11-j DR. J. MERCER: STURM-LIOUVILLK SERIES OF NORMAL 



contemplated by HILBBRT and SCHMIDT. KNESER'S paper* is of importance as 

 marking the first step in this direction, but his results are far less general than those 

 obtained by HOBSOK and published last year in the 'Proceedings of the London 

 Mathematical Society.'t As one of many interesting applications of a general 

 convergence theorem, the latter* has been able to show that any Sturm-Liouville 

 series corresponding to an assigned function converges at a point, provided that the 

 function has a Lebesgue integral in the interval of representation, and is of limited 

 totiil Hurt nation in an arbitrarily small neighbourhood of the point in question. 

 Taken in conjunction with other results of a similar kind, this cannot fail to suggest 

 the possibility of extending most of the well-known theorems on FOURIER'S series to 

 the whole class of Sturm-Liouville expansions. It is the purpose of this memoir to 

 show that all the more important theorems are capable of this extension. 



It will not be necessary to give here a detailed account of the results obtained, 

 seeing that those of importance have from time to time been summarised as formal 

 theorems printed in italics. It may, however,, be useful to say a few words as to the 

 plan on which the memoir has been written. The first section is devoted to the proof 

 of two theorems of convergence which find repeated applications in the sequel. In 

 4 of the second, a theorem relative to the expansion of a function as a series of 

 normal functions is established. The theorem has reference to a very wide class 

 of expansions. The only obstacle which can hinder its application in any given case 

 is the difficulty of determining an asymptotic formula for the solving function K x (s, t), 

 when X is negative and numerically large. At the commencement of the third section 

 a formula of this kind is obtained which makes it possible to apply the theorem to 

 what I have called canonical Sturm-Liouville series (III., 20). The latter portion 

 of the section is devoted to this application. The results obtained are extended so as 

 to apply to the most general form of Sturm-Liouville series. The fourth, and 

 remaining, section is given up to a discussion of questions of convergence. It is here 

 that the properties of orthogonal functions, proved in the latter half of the second 

 section, find their application. 



In conclusion, I may say that in later memoirs I hope to further develop the ideas 

 which have here been made use of. With such modifications as are necessary from 

 the fact that the characteristic function is no longer limited, I hope especially to apply 

 them to expansions in LEGENDRE'S and in BESSEL'S functions. 



I- THE THEOREMS OF CONVERGENCE. 



, In the following pages we shall find it necessary to make frequent use of two 

 theorems of convergence. These properly belong to the general theory of series (or 



* ' Math. Ann.,' vol. 63, pp. 477-524. 

 t Series 2, vol. 6, pp. 349-395. 

 I Pp. 386-387. 



