Fourier Series Notes

Notes for Fourier Series

Periodic functions:

A periodic function is one which repeats regularly after a certain interval. The repetition may occur in space or time, or both. Such a function is termed ‘oscillatory’, and has the general form:

f(x+P)=f(x)

For all x, where P is a period of the function – an interval over which f(x) can be translated, and brought back into coincidence with itself.

Least period:

The least period of a periodic function is the smallest value of P for which the above relation holds. It is usually simply referred to as the period of the function.

Trigonometric periodic functions and Fourier series:

The functions sinx and cosx are periodic functions with least period 2π:

• A Fourier Series is a series which can be used to approximate almost any periodic function as a sum of sines and cosines.

• Fourier space is a space for which the basis vectors are sines and cosines.

Functions as bases:

Like vectors, functions can form a basis for an infinite-dimensional space of functions. Any function in the space can be represented as a linear combination of basis functions. As for vectors, orthogonal and orthonormal bases are particularly nice to work with; sets of functions which are mutually orthogonal are therefore useful as bases. On any interval [a,b], there may be many sets of functions that are orthogonal.

Orthogonality of trigonometric functions:

The trigonometric functions Re{e^inx}=1,cosx,cos2x,cos3x… andIm{e^inx}=sinx,sin2x,

sin3x… are mutually orthogonal over the interval [π,π]. Since they are also invariant under translation by integer multiples of their least period 2π, orthogonality over this interval implies orthogonality over all space:

Where orthogonality of two functions is defined by their inner product, as above. Negative values of n need not be considered, since cosines and sines are symmetric (cos(nx)=cos(nx)) and antisymmetric (sin(nx)=sin(nx)), respectively.

Trigonometric functions therefore form an orthogonal basis which can be normalised to form an orthonormal basis by dividing the sines by $\sqrt{\pi}$ and the cosines by $\sqrt{2\pi}$.

Proof of mutual orthogonality:

Demonstrate individual orthogonality:

$$\int_{- \pi}^{\pi}{\sin{(nx)}}dx = \left\lbrack - \frac{1}{n}\cos{(nx)} \right\rbrack_{- \pi}^{\pi} = 0$$

$$\int_{- \pi}^{\pi}{\cos{(nx)}}dx = \left\lbrack \frac{1}{n}\sin(nx) \right\rbrack_{- \pi}^{\pi} = 0,\ \ \int_{- \pi}^{\pi}{\cos{(0x)}}dx = \lbrack 1\rbrack_{- \pi}^{\pi} = 2\pi$$

Extend to mutual orthogonality:

Combination of two sines: sin(nx) and sin(mx),n,mℤ:

$$\int_{- \pi}^{\pi}{\sin{(nx)\sin{(mx)}}}dx = \int_{- \pi}^{\pi}{\frac{1}{2}\left( \cos(m - n) - \cos(m + n) \right)}dx$$

Since mnand m+n are both integers:

$$\int_{- \pi}^{\pi}{\sin{(nx)\sin{(mx)}}}dx = \{\begin{matrix} 0,\ \ m \neq n \\ \pi,\ \ m = n \neq 0 \\ 0,\ \ m = n = 0 \\ \end{matrix}$$

Combination of two cosines: cos(nx) and cos(mx),n,mℤ:

$$\int_{- \pi}^{\pi}{\cos{(nx)\cos{(mx)}}}dx = \int_{- \pi}^{\pi}{\frac{1}{2}\left( \cos(m - n) + \cos(m + n) \right)}dx$$

Since mn and m+n are both integers:

$$\int_{- \pi}^{\pi}{\cos{(nx)\cos{(mx)}}}dx = \{\begin{matrix} 0,\ \ m \neq n \\ \pi,\ \ m = n \neq 0 \\ 2\pi,\ \ m = n = 0 \\ \end{matrix}$$

Combination of a sine and a cosine: cos(nx) and sin(mx),n,mℤ:

$$\int_{- \pi}^{\pi}{\cos{(nx)\sin{(mx)}}}dx = \int_{- \pi}^{\pi}{\frac{1}{2}\left( \sin(m - n) + \sin(m + n) \right)}dx$$

Since mn and m+n are both integers:

$$\int_{- \pi}^{\pi}{\cos(nx)\sin{(mx)}}dx = \{\begin{matrix} 0,\ \ m \neq n \\ 0,\ \ m = n \neq 0 \\ 0,\ \ m = n = 0 \\ \end{matrix}$$

Generalising orthogonality to the interval [L,L]:

For orthogonality over the interval [L,L], trigonometric functions must have least period 2L, thus be of the form $\sin\left( \frac{m\pi x}{L} \right)$ and $\cos\left( \frac{n\pi x}{L} \right)$ for n,mℤ. In this case, the orthogonality relations are:

And the relations hold over any interval [a,b] such that ba=2nL,nℤ.

Fourier series:

Fourier analysis is based on the result that trigonometric functions form a basis for a space of periodic functions with least period 2L. Any periodic function of period 2L can be uniquely written as a linear combination of trigonometric functions:

$$\boxed{\mathbf{f}\left( \mathbf{x} \right)\mathbf{=}\frac{\mathbf{a}_{\mathbf{0}}}{\mathbf{2}}\mathbf{+}\sum_{\mathbf{n = 1}}^{\mathbf{\infty}}\left\lbrack \mathbf{a}_{\mathbf{n}}\mathbf{\cos}\left( \frac{\mathbf{n\pi x}}{\mathbf{L}} \right)\mathbf{+}\mathbf{b}_{\mathbf{n}}\mathbf{\sin}\left( \frac{\mathbf{n\pi x}}{\mathbf{L}} \right) \right\rbrack}$$

Where a_n and b_n are known as Fourier coefficients and the factor of $\frac{1}{2}$ on the first term is included for later convenience.

Fourier coefficients:

The Fourier coefficients represent the weighting of each basis function in the overall function. They are analogous to the components of a vector and can be determined in the same way – by taking the inner product of the overall function with each of its basis functions, just as vector components are found by taking the projection of the overall vector onto each of its basis vectors:

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