Math 104A代写、代做algorithm、代写Python语言、Python编程设计调试

日期：2020-03-14 10:46

Math 104A Final Projects?

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 (corresponding

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).

Prove that if the fj , for j = 0, 1, ..., N ? 1 are real numbers then c0 is real and cN?k = ˉck,

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

array fj at the equidistributed nodes xj = j(2π/N), for j = 0, 1, ..., N ? 1, i.e.

fj sin kxj , for k = 0, 1, ..., N/2 ? 1.

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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. Let fj = esin xj

, xj = j2π/N for j = 0, 1, ..., N ?1. Take N = 8. Using your fft package obtain

P8(x) and find a spectral approximation of the derivative of esin x at xj for j = 0, 1, ..., N ? 1

by computing P(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,

(a) Let Pn be the space of polynomials of degree at most n. Prove that the error f ? Pn is

orthogonal to this space, i.e. hf ? Pn, qi = 0 for any q ∈ Pn.

(b) Using the analogy of vectors interpret this result geometrically (recall the concept of

orthogonal projection).

6. (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.

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