it is hard to give a satisfying high level explanation but i would love to see someone try. a Dirichlet kernels proof I just looked up that works for integrabl

it is hard to give a satisfying high level explanation but i would love to see someone try. a Dirichlet kernels proof I just looked up that works for integrable periodic functions is accessible to a really smart high schooler, but still involves a lot of minutae. http://math.uchicago.edu/~may/REU2012/REUPapers/Cuddy.pdf http://math.uchicago.edu/~may/REU2012/REUPapers/Cuddy.pdf . You can define a function called a "n-th order Dirichlet kernel" which when you convolve it with f yields the n-th order Fourier series approximation for f. I guess the missing intuition is that Dirichlet kernels behave more and more like a delta as order goes up, which you can prove using algebra and calculus. Intuitively you know what convolution with a delta looks like, so it is easy to believe that convolution with something near a delta is numerically similar. There is a nice visual on the wiki article for a sequence of Dirichlet kernels looking more and more like a delta.