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Course: MATH 7337, Fall 2008
School: Georgia Tech
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Word Count: 711

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7337 FINAL MATH EXAM DUE: Noon, December 13, 2006 Work the following problems and hand in your solutions. YOU MUST WORK INDEPENDENTLY. You may use any book or your notes, but you may not use anyone elses notes. You may not discuss this exam with anyone except the professor. Please identify any sources that you consulted. Write LEGIBLY on the FRONT side of the page only, and STAPLE your pages together. 1. Let f...

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7337 FINAL MATH EXAM DUE: Noon, December 13, 2006 Work the following problems and hand in your solutions. YOU MUST WORK INDEPENDENTLY. You may use any book or your notes, but you may not use anyone elses notes. You may not discuss this exam with anyone except the professor. Please identify any sources that you consulted. Write LEGIBLY on the FRONT side of the page only, and STAPLE your pages together. 1. Let f , g L2 (R) be given. We know that Vg f is continuous; prove that Vg f C0 (R2 ), i.e., > 0, compact K R2 such that (x, ) K = |Vg f (x, )| < . / Denition 1. A set of vectors {fn }nN in a Hilbert space H is a tight frame for H if there exists a number A > 0 so that f H, n=1 | f, fn |2 = A f 2 . The number A is the frame bound. The frame operator is the mapping S : H H dened by Sf = n=1 f, fn fn for f H. We proved in class that this series converges for each f (this is a consequence of the fact that | f, fn |2 < ). 2. Assume that {fn }nN is a tight frame with frame bound A. a. Show directly that the frame operator S and the operator S AI are self-adjoint, where I is the identity operator on H. b. Show that S = AI. c. Show that the following three statements are equivalent, i.e., each implies the other. i. fn 2 = A for every n. ii. {fn }nN is an orthogonal (but not necessarily orthonormal) sequence with no zero elements. That is, fm , fn = 0 if m = n, and every fn = 0. iii. {fn }nN is a basis for H. That is, given f H there exist unique scalars cn (f ) such that f = cn (f ) fn . n=1 1 2 1 1 3. Recall that {e2inx }nZ is an orthonormal basis for L2 [ 2 , 2 ]. Given a xed T > 0 dene enT (x) = T 1/2 e2inT x . 1 a. Show that if T > 1, then {enT }nZ is incomplete in L2 [ 1 , 2 ]. 2 Hint: There are at least two ways to do this: rst, show directly that the nite linear span is not dense, and second, nd a nonzero function that is orthogonal to each enT exponential . Remark: As a consequence, there exist distinct nonzero functions f and g in L2 [ 1 , 1 ] 2 2 such that f, enT = g, enT for every n (this is aliasing in engineering terminology). b. Now let 0 < T < 1 be xed for the remainder of this problem. Show that {enT }nZ is 1 a tight frame for L2 [ 2 , 1 ]. 2 1 c. Show directly that {enT }nZ is not an orthogonal sequence in L2 [ 2 , 1 ]. Also show 2 1 1 directly that it is not a basis for L2 [ 2 , 2 ] by nding two dierent ways to write the constant function 1 as an innite linear combination of the exponentials enT . 1 d. Suppose that f L2 (R) is bandlimited to [ 1 , 2 ], i.e., supp(f ) [ 1 , 1 ]. Find an 2 2 2 explicit value for f , enT in terms of f . e. Use the fact that {enT }nZ is a tight frame for L2 [ 1 , 1 ] to formulate and prove a result 2 2 1 1 stating that a function f L2 (R) that is bandlimited to [ 2 , 2 ] can be recovered from its samples {f (nT )}nZ . Note: This is the Classical (Shannon) Sampling Theorem. Hint: f is 1 1 2 an element of L [ 2 , 2 ]. 4. a. Prove the following perturbation result for frames. Let {fn } be a frame for a Hilbert space H...

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