Fourier Series

1.3 Dirichlet ConditionsClearly the FS representation is useful only when the series converges and the value towhich the series converges is known. It is desirable to know for what class or classes offunction this is so. We will only prove convergence for the most simple case and statefurther results as necessary.1.3.1 Simplified Fourier theoremIf fx is a periodic function with period 2 and if fx is continuous, with continuous firstand second derivatives, then the Fourier seriesSx 1 2 a 0 ∑n1a n cosnx b n sinnxis convergent at each point of the interval −,.ProofConsider the coefficient a n and look at this in terms of the given conditions and integrateby partsa n 1 fxcosnx dx 1 fx− n 1 f ′ xn 2cosnx−− 1n 2sinnx−− 1 −f ′ xn ′′ f xcosnx dx − 1−n 2sinnx dx ′′ f xcosnx dx .−In this evaluation, the endpoint evaluations are zero due to the strict requirements ofperiodicity and continuity.As fx,f ′ x and f ′′ x are continuous, there exists M 0 such that |f ′′ x| M for allx ∈ −,. We know that |cosnx| ≤ 1andso|a n | 1 f ′′ xcosnx dx 1n 2 −n 2Similarly |b n | 2M .n 2 Mdx 2M .− n 2Now collect up all terms in the series and consider the modulus at an arbitrary value of x17