# MultiIndices#

## Background#

A multiindex is simply a length $$D$$ vector of nonnegative integers $$\mathbf{p}=[p_1,p_2,\dots,p_D]$$. Multiindices are commonly employed for defining multivariate polynomial expansions and other function parameterizations. In these cases, sets of multiindices define the form of the expansion.

Example: Multivariate Polynomial

A multivariate polynomial $$\Phi_{\mathbf{p}}(\mathbf{x}) : \mathbb{R}^D\rightarrow R$$ can be defined as the product of $$D$$ univariate polynomials. Using monomials for example,

$\Phi_{\mathbf{p}}(\mathbf{x}) = \prod_{i=1}^D x_i^{p_i}$

A multivariate polynomial expansion can then be written succinctly as

$f(\mathbf{x}) = \sum_{\mathbf{p}\in\mathcal{S}} c_{\mathbf{p}} \Phi_{\mathbf{p}}(\mathbf{x})$

where $$\mathcal{S}$$ is a set of multiindices and $$c_{\mathbf{p}}$$ are scalar coefficients.

Example: Wavelets

Multivariate polynomials are constructed from a tensor product of one-dimensional functions and each one-dimensional function depends on a single integer: the degree of the one-dimensional polynomial. This is a common way to define multivariate functions from indexed families of one-dimensional basis functions. In a general setting, however, the one-dimensional family does not need to be index by a single integer. Families of one-dimensional functions indexed with multiple integers can also be “tensorized” into multivariate functions. Wavelets are a prime example of this.

A one dimensional wavelet basis contains functions of the form

$\psi_{j,k}(x) = 2^{j/2}\psi(2^jx -k)$

where $$j$$ and $$k$$ are integers and $$\psi :\mathbb{R}\rightarrow \mathbb{R}$$ is an orthogonal wavelet. Unlike polynomials, two integers are required to index the one-dimensional family. Nevertheless, a multivariate wavelet basis can be defined through the tensor product of components in this family:

$\Psi_{\mathbf{p}}(\mathbf{x}) = \prod_{i=1}^{D/2} \psi_{p_{2i},p_{2i+1}}(x_i)$

where $$\Psi_{\mathbf{p}} : \mathbb{R}^{D/2}\rightarrow \mathbb{R}$$ is a multivariate wavelet basis function in $$D/2$$ dimensions and $$\mathbf{p}$$ is a multiindex with $$D$$ components.