14 
Indiana University Studies 
the function f(x) which corresponds to an arbitrary set of coefficients, 
Ci, c 2 , . . . . , c n , . . . , unique? 
* Altho these questions have been the subject of numerous in- 
vestigations since the work of Fourier, Sturm and Liouville first 
brought the problem to the attention of mathematicians early in the 
nineteenth century, E. Schmidt 428 in his dissertation in 1905 opened 
up a new and fruitful field of investigation in this direction. In his 
theory of orthogonal and bi-orthogonal systems of functions connected 
with integral equations and his development of the properties of an 
equation with a symmetric kernel, he has set up a model which has 
guided investigators to some of the most elegant theorems in the 
subject. The work of Mrs. Pell 369 on bi-orthogonal functions, and 
that of M. Plancherel, 424 A. Kneser, 413 W. Myller-Lebedeff, 421 and 
H. Weyl 430 on various representations of arbitrary functions should be 
especially mentioned. 
Closely associated with this line of research is the Fischer- 
Riesz theorem, independently discovered by E. . Fischer 364 and F. 
Riesz 373 in 1907, which is fundamental in the theory of the closure 
properties of the kernel of the Fredholm equation of first kind. In 
this connection, also, we should mention the existence theorem for the 
homogeneous equation of first kind which has been, in particular, the 
subject of investigations by G. Lauricella 219 and C. Severini 239 . 
After the problem of the inversion of indefinite integrals had 
been brought to so successful a solution by Volterra, and with the 
work of Schmidt to serve as a guide, it was natural to expect develop- 
ments next in the theory of the Fredholm equation of first kind. 
Altho there had been previous contributions to the subject in the 
form of isolated examples, the real foundation was laid by E. Picard 
in a series of articles the first of which appeared in 1909. 228, 60 
The names of L. Amoroso, 201 G. Lauricella, 219 and A. Verger io 246 are 
among the important contributors to the recent theory of this 
equation. 
From about 1907 the number of memoirs on integral equations 
rapidly increased. Different . types of kernel were studied, and 
specific solutions for a number of equations important in practice 
were recorded. In this connection the papers of H. Bateman and 
G. H. Hardy made noteworthy contributions. J. Mercer 196 did 
fundamental work with functions of positive and negative type and 
showed how the classical theorems on definite quadratic forms have 
important generalizations in the theory of integral equations. 
E. Goursat 139 and H. Lebesgue 151 developed the Fredholm theory 
for bilinear kernels. The former gave a preliminary discussion of the 
subject and the latter extended it to the limiting case where the 
