Asymptotic estimation of oscillating functions using an interval calculus

Abstract

We consider the problem of estimating the asymptotic growth of functions defined by expressions involving exponentials, logarithms, algebraic operations and also sine functions. Modulo the assumption that zero-equivalence can be decided on the set of constant terms, an algorithm exists for the case when there are no trigonometric functions in the expression; see [21].

If we admit the sine function but with arguments restricted to expressions with finite limits then our objects of study lie within a Hardy field, i.e. a field of functions defined on deleted neighbourhoods of infinity in R which is closed under the operation of differentiation, [17], [3]. It follows that an order can be defined on the class of expressions by setting f>g whenever f(X)>g(X) for sufficiently large X. In the particular case under consideration, the methods of [21] can be adapted with relatively little change to give an algorithm for deciding > modulo an oracle for zero-equivalence of constant terms. This is presented in Section 2, following an introductory Section 1.

For the full class of expressions far less can be expected. The relation > cannot even be defined let alone decided. Moreover zero-equivalence of the expressions is known to be recursively undecidable, [15], [6] and so, since division belongs to the signature, it is likewise undecidable whether an expression is everywhere undefined as a function. However, in many cases, it is nonetheless possible to obtain useful bounds, for example in the form of upper and lower estimates. In Section 3, we present a calculus of intervals which is suitable for this purpose, and which differs from the standard interval-analysis treatment in a number of respects. This is applied in the next section where we show how to attach a finite set of intervals to any expression in such a way that if the expression defines a function, except perhaps for a countable set of values of X, the function values lie in the union of the intervals. The end points of the intervals will either be estimate forms similar to those of [21] or one of the symbols ∞, - ∞.

In Section 5, a possible improvement to deal with cancellation between trigonometric expressions is discussed.