Staircase Extensions of Directional Integral Inequalities in Normed Vector Spaces

This paper establishes directional integral inequalities and associated error estimates along ordered staircases generated by a finite family of directions in a real normed vector space. The endpoint increment along the corresponding staircase is represented as a sum of directional integrals. Exact integral identities are obtained for the errors of the trapezoid and midpoint approximations to this increment. When the absolute value of the derivative of each directional function is convex, both approximations satisfy error estimates with coefficient 1/8. Consequences are given for the difference between the two approximations, for uniformly bounded directional derivatives, and for the case in which the directional functions are affine. Applications to successive coordinate updates, quadratic growth, anharmonic potentials, and finite directional staircases in ℓ2 are presented.