EXPANSION PROBLEMS WITH IRREGULAR BOUNDARY 



CONDITIONS. 



By Dunham Jackson. 

 Keceived, May 12, 1915. 



The expansion problems which form the subject of the following 

 discussion are associated with ordinary linear homogeneous differen- 

 tial equations having no singular points in the interval of variation of 

 the independent variable. In nearly all the problems of this sort 

 that have been studied at length, the boundary conditions are of the 

 type which Birkhoff ^ designates as regular. A notable exception 

 occurs in the papers in which Liouville ^ shows that a considerable 

 part of the classical Sturm-Liouville theory of equations of the second 

 order, including the formal expansion of a given function in series of 

 characteristic functions, is capable of extension to certain special 

 systems of higher order. It is readily seen that ^ the boundary condi- 

 tions there considered are irregular.* Consequently the question of 

 the convergence of Liouville's special series, which Liouville himself 

 does not discuss at all, is still left open by Birkhoff's general theory. 



The present paper is devoted to a study of this question, to which 

 the author's attention was called by Professor Bocher. No attempt 

 is made to attain the greatest possible generality in the coefficients 

 of the differential equation; in the matter of boundary conditions, on 

 the other hand, the hypotheses are sufficiently general to include all 



1 Reference will be made to three papers of Birkhoff by means of Roman 

 numerals as follows: 



I. Transactions of the American Mathematical Society, 9, 219-231 

 (1908). 



II. Transactions of the American Mathematical Society, 9, 373-395 

 (1908). 



III. Rendiconti del Circolo Matematico di Palermo, 36, 115-126 (1913). 

 The definition of regular boundary conditions is introduced in II, 382-383. 

 In connection with these papers of Birkhoff, see also the following articles 



by Tamarkine: 



I. Rendiconti del Circolo Matematico di Palermo, 34, 345-382 (1912). 



II. Rendiconti del Circolo Matematico di Palermo, 37, 376-378 (1914). 



2 I. Journal de I'Ecole Royale Polytechnique, 15, cahier 25, 85-117 (1837). 

 II. Journal de Mathematiques pures et appliquees, 3, 561-614 (1838). 



3 When the differential equation is of order higher than the second. 



4 The boundary conditions are given by Liouville in non-homogeneous form, 

 but it is immediately apparent that they are equivalent to a set of homogene- 

 ous conditions. 



