Fourier series

Fourier series

A special kind of trigonometric series. French mathematician J.-B.-J. Fourier proposed it when studying the boundary value problem of partial differential equations. Thus greatly promoting the development of the theory of partial differential equations. At home, Cheng Minde first systematically studied multivariate trigonometric series and multivariate Fourier series. He first proved the uniqueness theorem of spherical sum of multivariate trigonometric series and revealed many characteristics of Riess-Bochner spherical average of multivariate Fourier series. Fourier series has greatly promoted the development of the theory of partial differential equations. It has important applications in mathematics, physics and engineering.

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Fourier series formula

Given a function x(t) with a period t, it can be expressed as an infinite series:

& lt Math & gtx(t)= \ sum _ {k =-\ infty} a _ k \ cdote e {JK (\ frac {2 \ pi}) t} < /math & gt； (j is an imaginary unit) (1)

Among them,

& lt Mathematics & GTA _ k = \ frac \ int _ x (t) \ cdote e {-JK (\ frac {2 \ pi}) t} < /math & gt； (2)

take notice of

Convergence of Fourier series

Convergence of Fourier series: Fourier series represented by periodic functions satisfying Dirichlet condition are convergent. Dilihri conditions are as follows:

X(t) must be absolutely integrable in any period;

In any finite interval, x(t) can only take a finite number of maxima or minima;

In any finite interval, x(t) can only have a finite number of discontinuous points of the first kind.

Gibbs phenomenon: at the nondifferentiable points of x(t), if only the finite items in the infinite series on the right side of (1) are taken as the sum of X(t), then X(t) will fluctuate at these points. A simple example is a square wave signal.

Orthogonality of trigonometric function families

The orthogonality of two different vectors means that their inner product is 0, that is, there is no correlation between the two vectors. For example, in three-dimensional Euclidean space, mutually perpendicular vectors are orthogonal. In fact, orthogonality is the abstraction and generalization of verticality in mathematics. A set of n mutually orthogonal vectors must be linearly independent, so it must be an n-dimensional space, that is, any vector in the space can be expressed linearly by them. The orthogonality of trigonometric function family is expressed by formula:

& lt Mathematics & gt \ int _ {2 \ pi} \ sin (NX) \ cos (MX) \, dx = 0；; & lt/math & gt；

& lt Mathematics & gt \ int _ {2 \ pi} \ sin (MX) \ sin (MX) \, dx = 0；; (m \ ne n)& lt； /math & gt；

& lt Mathematics & gt \ int _ {2 \ pi} \ cos (MX) \ cos (MX) \, dx = 0；; (m \ ne n)& lt； /math & gt；

& lt Math & gt \ int _ {2 \ pi} \ sin (NX) \ sin (NX) \, dx = \ pi& lt/math & gt；;

& lt Mathematics & gt \ int _ {2 \ pi} \ cos (NX) \ cos (NX) \, dx = \ pi& lt/math & gt；;

Odd and even functions

Odd number function < math > f _ o (x) <; /math & gt； It can be expressed as a sine series or even a function.

& lt Mathematics & GTF _ o (x) = \ sum _ {-\ infty} {+\ infty} b _ k \ sin (kx); & lt/math & gt；

& lt Mathematics & GTF _ e (x) = \ frac+\ sum _ {-\ infty} a _ k \ cos (kx); & lt/math & gt； Just pay attention to Euler's formula:

Generalized Fourier series

Any orthogonal function system

& lt Mathematics & gt \ int _ f 2 (x) \, dx = \ sum _ {k =1} {\ infty} c _ < /math & gt； (4),

Then the series.

& lt Mathematics & GTC _ n = \ int _ f (x) \ phi _ n (x) \, dx < /math & gt； (6)。

In fact, whether (5) converges or not, we always have:

& lt Mathematics & gt \ int _ f 2 (x) \, dx \ ge \ sum _ {k =1} {\ infty} c _ < /math & gt； This is the so-called Bessel inequality. In addition, formula (6) can be easily deduced from orthogonality, because for any unit orthogonal basis,