Let $(\mathcal{X}, \pi)$ be a probability space such that $\pi$ has full support. Consider $L^2(\mathcal{X},\pi)$ to be the inner product space of function $f: \mathcal{X}^n \to \mathbb{R}$, with inner product $\langle f,g\rangle = \mathbf{E}_{x \sim \pi^{\otimes n}} [f(x) g(x)]$ (here $\pi^{\otimes n}$ is the product probability distribution on $\mathcal{X}^n$). A Fourier basis for this space is a set of orthonormal basis $\phi_0, \phi_1, \ldots$ such that every function could be uniquely represented as a linear combination of the basis functions (and the Fourier coefficients could be calculated explicitly).

For example, it is known that parity functions form an orthonormal basis for the set of Boolean functions $f: \{0,1 \}^n \to \{0,1\}$ denoted as $L^2(\{0,1\}^n, \pi^{\otimes n}_p)$ (where $\pi_p$ is the Bernoulli distribution with probability $p$. Similarly Hermite polynomials form orthonormal basis for $L^2(\mathbb{R}^n, \pi^{\otimes n}_g)$ where $\pi_g$ is the Gaussian distribution.

Are there other known cases where one could compute the basis functions explicitly ? Existence could be shown and Gram-schmidt could be done to get them on a case by case basis. Are there general results for like say sub-Gaussian families of distributions, i.e., $L^2(\mathbb{R}^n, \pi^{\otimes n}_{sg})$ where $\pi_{sg}$ is a sub-Gaussian distribution ?



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