ANALYSIS QUALIFYING EXAM PROBLEMS BRIAN LEARY ...

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$Math 33B/1 S00 K. Solutions to Midterm Exam G1, Version B If you ...$

$Math 3C Homework 3 Solutions$

8 <strong>ANALYSIS</strong> QUALS Problem 2: Prove the following form of Jensen’s inequality: If f : [0, 1] → R is continuous, then 1 e f(x) 1 dx ≥ exp f(x)dx . 0 Moreover, if equality occurs then f is a constant function. Solution. Let a = min [0,1] f, b = max [0,1] f, and t = 1 f(x)dx. Let 0 Then for a < s < t, we have that e β = sup a
<strong>ANALYSIS</strong> QUALS 9 But by the geometric series, the LHS is equal to 1 N and since α /∈ Q, the denominator is never 0. Hence, this approaches 0 as N → ∞, and 1 0 e2πikxdx = 0, so the statement is true for all trigonometric polynomials. 1 − e 2πiNαk 1 − e 2πiαk Now let ε > 0, f a continuous function from R/Z to R. As trigonometric polynomials are uniformly dense in the space of continuous periodic functions, we may choose P (x) trigonometric such that sup |f(x) − P (x)| < x∈R/Z ε 3 . Then for N sufficiently large, we have that N−1 1 1 ε P (nα) − P (x)dx < N 3 . n=0 Then N−1 1 1 f(nα) − f(x)dx N n=0 0 ≤ 1 N−1 N−1 1 1 |f(nα) − P (nα)| + P (nα) − P (x)dx N N + 0 n=0 n=0 0 0 < ε. Hence, the statement holds for general f. (b) Choose f + ε , f − ε continuous on the torus so that f ( εx) ≤ χ [a,b](x) ≤ f + ε (x) and 1 b − a − 2ε ≤ 0 f − 1 ε (x)dx, 0 f + ε (x)dx < b − a + 2ε. Let SN = 1 N N−1 n=0 χ [a,b](nα). Then N−1 1 f N − ε (nα) ≤ SN ≤ 1 N−1 N n=0 n=0 f + ε (nα). 1 |P (x) − f(x)| dx But the left side approaches 1 0 f − ε (x)dx and the right side approaches 1 0 f + ε (x)dx. Thus b − a − 2ε ≤ lim N→∞ SN ≤ b − a + 2ε. As this holds for all ε > 0, we have that lim n→∞ SN = b − a = 1 0 χ [a,b](x)dx. Problem 6: Let D := {z ∈ C : |z| ≤ 1} and consider the (complex) Hilbert space ∞ H := f : D → C|f(z) = ˆf(k)z k with f 2 ∞ := (1 + |k| 2 )| ˆ f(k)| 2 < ∞ k=0 k=0
Page 1 and 2: ANALYSIS QUALIFYING EXAM PROBLEMS B
Page 3 and 4: ANALYSIS QUALS 3 Proof. (a) First,
Page 5 and 6: ∂∆n ANALYSIS QUALS 5 0, 1, 2, .
Page 7: ANALYSIS QUALS 7 Thus, as |h ′ (0
Page 11 and 12: Then Γ (j) k ANALYSIS QUALS 11 f(
Page 13 and 14: ANALYSIS QUALS 13 = 1 √ |f(w)|
Page 15 and 16: ANALYSIS QUALS 15 Then B has the s
Page 17 and 18: ∂Q ANALYSIS QUALS 17 Problem 8: L
Page 19 and 20: ANALYSIS QUALS 19 Consider ˆxn ∈
Page 21 and 22: ANALYSIS QUALS 21 b) Suppose S −
Page 23 and 24: ANALYSIS QUALS 23 Problem 12: Let f
Page 25 and 26: ANALYSIS QUALS 25 b) I present the
Page 27 and 28: ANALYSIS QUALS 27 Solution. Picard
Page 29 and 30: ANALYSIS QUALS 29 Problem 2: Is eve
Page 31 and 32: δ 0 Then f(ξ) δf(w) = δ dξ
Page 33 and 34: ANALYSIS QUALS 33 But as D is compa
Page 35 and 36: so y ∈ B(x, ε), so y ∈ B(x (n1
Page 37 and 38: Problem 10: Let the power series f(
Page 39 and 40: ANALYSIS QUALS 39 where the inequal
Page 41 and 42: ANALYSIS QUALS 41 Problem 8: Let f
Page 43 and 44: ANALYSIS QUALS 43 Winter 2007 Probl
Page 45 and 46: Then for N sufficiently large, we h
Page 47 and 48: ANALYSIS QUALS 47 see that on ( √
Page 49: ANALYSIS QUALS 49 If z0 ∈ K, choo

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