Your function is not defined for x in [-1, 0) or (0,1]?
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