Abstract
In this article, the properties of Fourier Series by discussing around the basic properties of integrable functions and kernel are discussed. With the discussion over the function f around discontinuities, it is found that f should at least be Reimann integrable functions to make Fourier series imitate the function successfully. More, about the filters, it is clear that the sharp filters will converge to f under a certain condition, with the analyzation over Dirichlet Kernel, and how the convergent becomes successful with the properties over good kernel.
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