# Generalized Fourier Coefficients

Assume we have an ON basis $$\{|e_i\rangle\}$$ for $$L_2^w(a,b)$$. We can write, $$|f\rangle = \sum_i f^i|e_i\rangle = \sum_i |e_i\rangle\langle e^i|f\rangle$$. Then, $$f(x)=\langle x|f\rangle = \sum_i \langle x|e_i\rangle \langle e^i|f\rangle = \sum_i f^i e_i(x)$$. $$f^i = \int_a^b dxw(x)e_i^*(x)f(x)$$.

$$f^i$$ are called fourier coefficients or generalized fourier coefficients.

Thus, $$\ell_2$$ and $$L_2^w(a,b)$$ are isomorphics, i.e. the same.

Hence, given an orthonormal basis, a vector in $$L_2^w$$ is characterized by an infinite sequence $$(f^1,f^2,\cdots)$$, i.e. a vector in $$\ell_2$$, for which their sum of swaures converges to a finite value.

Therefore, all Hilbert spaces are isomorphic to $$\ell_2$$.

(A bit handwavy since we have infinite dimensional vector spaces).

## Identity from Generalized Fourier Coefficients

$$\mathbb{I} = \sum_i |e_i\rangle\langle e_i|$$.

$$\langle x|\mathbb{I}|x'\rangle = \sum_i e_i^*(x)e_i(x') = \delta(x-x')/w(x)$$.

Created: 2023-06-25 Sun 02:23

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