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An approximate integration formula that guarantees the minimum error for a given class of functions, relative to all formulas of a specified type. As an example, consider the quadrature formula

$$\tag{* } \int\limits _ { a } ^ { b } \rho (x) f (x) dx = \ \sum _ {k = 1 } ^ { n } \sum _ {i = 0 } ^ { m } p _ {ki} f ^ { (i) } (x _ {k} ) + R (f),$$

where $\rho (x)$ is a weight function. The remainder (error) term $R (f) = R (f, X _ {n} , P _ {nm} )$ depends both on the function $f (x)$, and on the vector $(X _ {n} , P _ {nm} )$ consisting of the interpolation nodes $x _ {k}$( it is usually assumed that $x _ {k} \in [a, b]$) and the coefficients $p _ {ki}$, $k = 1 \dots n$; $i = 0 \dots m$. Fixing $n \geq 1$ and $m \geq 0$, let $A$ denote some set of vectors $(X _ {n} , P _ {nm} )$( and hence also some set of quadrature formulas), defined by some restrictions on the interpolation nodes and coefficients (in particular, one might consider the set $A = A ( \overline{X}\; _ {n} )$ of coefficients $p _ {ki}$ for a fixed node vector $\overline{X}\; _ {n}$). Let $\mathfrak M$ be some class of functions $f (x)$, it being assumed that the integral and the sum in (*) exist. The best quadrature formula of type (*) for the class $\mathfrak M$ relative to the set $A$ is defined by a vector $(X _ {n} ^ {*} , P _ {nm} ^ {*} )$ for which

$$\sup _ {f \in \mathfrak M } | R (f, X _ {n} ^ {*} , P _ {nm} ^ {*} ) | =$$

$$= \ \inf _ {(X _ {n} , P _ {nm} ) \in A } \sup _ {f \in \mathfrak M } | R (f, X _ {n} , P _ {nm} ) | .$$

The construction of best quadrature formulas is intimately connected with certain problems in spline approximation; in many cases it reduces to minimizing the norm of a monospline (see [1]). Best quadrature formulas, together with sharp estimates for the remainder term, are known for many important classes of continuous and differentiable functions. From a more general point of view, the problem of finding best quadrature formulas and the corresponding errors for a class $\mathfrak M$ may be viewed as the problem of optimal recovery of a functional

$$J (f) = \int\limits _ { a } ^ { b } \rho (x) f (x) dx,$$

where $f \in \mathfrak M$, on the basis of the information $\{ f ^ { (i) } (x _ {k} ) \}$, $k = 1 \dots n$; $i = 0 \dots m$. The concept of a best quadrature formula generalizes in a natural way to functions of several variables (cubature formulas).

#### References

 [1] S.M. Nikol'skii, "Quadrature formulae" , H.M. Stationary Office , London (1966) (Translated from Russian) [2] N.M. Krylov, "Approximate calculation of integrals" , Macmillan (1962) (Translated from Russian) [3] P.J. Laurent, "Approximation et optimisation" , Hermann (1972) [4] A.A. Zhensykbaev, "Monosplines of minimal norm and quadrature formulas" Uspekhi Mat. Nauk , 36 : 4 (1981) pp. 107–159 (In Russian)

The terminology "best formula" is often encountered in the literature on numerical analysis, but, as was observed in [a2], p. 75, it should be taken with a large dose of salt, because, after all, any quadrature formula, no matter how the weights $p _ {k i }$ and the nodes $x _ {k}$ are chosen, will exactly integrate an infinite-dimensional family of functions.