Chapter 3 Fourier series 534 Convergence(收敛) of the fourier series 1. Approximation(近似性) joo en(=x(0)-x(t)-Error EN=k、O)2m=J,、0(M (1)N↑→EN↓N→0EN>0 )=4=1[nx)%冷最小 11
11 Chapter 3 Fourier Series §3.4 Convergence（收敛） of the Fourier Series 1. Approximation(近似性) ( ) j k t k N k N N x t a e 0 ˆ + =− = e (t) x(t) x (t) N = − N ——Error (1) N EN ( ) x(t)e dt T a a j k t T k k 0 1 2 ˆ − = = E e (t) dt N T N 2 = EN最小 N → EN →0 e (t)e (t)dt T N N =
Chapter 3 Fourier series 2. Dirichlet conditions: Condition 1 x()a< 2<∞ <T> ( 0 x()=1/t,0<t≤1,T=1 12
12 ak Chapter 3 Fourier Series 2. Dirichlet Conditions: Condition 1 ( ) x t dt T x(t)=1/ t , 0 t 1 , T=1
Chapter 3 Fourier series Condition 2 In any finite interval, x(t) is of bounded variation x()=sin(2z/),0<t≤1,T=1
13 Chapter 3 Fourier Series Condition 2. In any finite interval , is of bounded variation. x(t) x(t)= sin(2 / t) , 0 t 1 , T=1
Chapter 3 Fourier series Condition 3 In any finite interval, there are only a finite number of discontinuities 8 16t 14
14 Chapter 3 Fourier Series Condition 3. In any finite interval , there are only a finite number of discontinuities
Chapter 3 Fourier series Gibbs phenomenon: Figure 3.9
15 Chapter 3 Fourier Series Gibbs Phenomenon: Figure 3.9