MATHEMATICS 5 



able," or, as Hobson calls it, integrable (D), although less in 

 absolute value than a function which is integrable (B), may not 

 itself be integrable (B). M. Boks concludes with a bibliography 

 of papers concerned with the modern notion of the integral. 



Yet another extension of integration is given by H, Hake 

 {Math. Ami., 83 (1921), 119-42). He introduces "upper" 

 and " lower " functions for a function f{x), and by means of 

 them defines integration S. A special case of this is shown 

 to be equivalent to integration (L), and the general case 

 applies to functions which are integrable (Di), (H) and (D). 



G. Mittag-Leffler {C.R., 173, 192 1, 1041-5) extends Cauchy's 

 Theorem to uniform functions, defined for sets of points every- 

 where dense, and satisfying conditions less restrictive than 

 monogeneity. 



The integral of \f{z)\'* taken round a circle is at least twice 

 as great as the same integral taken along any diameter if 

 f{z) is analytic and regular over the closed circle. F, Fejer and 

 F. Riesz {Math. Zs., 11, 1921, 305-14) show that there is no 

 better inequality. 



The monogenic function ^ ^ may have some of its 



2 — a^ 



apparent poles a^ as regular points, but they will all be true 

 poles if the series ^l-^^.l converges rapidly enough (Borel). 

 J. Wolff {C.R., 173, 192 1, 1056-8) proves by an example that 

 simple convergence is not sufficient, and E. Borel {ibid.) points 

 out the importance of this result. 



The problem of the approximate representation of an 

 arbitrary function by means of potynomials or by means of 

 finite trigonometric sums has been treated to a considerable 

 extent during the last twenty years. An interesting report 

 thereon has been presented to the American Mathematical 

 Society by Dunham Jackson {Bull. Amer. Math. Soc, 27, 1921, 

 415-31). The theory goes back to Tchebj'chef, who between 

 1850 and i860 discussed the problem of determining a poly- 

 nomial Pn{x), of given degree w, to approximate to a given 

 continuous function f{x), in such a way that the maximum of 

 the absolute value of the error, 



Max. \f{x)-P,{x)\, 



shall be as small as possible. An account of the early work is 

 to be found in de la Vallee Poussin : Legons sur I' approximation 

 des fonctions d'mte variable reelle (19 19). Later investigations 

 have been concerned with the degree of approximation obtained, 

 and the most important papers are by de la Vallee Poussin and 

 S. Bernstein. 



