E-Book Content
284
MA THEMA TICS: C. N. MOORE
PROC. N. A. S.
'Sylvester, Camb. and Dublin Math. J., 8, 1858, 62-64. 2 Gordan, Math. Ann., 5, 1872, 95-122. 3 Shenton, Amer. J. Math., 37, 1915, 247-271. 4Rowe, Trans. Am. Math. Soc., 12, 1911, 295-310. Maschke, Trans. Amer. Math. Soc., 4, 1903, 446, 448. 6 Glenn, Theory of Invariants, 1915, 167. Encyc. der math. Wiss., 1 (IB2), 385. 8 Meyer, Apolaritdt und Rat. Curv., 1883, 18. ON THE APPLICATION OF BOREL'S METHOD TO THE SUMMATION OF FOURIER'S SERIES' By CHARLzs N. MOORS DEPARTMSNT OF MATHZMATICS, UNIVZRSIrY OF CINCINNATI
Communicated April 18, 1925
There are two ways of studying the relationship between various methods for the stummation of divergent series. One consists in the attempt to determine directly whether or not one is more general than the other, and if this is not the case to determine under what conditions both methods apply and give the same sum to the series that is used. Another method involves the determination of the relative scope of the various processes in summing certain general tvpes of series that are of fundamental importance in analysis. The former method is more exhaustive from the theoretical point of view; the latter is, perhaps, of greater practical interest. The two most important types of series in analysis at the present time are power series and Fourier's series. It is well known that Cesiro's method will not sum a power series outside of its circle of convergence, whereas Borel's method applies everywhere within the polygon of summability. However, in the case where the circle of convergence is a natural boundary, Cesaro's method may be applicable at points on the circle of convergence where Borel's method fails. This phase of the relationship between the two methods may well be descrbed by an illuminating remark made by G. H. Hardy2 in another connection, namely, that "Borel's method, although more powerful than Cesaro's, is never more delicate, and often less so." Ceskro's method has been found to be admirably adapted to the study of Fourier's series. It will give the proper sum for the Fourier's series of any continuous function at all points, and will sum the Fourier's series of any function having a Lebesgue integral to the value of the function, except perhaps at a set of points of measure zero. Since there is considerable similarity in the behavior of Fourier's series and the behavior of power series on the circle of convergence, it is natural to expect that Borel's
VOL,. 11, 1925
MA THEMA TICS: C. N. MOORE
285
method will not be as effective as CesAro's in dealing with the former type of series. That this is the case is proved by the results of the present paper. The application of Borel's method to the summing of Fourier's series leads to the study of the behavior of the following integral3
[f(x + 2t) +f(x-2t)]e-X'2 sint Xtdt 7rJ0 as X becomes infinite. This integral is of the type termed by Lebesgue singular integrals. Its kernel o(X, t) = 2
7r
-xk2 sin-Xt t
is such that
LB(X)
= fk
(X t) I dt
(1)
does not remain bounded as X becomes infinite. It follows therefore from a general theorem due to Lebesgue4 that Borel's method will not sum the Fourier's series of every function having a Lebesgue integral or even of every continuous function. Having found that Borel's method is less effective than Ceshro's in su'mming Fourier's series, we next wish to determine if it is more effective than ordinary convergence. The kernel of the singular integral that arises in the study of convergence, the well known Dirichlet's integral, is
iV(n, t) =2
sin (2n + sin t
l)t
The values of
L(n)
=f1
(n, t) I dt
(2)
for successive values of n are termed Lebesgue co