The singular values and singular functions of the convolution operator Ko= fo'K(x-y)'dy, O<-x <-I.
are studied under the conditions that K(u) is mildly smooth and K(0) ~ 0. It is shown that these singular values and functions are asymptotic to those of the operator with K(u) -= I. A study of the kernel K(u) = e"" reveals that the results obtained are the best possible. Numerical and computational implications for the solution of convolution integral equations of the first kind, g = Kf, are briefly discussed.
I. INTRODUCTION
Convolution integral equations of the first kind
arise frequently in practice. Usually one assumes 0 ~ x < oc. However, in many contexts this is unrealistic. For instance, if (l.1) models an experiment done over time, x --t, one must recognize that only a finite time period elapses during the experiment. Thus x = t -< T and it is appropriate to consider both f and g as having domains (0, T). We shall study (l.1) under such a restriction. Attempts to solve (1.1) numerically usually give rise to triangular matrix operators, with K(0) occurring along the diagonal. If K(0) --0 a more sophisticated approach is called for since the matrix is then singular. This is not just a numerical difficulty. It is shown in [1] that the behavior of the singular values of the integral operator K associated with (l.1) is very dependent upon the properties of K(u) near u = 0. In this paper the assumption K(0) ~ 0 is made everywhere.
Numerical studies of a large number of problems revealed that the nth singular value of K seemed to behave like l/n. The singular values associated with Ko(u) = l, T = 1, are precisely [(n + ยฝ)'rr]-'. Calculations also showed that the singular functions for quite general K's seemed to approach those of K0 as n increased, though this approach was less dramatic than with the singular values.
An analytic study of the kernels K(u) = e '~ (reproduced in Sec. 8) confirmed these observations. These computational and analytical investigations led to obvious conjectures, which will be verified in this paper.
It must be noted that similar observations have been made in the past. In [2] attention is turned at once to matrix operators associated with K. A plausibility argument is made there concerning the properties of the singular values and singular vectors of these Toeplitz matrices. These properties are the discrete analogues of those we shall discuss for the integral operator.
In Sec. 2 we pose the problem more precisely and show that the original integral operator, K, can easily be replaced by a more tractable self-adjoint operator, Ks. (This is the continuous analogue of a matrix operator introduced in [2].) In Sec. 3 the problem of finding the ranges of K and Ks, under various smoothness restrictions on their associated kernels, is investigated. This information is very important in Secs. 4 and 5, which set the stage for the study of the eigenfunctions and eigenvalues of Ks. Asymptotic properties of these eigenfunctions and eigenvalues are focused on in Secs. 6 and 7, respectively. The results provide quite complete information on the asymptotic behavior of the singular functions and singular values of the