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An archived instance of discourse for discussion in undergraduate PDE.

Problem 2 Homework Set

rjensen

The Problem states:

If cn are the Fourier coefficients for f using the orthonormal set {fn}n=1, show:
n=1cnfn,fn=1cnfn=0.

I am assuming f equals n=1fn not, n=1cnfn since the right side of the inner product would immediately reduce to zero? My guess is that it isn't that simple.

cromer

From what I understand, the fn's are elements of a set of orthonormal "vectors" which form a basis. To write down f in this basis, each basis "vector" must have the appropriate coefficient cn. On the other hand, I do not know if we are to assume that the sum ncnfn does converge to f or not.

rjensen

I just realized they have a general definition for Fourier Coefficients in Theorem 3.6 in the textbook so I'll give your suggestion a try.