Xingde Dai

The existence of subspace wavelet sets

Let H be a reducing subspace of L2(Rd), that is, a closed subspace of L2(Rd) with the property that f(Amt - l) ∈ H for any f ∈ H, m ∈ Z and l ∈ Zd, where A is a d × d expansive matrix. It is known that H is a reducing subspace if and only if there exists a measurable subset M of Rd such that AtM = M and F(H) = L2(Rd) ċ χM. Under some given conditions of M, it is known that there exist A-dilation subspace wavelet sets iwith respect to H. In this paper, we prove that this holds in general.

THE MINIMUM OF KNOT ENERGY FUNCTIONS

In this paper we discuss some fundamental issues regarding knot energy functions. These include the existence of minimum values of energy functions of smooth knots and energy functions of polygonal knots within a knot type, the convergence of these minimum values in the case of polygonal knot energy and the convergence of the corresponding polygons where these minimum values are attained. When the polygonal knot energy is derived from a smooth knot energy, will the minimal polygonal knot energies converge to the infimum of the smooth knot energy? Do the corresponding polygons converge to a smooth knot at which the smooth energy achieves its minimal value? We show that one cannot expect these to be true in general and outline certain conditions that would ensure a positive answer to some of the above questions.