New cubature formulas and Hermite–Hadamard type inequalities using integrals over some hyperplanes in the d -dimensional hyper-rectangle

Abstract

This paper focuses on the problem of approximating a definite integral of a given function f when, rather than its values at some points, a number of integrals of f over certain hyper- plane sections of a d -dimensional hyper-rectangle C d are only available. We develop several families of integration formulas, all of which are a weighted sum of integrals over some hyperplane sections of C d , and which contain in a special case of our result multivariate analogs of the midpoint rule, the trapezoidal rule and Simpson’s rule. Basic properties of these families are derived. In particular, we show that they satisfy a multivariate version of Hermite–Hadamard inequality. This latter does not require the classical convexity assump- tion, but it has weakened by a different kind of generalized convexity. As an immediate consequence of this inequality, we derive sharp and explicit error estimates for twice con- tinuously differentiable functions. More precisely, we present explicit expressions of the best constants, which appear in the error estimates for the new multivariate versions of trapezoidal, midpoint, and Hammer’s quadrature formulas. It is shown that, as in the uni- variate case, the constant of the error in the trapezoidal cubature formula is twice as large as that for the midpoint cubature formula, and the constant in the latter is also twice as large as for the new multivariate version of Hammer’s quadrature formula. Numerical ex- amples are given comparing these cubature formulas among themselves and with uniform and non-uniform centroidal Voronoi cubatures of the standard form, which use the values of the integrand at certain points.