Concept

Fejér kernel

In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959). The Fejér kernel has many equivalent definitions. We outline three such definitions below:

The traditional definition expresses the Fejér kernel in terms of the Dirichlet kernel: where is the kth order Dirichlet kernel.
The Fejér kernel may also be written in a closed form expression as follows This closed form expression may be derived from the definitions used above. The proof of this result goes as follows. First, we use the fact that the Dirichlet kernel may be written as: Hence, using the definition of the Fejér kernel above we get: Using the trigonometric identity: Hence it follows that:
The Fejér kernel can also be expressed as: The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is with average value of . The convolution Fn is positive: for of period it satisfies Since , we have , which is Cesàro summation of Fourier series. By Young's convolution inequality, Additionally, if , then a.e. Since is finite, , so the result holds for other spaces, as well. If is continuous, then the convergence is uniform, yielding a proof of the Weierstrass theorem. One consequence of the pointwise a.e. convergence is the uniqueness of Fourier coefficients: If with , then a.e. This follows from writing , which depends only on the Fourier coefficients. A second consequence is that if exists a.e., then a.e., since Cesàro means converge to the original sequence limit if it exists.

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Related concepts (1)

Fejér's theorem

In mathematics, Fejér's theorem, named after Hungarian mathematician Lipót Fejér, states the following: Explicitly, we can write the Fourier series of f as where the nth partial sum of the Fourier series of f may be written as where the Fourier coefficients are Then, we can define with Fn being the nth order Fejér kernel. Then, Fejér's theorem asserts that with uniform convergence.

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The traditional definition expresses the Fejér kernel in terms of the Dirichlet kernel: where is the kth order Dirichlet kernel.
The Fejér kernel may also be written in a closed form expression as follows This closed form expression may be derived from the definitions used above. The proof of this result goes as follows. First, we use the fact that the Dirichlet kernel may be written as: Hence, using the definition of the Fejér kernel above we get: Using the trigonometric identity: Hence it follows that:
The Fejér kernel can also be expressed as: The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is with average value of . The convolution Fn is positive: for of period it satisfies Since , we have , which is Cesàro summation of Fourier series. By Young's convolution inequality, Additionally, if , then a.e. Since is finite, , so the result holds for other spaces, as well. If is continuous, then the convergence is uniform, yielding a proof of the Weierstrass theorem. One consequence of the pointwise a.e. convergence is the uniqueness of Fourier coefficients: If with , then a.e. This follows from writing , which depends only on the Fourier coefficients. A second consequence is that if exists a.e., then a.e., since Cesàro means converge to the original sequence limit if it exists.