ON THE THEORY OP INTEGRAL EQUATIONS. 



385 



form an orthogonal system for the interval (— °°, + >*>). The poly- 

 nomials of Hermite and Laguerre have been studied in connection with 

 integral equations by Wera Lebedeff 1 and Weyl. 2 Hermite's poly- 

 nomials occur in solutions of the equation of the conduction of heat and 

 also in solutions of the equation of the parabolic cylinder. Various 

 definite integral representations of them have been obtained by 

 Whittaker 3 and Watson. 4 



Another class of normal functions, called by K. Pearson the w-iunc- 

 tions, can be derived from the equation of the conduction of heat ; they 

 occur in the theory of statistics. The expansions in series of w-functions 

 have been discussed by E. Cunningham. 5 



In addition to the orthogonal functions mentioned above, there are 

 the trigonometrical functions sin a h x where the quantities a k satisfy a 

 transcendental equation, the Bessel's functions J„ (« k x) under like 

 conditions, and functions such as the Legendre polynomial, spherical 

 harmonics, the functions associated with the equation of the elliptic 

 cylinder, and Lame's functions. Also there are functions such as 



\' X 



which form an Orthogonal system for the unlimited interval (—00,00). 



12. Expansions in Series of Orthogonal Functions. 



When a function f(x) can be expanded in a series of the type 



/(a) = 2aA(a>) W 



which is uniformly convergent in the interval for which the orthogonal 

 functions are defined, then the coefficients may be determined by the 

 so-called Fourier's rule 





 o« - \f(n)M x ) dai 



(2) 



and similarly for expansions in series of functions belonging to 

 biorthogonal system. 



The Fourier's constants a„ may often be used to specify a function 

 f{x) even when the series for f(x) is not convergent, and there is a 

 remarkable theorem due to de la Vallee Poussin and Liapounoff for 

 Fourier's series, which enables us to express the integral of the product 

 of two functions f\x), g(x) in terms of their Fourier's constants. The 

 formula is 



f f{x)g(x)dx - S a n b n , 



a 



1 Dissertation Gottingen, 1906 ; Math. Ann., Ed. lxiv , p. 400. 



2 Distertation Gottingen, 1908. 



• Proc. Lond. Math. Soc., vol. xxxv. (1903), p. 417. * Ibid-, 1910, vol. vm. 

 8 Proc. Roy. Soc. A., vol. Ixxxi. (1908), p. 310. 



* Various expansions of this type are considered by A. Stephenson, Mess, of 

 Math., p. 1 (1903), April (1904), and the author, Camb. Phil. Trans., vol. xx., p. 281. 



