minimality property of Fourier coefficients

Let $f$ be a Riemann integrable periodic real function with period $2\pi$ and $n$ a positive integer.  Among all “trigonometric polynomials”

$$\phi (x):=\frac{{\alpha }_0}{2}+\sum _{j=1}^n({\alpha }_j\cos jx+{\beta }_j\sin jx),$$

the polynomial with the coefficients ${\alpha }_j$ and ${\beta }_j$ being the Fourier coefficients

$${\alpha }_j=a_j:=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\cos jxdx$$

and

$${\beta }_j=b_j:=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\sin jxdx$$

for the Fourier series of $f$ produces the minimal value of the mean square deviation

$$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{[f(x)-\phi (x)]}^2𝑑x.$$

Proof.  For any fixed number $n$, it’s a question of giving the least value to the definite integral

$m:={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}{\left[f(x)-{\displaystyle \frac{{\alpha }_0}{2}}-{\displaystyle \sum _{j=1}^n}({\alpha }_j\cos jx+{\beta }_j\sin jx)\right]}^2𝑑x(\geqq 0).$ (1)

Expanding $m$ and integrating termwise yields

$m=$ ${\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}{(f(x))}^2𝑑x-{\displaystyle \frac{{\alpha }_0}{2\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)𝑑x$

$-{\displaystyle \frac{1}{\pi }}{\displaystyle \sum _{j=1}^n}{\alpha }_j{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\cos jxdx-{\displaystyle \frac{1}{\pi }}{\displaystyle \sum _{j=1}^n}{\beta }_j{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\sin jxdx+{\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{{\alpha }_0^2}{4}}{\displaystyle {\int }_{-\pi }^{\pi }}𝑑x$

$+{\displaystyle \frac{1}{2\pi }}{\displaystyle \sum _{j=1}^n}{\alpha }_j^2{\displaystyle {\int }_{-\pi }^{\pi }}{\cos }^{2}jxdx+{\displaystyle \frac{1}{2\pi }}{\displaystyle \sum _{j=1}^n}{\beta }_j^2{\displaystyle {\int }_{-\pi }^{\pi }}{\sin }^{2}jxdx$

$+{\displaystyle \frac{{\alpha }_0}{2\pi }}{\displaystyle \sum _{j=1}^n}{\alpha }_j{\displaystyle {\int }_{-\pi }^{\pi }}\cos jxdx+{\displaystyle \frac{{\alpha }_0}{2\pi }}{\displaystyle \sum _{j=1}^n}{\beta }_j{\displaystyle {\int }_{-\pi }^{\pi }}\sin jxdx+{\displaystyle \frac{1}{\pi }}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k=1}^n}{\alpha }_j{\beta }_k{\displaystyle {\int }_{-\pi }^{\pi }}\cos jx\sin kxdx$

$+{\displaystyle \frac{1}{\pi }}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k\ne j}}{\alpha }_j{\alpha }_k{\displaystyle {\int }_{-\pi }^{\pi }}\cos jx\cos kxdx+{\displaystyle \frac{1}{\pi }}{\displaystyle \sum _{j=1}^n}{\displaystyle \sum _{k\ne j}}{\beta }_j{\beta }_k{\displaystyle {\int }_{-\pi }^{\pi }}\sin jx\sin kxdx.$

Here, we have the Fourier coefficients

$$\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)𝑑x=a_0,\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\cos jxdx=a_j,\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\sin jxdx=b_j.$$

Furthermore,

$${\int }_{-\pi }^{\pi }{\cos }^{2}jxdx={\int }_{-\pi }^{\pi }{\sin }^{2}jxdx=\pi ,{\int }_{-\pi }^{\pi }\cos jx\sin kxdx=\mathrm{\hspace{0.33em}0}$$

and

$${\int }_{-\pi }^{\pi }\cos jx\cos kxdx={\int }_{-\pi }^{\pi }\sin jx\sin kxdx=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}\text{for }k\ne j.$$

Using all these we can write

$$m=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{(f(x))}^2𝑑x-\frac{{\alpha }_0{a}_0}{2}-\sum _{i=1}^n({\alpha }_i{a}_i+{\beta }_i{b}_i)+\frac{a_0^2}{4}+\frac{1}{2}\sum _{i=1}^n({\alpha }_i^2+{\beta }_i^2).$$

Adding and subtracting still the sum  $\frac{a_0^2}{4}+\frac{1}{2}{\sum }_{i=1}^n(a_i^2+b_i^2)$  yields finally the form

$$m=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{(f(x))}^2𝑑x-\frac{a_0^2}{4}-\frac{1}{2}\sum _{i=1}^n(a_i^2+b_i^2)+\frac{1}{4}{({\alpha }_0-a_0)}^2+\frac{1}{2}\sum _{i=1}^n[{({\alpha }_i-a_i)}^2+{({\beta }_i-b_i)}^2].$$

The three first addends of this sum do not depend on the choice of the quantities ${\alpha }_i$ and ${\beta }_i$.  The other addends are non-negative, and their sum is minimal, equal 0, when

$${\alpha }_i=a_i,{\beta }_i=b_i\mathit{\hspace{1em}}\forall i.$$

Accordingly, the mean square deviation $m$, i.e. (1), is minimal when one uses the Fourier coefficients. Q.E.D.

Title	minimality property of Fourier coefficients
Canonical name	MinimalityPropertyOfFourierCoefficients
Date of creation	2013-03-22 18:22:00
Last modified on	2013-03-22 18:22:00
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Theorem
Classification	msc 42A16
Classification	msc 42A10
Related topic	CommonFourierSeries
Related topic	UniquenessOfFourierExpansion