Assignments - Math 667

Assignment 1

1. Show that the sum of |c(n)| ² (1+n²)^k is equal to the integral of f(t)^*[1-d²/dt²]^kf(t) over the interval [0,2*pi]. Use integration by parts to put this in a more symmetric form. For example, when k = 1, the sum equals the the integral of |f(t)| ² + |f'(t)| ²

2. Let f be the 2*pi periodic function that is given by
    
   f(t) = (360-7*pi⁴)/360 + pi²(t- pi)² /12 - (t- pi)⁴ /24
    
   when t is in the interval [0,2*pi]. Verify that f is twice continuously differentiable (check that the values at the ends match), and has a piecewise smooth third derivative with a jump discontinuity at multiples of 2*pi. Use MATLAB to find the approximation rate for the partial sums, just as we did in class.

Assignment 2

1. Find the Fourier transform of exp(-½t²) without using path integration techniques from complex analysis.

2. Prove the convolution theorem.

Assignment 3

1. Prove the convolution theorem for the DFT.

2. An N×N matrix A is called a circulant if all of its diagonals (main, super, and sub) are constant, and the indices are interpreted ``modulo N''. For example, this 4×4 matrix is a circulant:

            [9 2 1 7
             7 9 2 1
             1 7 9 2
             2 1 7 9]

   Prove that the DFT diagonalizes all circulant matirices; that is, that U^H_N A U_N = D. What are the diagonal entries of D? - i.e., what are the eigenvalues of A?

3. Find the Z-transform for the sequence x = (... 0 0 1 ½ ¼ ... ½ⁿ ... ).

Assignment 4

1. Let f be a function continuous on [0,1]. Let H_k(t) = N^½ H(Nt-k), where H(t) is 1 for t in [0,1), and 0 otherwise. Form the projection of f onto the span of the H_k's,

   f^N = < f , H₀ > H₀ + ... + < f , H_N-1 > H_N-1.

   Show that f^N converges uniformly to f on [0,1].

2. For f(t)=1-t², use MATLAB to find the Haar wavelet decomposition (on [0,1]) for N=4, 8, and 16. Plot the results.

Assignment 5

1. Exercises 1, 7, and 10 in §4.5.

2. (Extra.) Apply the Strang-Fix conditions to the Haar approximation spaces, V_j. What is their approximation power?

Assignment 6

1. Verify that the iterative equations for the scaling function are equivalent to the product form

   F_n+1(w)=m₀(w/2)F_n(w/2), where F_n is "phi hat"_n.

2. Show: If the p_k's in the scaling relation are all zero for k>N and k<0, and if the iterates phi_n converge to phi, then phi has compact support. Find the support of phi in terms of N.

3. Exercise 8, §5.4.

Assignment 7

1. Exercises 9 and 10, §5.4.

2. Show that if p₀, p₁, ... , p_N-1 are filter coefficients for an orthonomal scaling function, then N must be even.