Fejér kernel

Plot of several Fejér kernels

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).

Definition

The Fejér kernel has many equivalent definitions. We outline three such definitions below:

1) The traditional definition expresses the Fejér kernel F n ( x ) {\displaystyle F_{n}(x)} in terms of the Dirichlet kernel: F n ( x ) = 1 n ∑ k = 0 n − 1 D k ( x ) {\displaystyle F_{n}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}D_{k}(x)}

where

D k ( x ) = ∑ s = − k k e i s x {\displaystyle D_{k}(x)=\sum _{s=-k}^{k}{\rm {e}}^{isx}}

is the kth order Dirichlet kernel.

2) The Fejér kernel F n ( x ) {\displaystyle F_{n}(x)} may also be written in a closed form expression as follows[1]

F n ( x ) = 1 n ( sin ⁡ ( n x 2 ) sin ⁡ ( x 2 ) ) 2 = 1 n ( 1 − cos ⁡ ( n x ) 1 − cos ⁡ ( x ) ) {\displaystyle F_{n}(x)={\frac {1}{n}}\left({\frac {\sin({\frac {nx}{2}})}{\sin({\frac {x}{2}})}}\right)^{2}={\frac {1}{n}}\left({\frac {1-\cos(nx)}{1-\cos(x)}}\right)}

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:[2]

D k ( x ) = sin ⁡ ( k + 1 2 ) x sin ⁡ x 2 {\displaystyle D_{k}(x)={\frac {\sin(k+{\frac {1}{2}})x}{\sin {\frac {x}{2}}}}}

Hence, using the definition of the Fejér kernel above we get:

F n ( x ) = 1 n ∑ k = 0 n − 1 D k ( x ) = 1 n ∑ k = 0 n − 1 sin ⁡ ( ( k + 1 2 ) x ) sin ⁡ ( x 2 ) = 1 n 1 sin ⁡ ( x 2 ) ∑ k = 0 n − 1 sin ⁡ ( ( k + 1 2 ) x ) = 1 n 1 sin 2 ⁡ ( x 2 ) ∑ k = 0 n − 1 [ sin ⁡ ( ( k + 1 2 ) x ) ⋅ sin ⁡ ( x 2 ) ] {\displaystyle F_{n}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}D_{k}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}{\frac {\sin((k+{\frac {1}{2}})x)}{\sin({\frac {x}{2}})}}={\frac {1}{n}}{\frac {1}{\sin({\frac {x}{2}})}}\sum _{k=0}^{n-1}\sin((k+{\frac {1}{2}})x)={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}[\sin((k+{\frac {1}{2}})x)\cdot \sin({\frac {x}{2}})]}

Using the trigonometric identity: sin ⁡ ( α ) ⋅ sin ⁡ ( β ) = 1 2 ( cos ⁡ ( α − β ) − cos ⁡ ( α + β ) ) {\displaystyle \sin(\alpha )\cdot \sin(\beta )={\frac {1}{2}}(\cos(\alpha -\beta )-\cos(\alpha +\beta ))}

F n ( x ) = 1 n 1 sin 2 ⁡ ( x 2 ) ∑ k = 0 n − 1 [ sin ⁡ ( ( k + 1 2 ) x ) ⋅ sin ⁡ ( x 2 ) ] = 1 n 1 2 sin 2 ⁡ ( x 2 ) ∑ k = 0 n − 1 [ cos ⁡ ( k x ) − cos ⁡ ( ( k + 1 ) x ) ] {\displaystyle F_{n}(x)={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}[\sin((k+{\frac {1}{2}})x)\cdot \sin({\frac {x}{2}})]={\frac {1}{n}}{\frac {1}{2\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}[\cos(kx)-\cos((k+1)x)]}

Hence it follows that:

F n ( x ) = 1 n 1 sin 2 ⁡ ( x 2 ) 1 − cos ⁡ ( n x ) 2 = 1 n 1 sin 2 ⁡ ( x 2 ) sin 2 ⁡ ( n x 2 ) = 1 n ( sin ⁡ ( n x 2 ) sin ⁡ ( x 2 ) ) 2 {\displaystyle F_{n}(x)={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}{\frac {1-\cos(nx)}{2}}={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}\sin ^{2}({\frac {nx}{2}})={\frac {1}{n}}({\frac {\sin({\frac {nx}{2}})}{\sin({\frac {x}{2}})}})^{2}}

