# Orthogonal functions and integrals.

1. Jun 28, 2007

### Klaus_Hoffmann

let be a set of orthonormal functions so $$(\phi _{i} , \phi_{j} ) =\delta _{ij}$$ then my question is if we can define:

$$g(y)=\int_{0}^{\infty}dxf(x) \phi_{y} (x)$$

$$f(x)=\int_{0}^{\infty}dy g(y) \phi_{y} (x)$$

in the sense that whenever the indices i and j are discrete (i,j=1,2,3,4,5,...) you have a Kronecker's delta, wereas if i and j can vary continously (any positive real) then the scalar product becomes

$$(\phi _{i} , \phi_{j} ) =\delta (i-j)$$