38G REPORTS ON THE STATE OF SCIENCE, 



and in particular 



\[/(x)]hlx = 2a,?. 



Kiesz l has shown that this theorem holds if f(x) and [/(a?)] 2 are 

 summable or integrable in the interval (a,b) and the system of ortho- 

 gonal functions \p„(x) is complete. The converse theorem is also true. 

 If -a' converges, then a function f(x) which is summable and whose 



square is also summable, exists such that the integral equation (2) is 

 satisfied. This theorem has been proved by Eiesz, 1 Fischer, 2 Hellinger, 3 

 and Weyl, 4 and forms the basis of the method by which Pi card 5 has 

 obtained necessary and sufficient conditions that the integral equation 







f(x) == L(x,t)<l>(t)(U 



a 



mny possess a solution f(t) which is summable and whose square is also 

 summable in the interval (a,b). 



In the treatment of series of orthogonal functions ^„(t) the following 

 identity which Schmidt attributes to Bassel 6 is of fundamental import- 

 ance : — 



b 



a it 



b b 



a a 



This gives rise to the inequality 



b b 



a a ■ 



which reduces to Schwarz's inequality when n = 1 . 



13. Expansions in Scries of Orthogonal Functions connected ivith a 

 Linear Differential or Integral Equation. 



The theory of these expansions now forms quite a large branch of 

 mathematics, as will be seen from Burkhardt's excellent report on expan- 

 sion in series of oscillating functions. Since the appearance of this work 



1 Comptes rendu s, 1907 ; Gbttinger Naehrichten, 1907. 



2 Comptes rendus, 1907. a Dissertation Oottingen, 1907. 



4 Math. Ann., Bd. lxxix., 1909. 



5 Comptes rendus, 1909, t. 148 ; Rend. Palermo, 1909. 



6 Astr. Naeh., 1828, vol. vi., p. 333. Bdclier points out that Bessel considers 

 sums instead of integrals, and refers to Plarr, Comptes rendus, 1857, vol. xliv., 

 p. 985. 



