Dirichlet kernel
In mathematical analysis, the Dirichlet kernel is the collection of functions
It is named after Peter Gustav Lejeune Dirichlet.
The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of Dn(x) with any function f of period 2π 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 diverges to infinity as n → ∞. One can estimate that
By using a Riemann-sum argument to estimate the contribute in the largest neighbourhood of zero in which is positive, and the 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.
Take the periodic Dirac delta function, which is not really a function, in the sense of mapping one set into another, but is rather a "generalized function", also called a "distribution", and multiply by 2π. We get the identity element for convolution on functions of period 2π. In other words, we have
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