Unformatted text preview: (b) Z [ n,n +1]  f  → 0 as n → 0. (c) √ n Z [ n,n +1]  f  → 0 as n → 0. (d) lim inf n →∞ √ n Z [ n,n +1]  f  = 0 3. Let f ∈ L 2 (0 , ∞ ) (Lebesgue measure). Prove the following: (a) ± ± ± Z x f ( t ) dt ± ± ± ≤ x 1 / 2 k f k 2 for x > 0. (b) lim x →∞ x1 / 2 Z x f ( t ) dt = 0. 4. Deﬁne f ( x,y ) = ( x4 / 3 sin( 1 xy ) if 0 < y < x < 1 otherwise Prove or disprove: Z 1 Z 1 f ( x,y ) dxdy = Z 1 Z 1 f ( x,y ) dydx . 1...
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This note was uploaded on 06/19/2011 for the course MATH 680 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.
 Spring '11
 NA

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