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Unformatted text preview: Math 8601: REAL ANALYSIS. Fall 2010 Homework #7. Problems and Solutions. #1. Let f ∈ L 1 ( R 1 ). Show that S n ( x ) = 1 n n 1 X j =0 f x + j n → S ( x ) = Z x +1 x f ( t ) dt in L 1 ( R 1 ) as n → ∞ , i.e. Z  S n ( x ) S ( x )  dx → 0 as n → ∞ . Proof. By Theorem 2.26, ∀ ε > 0 there are a constant A > 0 and a continuous function g ( x ) on R 1 , such that Z R 1  f ( x ) g ( x )  dx < ε, and g ( x ) ≡ 0 for  x  ≥ A. We can write Z R 1  g ( x + t ) g ( x )  dx = A +1 Z A 1  g ( x + t ) g ( x )  dx for  t  ≤ 1 . By the dominated convergence theorem (Theorem 2.27(a)), Z R 1  g ( x + t ) g ( x )  dx → 0 as t → . Hence Z R 1  f ( x + t ) f ( x )  dx ≤ Z R 1  f ( x + t ) g ( x + t )  dx + Z R 1  g ( x + t ) g ( x )  dx + Z R 1  g ( x ) f ( x )  dx = Z R 1  g ( x + t ) g ( x )  dx + 2 Z R 1  g ( x ) f ( x )  dx < Z R 1  g ( x + t ) g ( x )  dx + 2 ε....
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This note was uploaded on 02/15/2012 for the course MATH 8601 taught by Professor Staff during the Fall '08 term at Minnesota.
 Fall '08
 Staff
 Math

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