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Dirichlet kernel
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(Definition)
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The Dirichlet kernel $D_n$ of order $n$ is defined as $$D_n(t)=\sum_{k=-n}^ne^{ikt}.$$ It can be represented as $$D_n(t)=\frac{\sin\left(n+\frac{1}{2}\right)t}{\sin\frac{t}{2}}.$$ Proof: It is
The Dirichlet kernel arises in the analysis of periodic functions because for any function $f$ of period $2\pi$ , the convolution of $D_N$ and $f$ results in the Fourier-series approximation of order $n$ : $$(D_N*f)(x)=\frac{1}{2\pi}\int_{-\pi}^\pi
f(y)D_n(x-y)dy=\sum_{k=-n}^n\hat{f}(k)e^{ikx}.$$
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"Dirichlet kernel" is owned by mathwizard.
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Cross-references: approximation, convolution, period, function, periodic functions, analysis, proof, order
There is 1 reference to this entry.
This is version 7 of Dirichlet kernel, born on 2004-02-26, modified 2007-07-07.
Object id is 5629, canonical name is DirchletKernel.
Accessed 7590 times total.
Classification:
| AMS MSC: | 26A30 (Real functions :: Functions of one variable :: Singular functions, Cantor functions, functions with other special properties) |
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Pending Errata and Addenda
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