Contributions to Annalen der Physik 1901-1922 by Einstein A.

By Einstein A.

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72 FOURIER SERIES FOURIER ANALYSIS AND GENERALISED FUNCTIONS Hence C(y) is continuous. Also, C(m/21) =Cm since, by theorem V(m-n)=o except when m=n, and V(o) = I. 21, NOTE. It is easily shown that C(y) is in fact a fairly good function, but this result is not needed below. Now, it was noted in connexion with definition 21 that a periodic functionf(x) could not have a finite number of singularities (unless the number were zero). However, theorem 29 shows that, provided only f(x) U(x/21) has a finite number of singularities, then the method of chapter 4 can be applied to determine the asymptotic behaviour of C(y) and hence of the cn's.

Hence, by theorem 24, we have equation (33) and hence equation (31) and hence the theorem. After this digression on convergence matters the main theorem follows at once. THEOREM 26. T. -) e-imrx/ldx. 21 PROOF. The second of equations (36) follows from theorems 25 and 23, and the first follows from it by theorem 15. NOTE. If in additionf(x) is absolutely integrable (as assumed in the classical theory of Fourier series) then we have simply c" = I21 II -I f(x) e-i"nx/ldx, since for such anf(x) we can write I ~ c,,=- },: f(2m+ll1 21m=_~ (2m-Ill I <0 =-},: 21m=_~ II -I fl(X) = dN+2fl(x) cixN+2 (-1 (I (N + 2)!

Hence ¢(y) G(y) is a good : / ~ co co llO (3 2 ) 'where on the right hand side equation (10) has been used. Now, WIth the same gloss regarding the term in curly brackets, If naw, for each n, x.. ,. - O(n) as n~OJ, and so co n) + U(21y + n)} G(y) dy, co :L lim um... :~:::h~~:::7:~:;'::~: bY (25)) such that Ie . - N. :N _cog(y) U(20'-n)G(y)dy= fco _cog(y)G(y)dy. or that it is < -e. :v's be N 1 , and then define the sequences N 8 , M 8 by mductlon as fonows. If N 1 , ... =--00 m-+oo '8=0 s LFA FOURIER ANALYSIS AND GENERALISED FUNCTIONS FOURIER SERIES function, by theorem I, and, by the definition of a regular sequence (definition 3), THEOREM 27.

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