# Integral of a multivariate polynomial from an orthogonal basis on a constrained domain

Let $\{ \Phi_i(\mathbf{Z}) \}_{i = 0}^n$ be a basis in a functional space where $\mathbf{Z} = \{ Z_i \}_{i = 1}^m$ and $\Phi_i(\mathbf{Z})$ are multivariate orthogonal polynomials with respect to some weight function $w(\mathbf{Z})$ with support $S \subseteq \bar{\mathbb{R}}^m$. The n-dimensional polynomial space is constructed as a tensor product of the one-dimensional counterparts.

The following holds:

$\int \cdots \int_{S} \Phi_i(\mathbf{Z}) \; w(\mathbf{Z}) \; d\mathbf{Z} = \delta_{0 i}$

where $\delta_{ij}$ is the Kronecker delta function. Now, suppose there are two real-valued functions given as linear combinations of the polynomials from the basis:

$f(\mathbf{Z}) = \sum_{i = 0}^n \alpha_i \Phi_i(\mathbf{Z})$

$g(\mathbf{Z}) = \sum_{i = 0}^n \beta_i \Phi_i(\mathbf{Z})$

such that $f(\mathbf{Z}) < g(\mathbf{Z})$, $\forall \mathbf{Z}$. Let $\tilde{S} = \{ \mathbf{Z}: f(\mathbf{Z}) \leq C < g(\mathbf{Z}) \}$ where $C$ is a constant.

The question is how to evaluate the following integral:

$\int \cdots \int_{\tilde{S}} \Phi_i(\mathbf{Z}) \; w(\mathbf{Z}) \; d\mathbf{Z} = \int \cdots \int_{S} 1_{\tilde{S}}(\mathbf{Z}) \; \Phi_i(\mathbf{Z}) \; w(\mathbf{Z}) \; d\mathbf{Z}$

where $1_A(a)$ denoted the indicator function of a set $A$. In particular, I am interested in the Hermite polynomial basis.

If the integration domains of each of $Z_i$ were independent, the multi-dimensional integral would boil down to a product one-dimensional integrals, which one could compute using, e.g., some quadrature rule.

I would be very grateful if you could share any thoughts about the problem. Thank you.

Regards, Ivan

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