i have a function

f(x)=

1 for x>0

0 for x=0

-1 for x<0

i need to find a series which belongs to C[T] for which fn(x) converges to f(x) in L2 norm

in other words

||fn(x)-f||_{L_2(T)} ->0 then n->infinity

how i tried:

i have dissamble the norm into the integral definition of L2(T)

so the integral is from -pi to +pi

and inside we have |fn-f|^2

then i splited the integral into to parts ,because f=1 for x>0 and f=-1 for x<0.

i tried to draw a formula that will make 1/n type in the end but

the continuety doesnt allow