- #1
eljose
- 492
- 0
Let,s suppose we have a function f(x) which is not on [tex] L^{2} [/tex] space but that we choose a basis of orthononormal functions so the coefficients:
[tex] c_{n}=\int_{0}^{\infty}dxf(x)\phi_{n}(x) [/tex] are finite.
would be valid to expand the series into this basis in the form:
[tex] f(x)=\sum_{n=0}^{\infty}\phi_{n}(x) [/tex] of course the sum:
[tex] \sum_{n=0}^{\infty}|c_{n}|^{2} [/tex] would diverge
[tex] c_{n}=\int_{0}^{\infty}dxf(x)\phi_{n}(x) [/tex] are finite.
would be valid to expand the series into this basis in the form:
[tex] f(x)=\sum_{n=0}^{\infty}\phi_{n}(x) [/tex] of course the sum:
[tex] \sum_{n=0}^{\infty}|c_{n}|^{2} [/tex] would diverge