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Orthogonal functions and integrals.

  1. Jun 28, 2007 #1
    let be a set of orthonormal functions so [tex] (\phi _{i} , \phi_{j} ) =\delta _{ij} [/tex] then my question is if we can define:

    [tex] g(y)=\int_{0}^{\infty}dxf(x) \phi_{y} (x) [/tex]

    [tex] f(x)=\int_{0}^{\infty}dy g(y) \phi_{y} (x) [/tex]

    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

    [tex] (\phi _{i} , \phi_{j} ) =\delta (i-j) [/tex]
  2. jcsd
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