TY - JOUR
T1 - On non-harmonic trigonometrical series
JF - Proceedings of the Royal Society of London. Series A
JO - Proc R Soc Lond A Math Phys Sci
SP - 237
LP - 249
M3 - 10.1098/rspa.1919.0005
VL - 95
IS - 668
AU -
Y1 - 1919/01/01
UR - http://rspa.royalsocietypublishing.org/content/95/668/237.abstract
N2 - §1. In preceding communications I have indicated the important consequences for the theory of series of functions of the consideration of what I have called Restricted Fourier series. This is due to the fact that a large proportion of the series actually employed in analysis can be replaced, to a first approximation, by trigonometrical series of the harmonic type. This is the case, for example, with series of Legendre coefficients, and with series of Bessel functions of rational order. There are, however, classes of series which can only be approximated to by non-harmonic trigonometrical series. This is, for example, the case with series of Bessel functions of irrational order. The typical term of the trigonometrical series is no longer cos n(x+α), where n is an integer, but cos kn(x+α), where, however, successive values of kn tend to differ by a constant quantity, which may, of course, without loss of generality, be taken to be unity.*
ER -