Prove that the space P of all entire functions of the form $$f(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi(t) e^{-izt} dt,$$ is a Hilbert Space, where $\varphi\in L^2[-\pi,\pi]$. The inner product of two functions f and g in P is defined to $$(f,g)=\int_{-\infty}^{\infty}f(t) g^*(t) dt$$ Any suggestions please?
Thanks.
Note that at first your definition is only suitable for $\phi\in L^1(\mathbb T) \cap L^2(\mathbb T)$, but can be continuously extended to $\phi\in L^2(\mathbb T)$.
The Operator taking $\phi$ to $f$ is precisely the fourier transform $\mathcal F$. It is a fundamental result of complex analysis that $\mathcal F: L^2(\mathbb T) \to L^2(\mathbb T)$ is an isometric isomorphism. All you need though is that it is an isomorphism, since this guarantees that $P = L^2(\mathbb T)$, wich is a hilbert space.