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For complex Fourier series applies following: "In the language of Hilbert spaces, the set of functions {$e_n = e^{inx}$; $n \in \mathbb{Z}$} is an orthonormal basis for the space $L^2([−\pi, \pi]$) of square-integrable functions of [−$\pi$, $\pi$]. " - Hilbert space interpretation But for power series( for example Taulor l. Laurent series) " the set of functions " that form Laurent series are what? In other words is basis of Laurent functions orthonormal in any sense( for example $L^2([-R,R])$?

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  1. Since $x^2$ and $x^4$ are both positive on $[-R,R]$, they cannot be orthogonal in any weighted Lebesgue space on real line, i.e., a space with inner product $\langle f,g\rangle = \int_a^b fgw$ where $w$ is a weight on $[a,b]$.

  2. In the complex plane the monomials $z^k$ are orthonormal with respect to the inner product $\langle f,g\rangle =\frac1{2\pi}\int_{|z|=1}f\bar g$, which is structure of the Hardy space $H^2$. However, a closer look will reveal that this is the same Fourier series stuff in different notation.

  3. We can try to stay within real numbers and define inner product of polynomials differently, using the vector of coefficients. That is, write polynomials as $p(x)=\sum_{k=0}^\infty a_k x^k$ and $q(x)=\sum_{k=0}^{\infty} b_k x_k$, where only finitely many coefficients are nonzero. Define $\langle p,q\rangle =\sum_{k=0}^\infty a_kb_k$. This is an inner product with respect to which the monomials $x^k$ form an orthonormal basis. The completion of this space is a subspace of the Hardy space $H^2$ which consists of the functions that are real on the real line. In other words, we are led to the complex plane whether we want it or not.

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