Dirichlet kernel

In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as where n is any nonnegative integer. The kernel functions are periodic with period . The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of Dn(x) with any function f of period 2pi is the nth-degree Fourier series approximation to f, i.e., we have where is the kth Fourier coefficient of f. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel. Of particular importance is the fact that the L1 norm of Dn on diverges to infinity as n → ∞. One can estimate that By using a Riemann-sum argument to estimate the contribution in the largest neighbourhood of zero in which is positive, and Jensen's inequality for the remaining part, it is also possible to show that: This lack of uniform integrability is behind many divergence phenomena for the Fourier series. For example, together with the uniform boundedness principle, it can be used to show that the Fourier series of a continuous function may fail to converge pointwise, in rather dramatic fashion. See convergence of Fourier series for further details. A precise proof of the first result that is given by where we have used the Taylor series identity that and where are the first-order harmonic numbers. The Dirichlet kernel is a periodic function which becomes the Dirac comb, i.e. the periodic delta function, in the limit with the angular frequency . This can be inferred from the autoconjugation property of the Dirichlet kernel under forward and inverse Fourier transform: and goes to the Dirac comb of period as , which remains invariant under Fourier transform: . Thus must also have converged to as . In a different vein, consider ∆(x) as the identity element for convolution on functions of period 2pi. In other words, we have for every function f of period 2pi.