The Snake Theorem for Unisolvent Families

Applications of the Snake Theorem.

“…Keener suspects th a t this assumption can be substantially weakened or even removed in this case. One may wish to affirm this suspicion, which would not only strengthen the results of [7], but also the results presented in this thesis.…”

“…Each of these problems were suggested by Keener in [7]. In the cases of approximation with constraints outside the interval of approximation and recipro cal approximation, we consider results in the literature.…”

“…In the case of maximin approximation, to our knowledge, all results are new. One theorem which uses this idea is presented in [7], The theorem may be viewed as a version of K arlin's oscillation theorem using a unisolvent approximating family of functions, paired with a related uniqueness theorem from [6, 7].…”

“…However, it has yet to be demon strated for all choices of varisolvent H. To ensure th at every best approximation p* to every / in [7] does not give rise to a constant error curve with respect to / , Keener introduces the following assumption. (A brief discussion of the necessity of this assumption when approximating using elements of unisolvent families appears in Chapter 5.…”

“…The necessity of Assumption A when approximating using elements of uni solvent families is questioned in [7]. Keener suspects th a t this assumption can be substantially weakened or even removed in this case.…”