This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 8601: REAL ANALYSIS. Fall 2010 Homework #7 (due on Friday, December 10). 50 points are divided between 5 problems, 10 points each. #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 as n . Hint. Using Theorem 2.26, show that ( h ) := sup  t  h Z R 1  f ( x + t ) f ( x )  dx as h + . Moreover, writing S n S in the form S n ( x ) S ( x ) = n 1 X j =0 1 /n Z f x + j n f x + j n + t dt, one can get Z R 1  S n ( x ) S ( x )  dx (1 /n ) . #2. Let f,f 1 ,f 2 ,...,f n ,... be measurable functions on a measure space ( X, M , ) with ( X ) = 1, such that f n ( x ) f ( x ) as n for all x X . Suppose that Z  f n  1+ d C for all n, with some constants > and C > . Show that Z  f n f  d as n ....
View
Full
Document
 Fall '08
 Staff
 Math

Click to edit the document details