# Riemann integrable

• May 8th 2011, 03:33 AM
PeaceSoul
Riemann integrable
Good morning to you all,

any genius in the house ? does anybody know how to solve this ?? please help. thank you :)

Ps: f_k is f sub k
f^ is f hat
f^(n) = nth Fourier coeffcient of f

suppose that {f_k } is a sequence where k goes from 0 to infinity, is a sequence of Riemann integrable functions on the interval [0; 1] such that

INTEGRAL FROM 0 TO 1 OF |f_k (x) - f(x)| dx -> 0 as k -> infinity

Show that f^_k(n) -> f^(n) uniformly in n as k -> infinity
• May 8th 2011, 06:19 AM
girdav
Quote:

Originally Posted by PeaceSoul
suppose that $\displaystyle \left\{f_k\right\}_{k\geq 0}$ is a sequence of Riemann integrable functions on the interval [0; 1] such that

$\displaystyle \lim_{k\to+\infty}\int_0^1|f_k (x) - f(x)| dx = 0$

Show that $\displaystyle \lim_{k\to+\infty}\hat{f_k}(n) =\hat{f}(n)$ uniformly in $\displaystyle n$.

What is the definition of $\displaystyle \hat f(n)$?
• May 8th 2011, 06:59 AM
PeaceSoul
^
f(n) which is the nth fourier coefficient of f

L is the length of the interval so L= b-a

interval is [a,b]

^
f (n) = (1/L ). integral from a to b of { f(x).e^ ( ( -2(pi)inx )/L )}dx

here is a better representation

http://s1138.photobucket.com/albums/...rrent=fhat.jpg
We only have information about the behavior on $\displaystyle (0,1)$ whereas $\displaystyle \widehat f(n)$ needs to know the integral on $\displaystyle (a,b)$.