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Let $d$ and $k$ be positive integers. Let $\mu$ be a positive Borel measureon $\mathbb{R}^2$ possessing finite moments up to degree $2d-1$. If the supportof $\mu$ is contained in an algebraic curve of degree $k$, then we show thatthere exists a quadrature rule for $\mu$ with at most $dk$ many nodes allplaced on the curve (and positive weights) that is exact on all polynomials ofdegree at most $2d-1$. This generalizes both Gauss and (the odd degree case of)Szeg\H{o} quadrature where the curve is a line and a circle, respectively, toarbitrary plane algebraic curves. We use this result to show that, without anyhypothesis on the support of $\mu$, there is always a cubature rule for $\mu$with at most $\frac32d(d-1)+1$ many nodes. In both results, we show that thequadrature or cubature rule can be chosen such that its value on a certainpositive definite form of degree $2d$ is minimized. We characterize the uniqueGaussian quadrature rule on the line as the one that minimizes this value orseveral other values as for example the sum of the nodes' distances to theorigin. The tools we develop should prove useful for obtaining similar resultsin higher-dimensional cases although at the present stage we can present onlypartial results in that direction.

Optimization approaches to quadrature: new characterizations of Gaussian  quadrature on the line and quadrature with few nodes on plane algebraic  curves, on the plane and in higher dimensions