The purpose of this problem is that I want to prove that for any λ...

Wisniewool

Wisniewool

Answered

2022-07-04

The purpose of this problem is that I want to prove that for any λ integrable function f on a bounded closed interval [a,b] holds
lim n [ a , b ] f ( x ) sin ( n x ) d λ = 0.
I have submitted a proof below.

Answer & Explanation

treccinair

treccinair

Expert

2022-07-05Added 18 answers

The smooth functions with compact suppport C 0 ( [ a , b ] ) functions are dense over the Lebesgue integral functions L 1 [ a , b ]. Then for every f there is a sequence ( f k ) k C 0 ( [ a , b ] ) such that f f k 1 k 0 . We observe that
| a b f k ( x ) sin ( n x ) d x | = | 1 n a b f k ( x ) cos ( n x ) d x | = 1 n 2 | a b f k ( x ) sin ( n x ) d x | M k n 2 n 0
Where M k = sup x [ a , b ] | f k ( x ) | ( b a ) < . Now
| a b f ( x ) sin ( n x ) d x | | a b f k ( x ) sin ( n x ) d x | + | a b ( f ( x ) f k ( x ) ) sin ( n x ) d x | M k n 2 + f k f 1 n f k f 1 k 0.

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