Analysis and Applications

This paper examines two properties of refinable functions. The first provides conditions under which a refinable function is strictly positive and the other presents a conjecture about cardinal interpolation by refinable functions. The conjecture is supported by some analysis and also computation.

Several asymptotic expansions of parabolic cylinder functions are discussed and error bounds for remainders in the expansions are presented. In particular, Poincaré-type expansions for large values of the argument z and uniform expansions for large values of the parameter are considered. It is shown how expansions can be derived by using the differential equation, and, for a special case, how an integral representation can be used. The expansions are based on those given in Olver (1959) and on modifications of these expansions given in Temme (2000). Computer algebra techniques are used for obtaining representations of the bounds and for numerical computations.

We consider a subRiemannian geometry induced by a step 3 subelliptic partial differential operator in ℝ³. Our main result is the characterization of a canonical submanifold through the origin, all of whose points are connected to the origin by infinitely many (subRiemannian) geodesics.