Math 240 (Driver) Qual Exam (5/22/2017)

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1 1 Name: I.D. #: Math 240 (Driver) Qual Exam (5/22/2017) Instructions: Clearly explain and justify your answers. You may cite theorems from the text, notes, or class as long as they are not what the problem explicitly asks you to prove. You may also use the results of prior problems or prior parts of the same problem when solving a problem this is allowed even if you were unable to prove the previous results. Make sure to indicate the results that you are using and be sure to verify their hypotheses. All 8 problems have equal value. Notation: m or dx is used to denote Lebesgue measure on the Borel -algebra R d where d may be 1, 2, or an arbitrary integer in N. For two functions, f,g, on R d,f g denotes their convolution when this makes sense. Measurable means Borel measurable on this test unless otherwise indicated. As usual k k p denotes the L p (µ)-norm for the measure space appearing in the given problem. Exercise 1.1. In each case below find L (allowing for values of ±1) and justify the calculations: Z 1 min (nx, 1) 1. L = lim dx, n!1 0 x Z L = lim n!1 1 x 3/2 einx dx 3. L = lim n!1 Z 1 0 e n2 x 2 cos e n2 x n 2 xdx.

2 2 Exercise 1.2. What is the closure of each of Re L 1 (R,m) in the Banach space norm, kfk 1 := R f dm in each of the R three cases listed below. Please briefly justify your answer. ( Z 1) = C c (R, R) 2) = {f 2 C c (R, R) :f (0) = 0}. 3) = f 2 C c (R, R) : [ 1,1] fdm =0 ).

3 3 Exercise 1.3. Let (H, h i) be a Hilbert space, {e n } 1 n=1 and {u n} 1 n=1 be orthonormal bases for H, and { n} 1 n=1 C with M := sup n n < Show Th := P 1 n=1 n hh e n i u n exists in H for all h 2 H. 2. Show kt k op apple M<1, where kt k op is the operator norm of T. 3. Show T is a compact operator if lim n!1 n =0.

4 4 Exercise 1.4. Answer the following true or false. For the true statements give a brief justification why it is true and for the false statements give a counter example. 1. If X is a Banach space and {' n } 1 n=1 X satisfies sup n ' n (x) < 1 for all x 2 X, then sup n sup kxk=1 ' n (x) < If X is a Banach space, D is a dense subspace of X, and {' n } 1 n=1 X satisfies sup n ' n (x) < 1 for all x 2 D, then sup n sup kxk=1 ' n (x) < Suppose that (,B,µ) is a measure space, n 2Bwith n " and µ ( n ) < 1 for all n 2 N. If f :! C is a measurable function such that then f 2 L 2 (µ). sup n2n Z n f1 f applen gdµ < 1 for all g 2 L 2 (µ),

5 5 apple t t Exercise 1.5. For t>0, let A t = 0 t 1 and for f : R 2! C let T t f (x) =f (A t x) for x 2 R Show kt t fk 2 = kfk 2 for all f 2 L 2 R 2,m. 2. Explain why lim t!0 kt t f fk 2 = 0 for all f 2 C c R Show lim t!0 kt t f fk 2 = 0 for all f 2 L 2 (R 2,m).

6 6 Exercise 1.6. Let ' 2 C 1 c (R, [0, 1)) satisfy R R 'dm = 1 and for ">0let " (x) = 1 " ' 1 " x. 1. If 1 <a<b<1 and h " (x) :=1 [a,b] ", show h 0 " (x) = " (x a) " (x b). 2. If f 2 C c (R, R) and g 2 L 1 (R,m) satisfy, Z fh 0 dm = R Z R ghdm for all h 2 C 1 c (R), (1.1) show f is absolutely continuous and f 0 = gm-a.e.

7 7 Exercise 1.7. Let g be a real valued function in L 1 ([0, 1],m) and h :[0, 1]! R be a strictly increasing function (i.e. h (x) <h(y) ifx<y) such that Z 1 0 g (x)[h (x)] n dm (x) = 0 for n 2 N. (1.2) 1. Under the further assumption that h is continuous and h (0) > 0, show g (x) = 0 m-a.e. x. 2. Is it still true that g (x) = 0 m-a.e. when h is continuous with h (1/2) = 0? [You must justify your answer!] Extra credit. What are the possible choices for g 2 L 1 ([0, 1],m) if we only assume h is strictly increasing (not necessarily continuous) but (1.2) now holds for n 2 N[{0}, where [h (x)] 0 1.

8 8 Exercise 1.8. Suppose that f 2 L 2 (R,m) is a function such that f (x) =0if x Show ˆf 2 C 1 (R, C) and sup k2r ˆf (n) (k) apple 1 p 2 2 2n +1 kfk Let {f n } 1 n=1 L2 (R,m) satisfy kf n k 2 apple 1 and f n (x) = 0 for x 1, shows for each 0 < M < 1 there exists n o 1 1 apple n 1 <n 2 <n 3 <... in N such that ˆfnk is uniformly convergent on [ M,M] to some g 2 C ([ M,M], C). k=1 Extra Credit. Find an explicit sequence {f n } 1 n=1 L2 (R,m) such that kf n k 2 = 1 and f n (x) =0if x ˆf n! 0 uniformly [ M,M] for any 0 <M<1. 1 such that