FNCM - Analysis Qual Key Ideas and Problems To-Do Will...

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Analysis Qual: Key Ideas and Problems To-Do Will Feldman, Brent Nelson, Nick Cook, Alan Mackey (Now with more Folland Problems!) (Red is To-Do) Spring 1970: R2. f Lebesgue measurable and continuous at 1. g is in L^1. Evaluate the limit as n goes to infinity of the integral from -n to n of f(1+x/n^2)g(x)dx. Pull the indicator function 1_[-n,n] inside and note that for large enough n the integrand is dominated by 2f(1)g(x). R3. Find the extreme points of the unit ball in L^2([0,1]). That is, find points h in the closed ball, such that h=.5(f+g) for f, g in the closed ball implies h=f=g. Write out the norm of h=.5(f+g) and Cauchy-Schwarz, then use ||f||,||g||<=1 to bound ||h||<=1. Then look back at conditions for equality in those inequalities, and they imply f=g=h. Fall 2001: R1. (a) 1,-1,0,0,0. ... new line 0, 1, -1, 0. .. etc. (b) Tonelli R2. Closed and bounded. b) determinant continuous and maps to {+1,-1} R4. 245B TA Section (density argument) R5. Consider number of faces, vertices and edges of unit ‘ball’ polytopes R6. 245b homework (linear functional mapping sequences to their limit) C3. (b) induction C4. See Sp02.11 C5. (a) uniform continuity (b) Sequence of coefficients is square summable: use orthogonality to write L2 norm of F(theta,r) for r fixed as sum |a_n|^2r^2n use mct as r goes up to 1 and continuity of F on the closed disk. Winter 2002: 1. Consider a sequence of “triangle” functions: f_n is zero on [1/n,1] and from [0,1/n] is a triangle of height 1 with peak at 1/2n. Then f_n converges to zero everywhere but not uniformly. [Counterexampled Prop 1.15), show induced measure is abs continuous w.r.t to Lebesgue measure and a] (Not sure this is the right number) 3. Apply Fubini to LHS. Must verify integrand is integrable using Tonelli. 4. Define a premeasure by mu( (a,b])=F(b)-F(a) (Follanpply Radon-Nikodym 5. 245b discussion: pf by contradiction, consider preimages of diadic annuli, infinitely many of which must have positive measure. Bound ||fg||_2 below by divergent sum for a carefully chosen g. (can also use operator stuff, Closed Graph theorem) 6. Use a desnity argument 7. First consider X=[0,\infty). Since X has both sets of arbitrarily large and arbitarily small measure, given any p*, L^p* is not contained in L^q for any q>p*, and is not contained in any L^r for any r<p*. So that means we have f_n in L^p* such that f_n is not in L^{p*+1/n} So we can make an f as 1
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follows: partition the kth unit interval into [0,1/2), [1/2,1/4),. .. and on each let f be 2^{-1}f_1, 2^{- 2}f_2,. . as it is on its kth interval of length 2^-n. This function is now in L^p* but not in L^p for any p>p*. (This construction is confusing) sorry We can define g_n and g in the same way to get a function in L^p* but not in L^p for any p<p*. Now define h on [0,\infty) by alternating f and g on each interval. Moving to R^n, do what you have to to make a radially symmetric function H out of our h.
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This note was uploaded on 01/30/2012 for the course MATH 131a taught by Professor Hitrik during the Spring '08 term at UCLA.

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FNCM - Analysis Qual Key Ideas and Problems To-Do Will...

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