Fourier Series

1.61.20.80.4000.20.40.60.811.21.4xFigure 1.8.1. Gibb’s phenomenon.1.8.3 Dirichlet KernelThere is also another informative process associated with the evaluation of Fourier seriesvia partial sums and which does not depend upon the form of fx. This employs theDirichlet Kernel and is approached from the definition of the partial sum, followed by somere-writing and re-arrangement,S N x 1 a 2 0 N∑n1 12− 12−ft dt fta n cosnx b n sinnx 1 N∑n1cosnx −ftcosnt dt sinnx −N1 2∑cosnxcosnt sinnxsinntn1Ndtftsinnt dt 12 ft 1 −2∑ cosnx − t dtn1 S N x 12 ft D N x − tdt (1.8.2)−where the integral representations for the coefficients and addition formulae have been usedas necessary. The function D N t is called the Dirichlet Kernel and is written43