We construct a uniformly stable family of bases for tensor product spline approximation on domains in R d . These bases are derived from the standard B-spline basis by normalization with respect to the L p -norm and a selection process relying on refined estimates for the de Boor-Fix functionals.

We study the classical and quantum mechanics of a three-dimensional stadium billiard. It consists of two quarter cylinders that are rotated with respect to each other by 90 degrees, and it is classically chaotic. The billiard exhibits only a few families of nongeneric periodic orbits. We introduce an analytic method for their treatment. The length spectrum can be understood in terms of the nongeneric and unstable periodic orbits. For unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory. For equal radii the billiard exhibits an additional symmetry. We investigated the effects of symmetry breaking on spectral properties. Moreover, for equal radii, we observe a small deviation of the level statistics from random matrix theory. This led to the discovery of stable and marginally stable orbits, which are absent for unequal radii.

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We provide explicit formulas for the joint spectral radius of certain classes of pairs of real matrices of order 2 with equal spectral radius.