Integration of Interval Functions
Technical summary rept.
WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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The lack of a theory of integration of interval functions has hampered the development of interval methods for the solution of integral equations, which constitute an important class of mathematical models of real situations. This report provides such a theory by defining integrals of interval functions to be an interval with the limits of integrals of step-functions respectively less and greater than the integrand as endpoints. As these lower and upper Darboux integrals of basic integration theory always exist, it follows that all interval functions are integrable. The simplicity of this theory is due to the fact that intervals, rather than real numbers only, are accepted as values of integrals, even if the integrand is a real function called a degenerate interval function in this context. Some useful properties of the interval integral defined in this report are derived, and its potential usefulness in connection with the numerical solution of integral equations is indicated.
- Numerical Mathematics