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 UHN A UN = 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 Hk(t) = N½ 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 Hk's,

    fN = < f , H0 > H0 + ... + < f , HN-1 > HN-1.

    Show that fN 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, Vj. What is their approximation power?

Assignment 6

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

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

  2. Show: If the pk's in the scaling relation are all zero for k>N and k<0, and if the iterates phin 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, ... , pN-1 are filter coefficients for an orthonomal scaling function, then N must be even.