s7 - Math 8601: REAL ANALYSIS. Fall 2010 Homework #7....

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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.

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s7 - Math 8601: REAL ANALYSIS. Fall 2010 Homework #7....

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