Math 104A Final Projects∗ Instructor: Xu Yang
General Instructions: Please follow TA’s instructions (on Gauchoapace) to turn it in. Write your own code individually. Do not copy codes!
The Discrete Fourier Transform (DFT) of a periodic array fj, for j = 0,1,…,N−1 (correspond- ing to data at equally spaced points, starting at the left end point of the interval of periodicity) is evaluated via the Fast Fourier Transform (FFT) algorithm (N power of 2). Use an FFT package, i.e. an already coded FFT (the functions fft and ifft in Matlab or numpy.fft in python).
1. Let
N−1
ck = fje−i2πkj/N.
j=0
Prove that if the fj, for j = 0,1,…,N − 1 are real numbers then c0 is real and cN−k = c ̄k,
where the bar denotes complex conjugate.
2. Which fft package are you using? Read the manual of your fft package, and write down the formula it’s using to return the coefficients. (Note: different packages may use different definitions of the DFT, so it is very important to figure out what your package is calculating before using it.)
3. Let PN(x) be the trigonometric polynomial of lowest order that interpolates the periodic arrayfj attheequidistributednodesxj =j(2π/N),forj=0,1,…,N−1,i.e.
1 N/2−1 1 N PN(x)=2a0+ (akcoskx+bksinkx)+2aN/2cos 2x ,
for x ∈ [0, 2π], where
ak = bk =
2 N−1
fj coskxj,
for k = 0,1,…,N/2,
fork=0,1,…,N/2−1.
k=1
N j=1 2 N−1
fjsinkxj, N j=1
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1
Write a formula that relates the complex Fourier coefficients computed by your fft package to the real Fourier coefficients, ak and bk, that define PN(x).
4. Letfj =esinxj,xj =j2π/Nforj=0,1,…,N−1.TakeN=8.Usingyourfftpackageobtain P8(x) and find a spectral approximation of the derivative of esin x at xj for j = 0, 1, …, N − 1 by computing P8′(xj). Compute the actual error in the approximation.
5.
The solution Pn(x) to the Least Squares Approximation problem of f by a polynomial of degree at most n is given explicitly in terms of orthogonal polynomials ψ0(x), ψ1(x), …, ψn(x), where ψj is a polynomial of degree j, by
Pn(x) =
(a) Let Pn be the space of polynomials of degree at most n. Prove that the error f − Pn is
(b) Using the analogy of vectors interpret this result geometrically (recall the concept of orthogonal projection).
(a) Obtain the first 4 Legendre polynomials in [−1, 1].
(b) Find the least squares polynomial approximations of degrees 1, 2, and 3 for the function
f(x) = ex on [−1,1].
(c) What is the polynomial least squares approximation of degree 4 for f (x) = x3 on [−1, 1]? Explain.
ajψj(x), aj = ⟨ψj,ψj⟩. orthogonal to this space, i.e. ⟨f − Pn, q⟩ = 0 for any q ∈ Pn.
n ⟨ f , ψ j ⟩
6.
j=0
2