Evaluation of product of Hermitian functions


I am working on expansion of a multivariate function f : R^k \xrightarrow{} R onto a basis of product of univariates Hermite functions:

f(x_1, x_2, \ldots, x_k) = \sum_{\boldsymbol{\alpha}} c_{\boldsymbol{\alpha}} \psi_{\boldsymbol{\alpha}}(x_1, x_2, \ldots, x_k) ,
where \boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_k) is a multi-index and \psi_{\boldsymbol{\alpha}}(x_1, x_2, \ldots, x_k) = \prod_{j=1}^k \psi_{\alpha_j}(x_j) and the \psi_{\alpha_j} are Hermite functions given by \psi_{\alpha_j}(x) = P_{\alpha_j}(x)\exp(-x^2/4).

Is there a Julia package that implements multivariate Hermite polynomials? k will usually be of order 50 and the degree of each polynomial will usually be less than 7-8, and I expect sparse decomposition so some multi-index coefficients \alpha_j will be zero.

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You can use an expansion over a Hermite space via https://github.com/JuliaApproximation/ApproxFun.jl

See the following documentation: https://juliaapproximation.github.io/ApproxFun.jl/stable/usage/spaces/ and https://juliaapproximation.github.io/ApproxFun.jl/stable/library/#ApproxFunOrthogonalPolynomials.Hermite

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