TY - JOUR
T1 - On the Fourier constants of a function
JF - Proceedings of the Royal Society of London. Series A
JO - Proc R Soc Lond A Math Phys Sci
SP - 14
LP - 24
M3 - 10.1098/rspa.1911.0016
VL - 85
IS - 575
AU -
Y1 - 1911/03/14
UR - http://rspa.royalsocietypublishing.org/content/85/575/14.abstract
N2 - 1. Considerable progress has been made lately in the study of the properties of the constants in a Fourier series, using this term in the most general sense possible consistent with the extended definition of integration due to Lebesgue. Thus we now know that these coefficients necessarily under all circumstances have the unique limit zero, as the integer denoting their place in the series increases indefinitely, and that the same is true if we substitute for that integer any other quantity which increases without limit. Further, we know that the series whose general term is bn/n, where bn is the typical coefficient of the sine terms, always converges, and we are able to write down its sum. That the series whose general term is an, where an is the typical coefficient of the cosine terms, converges when the origin is an internal point of an interval throughout which the function has bounded variation, and that accordingly the series whose general term is an/nq, (0<q), converges, is an immediate consequence of known results. Should the function have its square summable, we know that the series whose general term is (an2 + bn2) converges, and we can write down its sum. We can also sum the series of the products of the Fourier coefficients of two such functions. From the property that Ʃ (an2 + bn2) converges, we can deduce that the series Ʃ an/nq and Ʃ bn/nq, (½<q) necessarily converge absolutely. Again, making use of a theorem recently proved, we may integrate the Fourier series of any summable function, after multiplying it term by term by any function of bounded variation, with a certainty that we shall obtain the same result as if the Fourier series converged to the function to which it corresponds, and such term-by-term integration were allowable.
ER -