傅里叶分析之一 傅里叶级数.pptx

Fourier Series;Content;Fourier Series;Fourier;Fourier Series;Fourier analysis;Fourier synthesis;Linear Operation;Fourier Series;Periodic Phenomenon;Periodic Phenomenon in time;Periodic Phenomenon in space;;;λ and v;More on spatial periodicity;The Mathematic Formulation;Why mathematics come?;Fourier Series; History of sine and cosine;Example:;Example1:;Example2:;Example3：;would you say it had frequency 1 Hz? I don’t think so. It has one period but you’d probably say that it has, or contains two frequencies, one cosine of frequency 1 Hz and one of frequency 2 Hz.;Periodic of sine and cosine;Question:;Answer：It All Adds Up;This is important！;Idea2: How complicate signal is?;Alternative formula：;Notes:;Using Euler’s Formula;Complex Form;Fourier Series;Introduction;Question is Solving for these coefficients. A direct approach:;Since the integral of the sum is the sum of the integrals, and the coefficients cn come out of each integral, all of the terms in the sum integrate to zero and we have a formula for the k-th coefficient: ; Similar to the following integral relations:;The cn are called the Fourier coefficients of f(t). They also denoted by : The sum is called a (finite) Fourier series. Also note that because of periodicity of f(t), any interval of length 1 will do to calculate f^(n);Question: What if the period isn’t 1?;Warning!;square wave;;How good a job do the finite sums do in approximating the triangle wave?;Notes:;Conclusion;The infinite Fourier series;The infinite Fourier series;The infinite Fourier series;Convergence is very hard！;Convergence in gerneral;Continues case : f(t);Smooth case: f(x);;More Detail:Pointwise Convergence vs. Uniform Convergence;Discontinuity Case:;General Case: ;Not completely general ;We want ;此时：;Fourier Series;Fundamental Result;General in integral;More Notes on L2[0,1];Orthogonality;Inner Product;Inner Product;A geometric approach to the inner product the projection of v onto the unit vector w/||w||, or, if you