This chapter covers the basics of cubature rules. Starting from the definition of polynomial spaces and cubature rules, it discusses interpolatory cubature rules, Tchakaloff’s theorem on positive cubature rules, and Sobolev’s theorem on invariant cubature rules. It provides product-type cubature rules on several regular domains, such as product domains, balls, and simplexes, as well as invariant cubature rules on these domains, and a brief section on constructing cubature rules numerically.

The chapter explores a connection between cubature rules and the discrete Fourier transform of exponential functions defined via lattice translation. Such discrete Fourier analysis yields cubature rules for exponential functions for the integral over the spectral set of the lattice, which become cubature rules on the fundamental domain of the spectral set for generalized cosine and sine functions, defined as certain symmetric and antisymmetric exponential sums. Furthermore, under appropriate transformation, these generalized trigonometric functions define Chebyshev polynomials that inherit the orthogonality of generalized sine and cosine functions, which lead to cubature rules for algebraic polynomials on the range of the fundamental domain under the transformation.

The chapter contains an extensive family of minimal or near-minimal cubature rules on the square and an unbounded domain. Those on the square extended minimal cubature rules for the Chebyshev weight functions in several aspects.

Cubature rules on triangles and simplexes are of special interest for the finite element method (FEM) and hp-FEM, which often employ triangulation of the domain. They also serve as a good specimen for constructing cubature rules on regular domains. However, due to difficulties in solving nonlinear systems of equations, it is often more practical to consider invariant cubature rules under the symmetric group for triangles and even more so for simplexes, as this approach can significantly reduce the size of the system, although the invariant cubature rules require more nodes. In this addendum, we discuss the structure of invariant cubature rules for integrals on triangles and simplexes.

Gauss cubature rules are straightforward extensions of the Gauss quadrature rules of one variable. A Gauss cubature rule of degree 2n-1 exists if, and only if, its nodes are common zeros of all orthogonal polynomials of degree n. They are the first example of minimal cubature rules but rarely exist. The chapter gives a comprehensive study that provides a complete characterization of the Gauss cubature rules in terms of the common zeros of the orthogonal polynomials and their structural relations, and it includes examples and counterexamples for the existence of Gauss cubature rules.

Lower bounds for the number of nodes determine the least number of nodes needed for a cubature rule of a given precision. They are essential for studying minimal cubature rules, especially for centrally symmetric integrals. All known lower bounds are collected and discussed in this chapter. These include Möller’s lower bounds for centrally symmetric integrals and wrapped product integrals and several lower bounds for spheres, balls, and simplexes.

The chapter explains another angle of looking at minimal cubature rules, using the language of ideal and variety in algebraic geometry. In essence, the existence of a cubature rule of degree m amounts to the existence of a polynomial ideal generated by m-orthogonal polynomials with zero-dimensional and real variety, and the codimension of the ideal equals the cardinality of the variety. The abstract point of view pinpoints the root of the difficulty in understanding the minimal cubature rules and indicates a theoretical roadmap.

The first minimal cubature rules beyond the Gauss cubature rules of degree 2n-1 are generated by zeros of orthogonal polynomials of degree n and quasi-orthogonal polynomials of degree n+1, which include those that attain Möller’s lower bound. This chapter provides a complete characterization of such cubature rules. The characterization is given in terms of a family of nonlinear equations derived through the structural relations of orthogonal polynomials. Several families of minimal or near-minimal cubature rules for the Chebyshev weight functions on the square are presented.

Many questions around cubature rules remain open. The chapter discusses two open problems; both are fundamental for further study. The first is about better lower bounds of the number of nodes, and the second discusses cubature rules of more than two variables.

The theory of quadrature rules has been developed extensively and is well understood. The chapter contains results and poofs that could be extended to several variables as a prelude to studying cubature rules. It also includes a characterization of positive quadrature rules and m-orthogonal polynomials.

Orthogonal polynomials play an important role in the study of cubature rules. Minimal cubature rules are often characterized via common zeros of orthogonal polynomials, and they are studied with the help of the structure relations of orthogonal polynomials. This chapter provides a concise and self-contained account of orthogonal polynomials in several variables.