3) The Fejér kernel can also be expressed as:

F n ( x ) = ∑ | k | ≤ n − 1 ( 1 − | k | n ) e i k x {\displaystyle F_{n}(x)=\sum _{|k|\leq n-1}\left(1-{\frac {|k|}{n}}\right)e^{ikx}}

Properties

The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is F n ( x ) ≥ 0 {\displaystyle F_{n}(x)\geq 0} with average value of 1 {\displaystyle 1} .

Convolution

The convolution Fn is positive: for f ≥ 0 {\displaystyle f\geq 0} of period 2 π {\displaystyle 2\pi } it satisfies

0 ≤ ( f ∗ F n ) ( x ) = 1 2 π ∫ − π π f ( y ) F n ( x − y ) d y . {\displaystyle 0\leq (f*F_{n})(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(y)F_{n}(x-y)\,dy.}

Since f ∗ D n = S n ( f ) = ∑ | j | ≤ n f ^ j e i j x {\displaystyle f*D_{n}=S_{n}(f)=\sum _{|j|\leq n}{\widehat {f}}_{j}e^{ijx}} , we have f ∗ F n = 1 n ∑ k = 0 n − 1 S k ( f ) {\displaystyle f*F_{n}={\frac {1}{n}}\sum _{k=0}^{n-1}S_{k}(f)} , which is Cesàro summation of Fourier series.

By Young's convolution inequality,

‖ F n ∗ f ‖ L p ( [ − π , π ] ) ≤ ‖ f ‖ L p ( [ − π , π ] )  for every  1 ≤ p ≤ ∞  for  f ∈ L p . {\displaystyle \|F_{n}*f\|_{L^{p}([-\pi ,\pi ])}\leq \|f\|_{L^{p}([-\pi ,\pi ])}{\text{ for every }}1\leq p\leq \infty {\text{ for }}f\in L^{p}.}

Additionally, if f ∈ L 1 ( [ − π , π ] ) {\displaystyle f\in L^{1}([-\pi ,\pi ])} , then

f ∗ F n → f {\displaystyle f*F_{n}\rightarrow f} a.e.

Since [ − π , π ] {\displaystyle [-\pi ,\pi ]} is finite, L 1 ( [ − π , π ] ) ⊃ L 2 ( [ − π , π ] ) ⊃ ⋯ ⊃ L ∞ ( [ − π , π ] ) {\displaystyle L^{1}([-\pi ,\pi ])\supset L^{2}([-\pi ,\pi ])\supset \cdots \supset L^{\infty }([-\pi ,\pi ])} , so the result holds for other L p {\displaystyle L^{p}} spaces, p ≥ 1 {\displaystyle p\geq 1} as well.

If f {\displaystyle f} 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 f , g ∈ L 1 {\displaystyle f,g\in L^{1}} with f ^ = g ^ {\displaystyle {\hat {f}}={\hat {g}}} , then f = g {\displaystyle f=g} a.e. This follows from writing f ∗ F n = ∑ | j | ≤ n ( 1 − | j | n ) f ^ j e i j t {\displaystyle f*F_{n}=\sum _{|j|\leq n}\left(1-{\frac {|j|}{n}}\right){\hat {f}}_{j}e^{ijt}} , which depends only on the Fourier coefficients.
  • A second consequence is that if lim n → ∞ S n ( f ) {\displaystyle \lim _{n\to \infty }S_{n}(f)} exists a.e., then lim n → ∞ F n ( f ) = f {\displaystyle \lim _{n\to \infty }F_{n}(f)=f} a.e., since Cesàro means F n ∗ f {\displaystyle F_{n}*f} converge to the original sequence limit if it exists.

See also

References

  1. ^ Hoffman, Kenneth (1988). Banach Spaces of Analytic Functions. Dover. p. 17. ISBN 0-486-45874-1.
  2. ^ Konigsberger, Konrad. Analysis 1 (in German) (6th ed.). Springer. p. 322.