Davis: Present Status of Integral Equations 13 
was due to G. D. Birkhoff. 89 In a paper published in 1908 Birkhoff 
discussed for a very general case the existence and distribution of 
characteristic numbers and the nature of the expansion of an arbitrary 
function in terms of the associated system of bi-orthogonal functions. 
This memoir has already become ' a classic . in connection with the 
Sturm-Liouville problem. 
The papers of Voiterra, Fredholm, and Hilbert now gave new 
stimulus to the study of infinite determinants and systems of 
equations in an infinite number of variables which had already been 
advanced a long way, notably by G. W. Hill, P; Appell, H. Poincare, 
and H. Von Koch. To this theory important additions have been 
made by D. Hilbert, E. Schmidt, H. Hellinger, 0. Toeplitz, E. Hilb, 
F. Riesz, and more recently by J. V. Walsh.* For an extended 
account of the development of this important subject and its 
numerous applications the reader is referred to the monograph of 
Riesz. f 
With the completion of the fundamental theorems in the theory 
of both the Voiterra and Fredholm equation, attention was next 
directed to a study of the characteristics of solutions as they depend 
upon the characteristics of the kernel and the development of arbi- 
trary functions in series of the orthogonal and bi-orthogonal functions 
which furnish solutions of the homogeneous Fredholm equation. 
The latter problem, frequently met with in applied mathematics, 
is that of developing an arbitrary function in terms of a system of 
functions, orthogonal in an interval (ab), i. e. functions 
ll! (x), u 2 (x), , U n (x), .... 
which satisfy the condition 
/: 
Ui (t) Uj (t) dt = 0, i % j. 
In a formal manner it may be assumed that an arbitrary function, 
f(x), can be expanded in terms of these functions 
f(x) = CiUi(x) + c 2 u 2 (x) + .....+ c n u n (x) + (6) 
and the coefficients calculated from the equation 
/: 
f(t) Ui (t) 
Various questions suggest themselves in this connection, however. 
First: can f(x) really be represented by series (6)? Second: to what 
extent does the series converge? Third: is the representation of a 
function by means of series (6), if it exists, unique? Fourth: is 
*Amer. Journal of Math., vol. 42 (1920), pp. 91-96. 1 
fLes systemes d’ equations lineaires a une infinite dunconnues. Gauthier-Villars, Paris (1913) 
182 pp. 
3-36790 
