case of the infinite system. For example, our diagonal dominance conditions (3.2) and (3.3) imply that A (t) has a bounded inverse for all /, but do not imply that A(t)A~(s) is even bounded when t 9^ s (cf. assumptions (B3) on p. 108 and (i) on p. 153 of [5]). We note that with some modifications, the I results in [6] can be extended to the cases of l (1 ^ p < 00 ) and c0. All results in the present paper are established for these sequence spaces. For other l results on infinite systems of linear differential equations, see [2] and [7].

Two-dimensional stationary phase approximation: stationary point at a corner

Asymptotic expansions are derived for the double integral \[ \iint_D {g(x,y)\exp (iNf(x,y))dx\,dy}, \] as

On a method of asymptotic evaluation of multiple integrals

N \to + \infty

Justification of the stationary phase approximation in time–domain asymptotics

, where

Asymptotic expansion of a integral involving a Bessel function

f(x,y)

Error bounds for multidimensional Laplace approximation

has a stationary point at a corner of the boundary of D. Two different methods are given, one for the case of local extrema and one for saddlepoints. In the case of saddlepoints, our method allows the boundary of D to be tangent to a level curve of f.

Multidimensional stationary phase approximation: boundary stationary point

In this paper, some of the formal arguments given by Jones and Kline [J. Math. Phys., v. 37, 1958, pp. 1-28] are made rigorous. In particular, the reduction procedure of a multiple oscillatory integral to a one-dimensional Fourier transform is justified, and a Taylor-type theorem with remainder is proved for the Dirac 8-function. The analyticity condition of Jones and Kline is now replaced by infinite differentiability. Connections with the asymptotic expansions of Jeanquartier and Malgrange are also discussed.

Asymptotic expansion of a multiple integral

A rigorous proof is supplied for the validity of an asymptotic approximation to the integral I(λ)=∫ab g(&kgr;)p {λ f(x)},dx, where f(x) and g(x) are sufficiently smooth functions on [a,b] and p(x) is a piece–wise smooth periodic function with mean zero. In addition, a two–dimensional generalization is given. Problems concerning coalescence of two stationary points and a stationary point near an end point are also considered.

Explicit Error Terms for Asymptotic Expansions of Stieltjes Transforms

Abstract An asymptotic expansion is derived for a quadruple integral involving the Bessel function J0[λ(xy + zw)], where each of the integration variables x, y, z and w belongs to the interval [0, 1] and the large variable λ tends to infinity. Corresponding results are also obtained for similar integrals in two and three dimensions.

Generalized Mellin convolutions and their asymptotic expansions

Abstract A numerical estimate is obtained for the error associated with the Laplace approximation of the double integral I ( λ ) = ∝∝ D g ( x , y ) e − λf ( x , y ) dx dy , where D is a domain in R 2 , λ is a large positive parameter, f ( x , y ) and g ( x , y ) are real-valued and sufficiently smooth, and ∝( x , y ) has an absolute minimum in D . The use of the estimate is illustrated by applying it to two realistic examples. The method used here applies also to higher dimensional integrals.