Tight frame oversampling and its equivalence to shift-invariance of affine frame operators

Let generate a tight affine frame with dilation factor , where , and sampling constant (for the zeroth scale level). Then for , oversampling (or oversampling by ) means replacing the sampling constant by . The Second Oversampling Theorem asserts that oversampling of the given tight affine frame generated by preserves a tight affine frame, provided that is relatively prime to (i.e., ). In this paper, we discuss the preservation of tightness in oversampling, where (i.e., and ). We also show that tight affine frame preservation in oversampling is equivalent to the property of shift-invariance with respect to of the affine frame operator defined on the zeroth scale level.

     

Localized nonlinear functional equations and two sampling problems in signal processing

Let 1 ≤ p ≤ ∞. In this paper, we consider solving a nonlinear functional equation f (x) = y, where x, y belong to ? ^p and f has continuous bounded gradient in an inverse-closed subalgebra of ? (?²), the Banach algebra of all bounded linear operators on the Hilbert space ? ². We introduce strict monotonicity property for functions f on Banach spaces ? ^p so that the above nonlinear functional equation is solvable and the solution x depends continuously on the given data y in ? ^p . We show that the Van-Cittert iteration converges in ? ^p with exponential rate and hence it could be used to locate the true solution of the above nonlinear functional equation. We apply the above theory to handle two problems in signal processing: nonlinear sampling termed with instantaneous companding and subsequently average sampling; and local identification of innovation positions and qualification of amplitudes of signals with finite rate of innovation.     

Hydrothermal synthesis of hollow silica spheres under acidic conditions

It is well-known that silica can be etched in alkaline media or in a unique hydrofluoric acid (HF) solution, which is widely used to prepare various kinds of hollow nanostructures (including silica hollow structures) via silica-templating methods. In our experiments, we found that sto?ber silica spheres could be etched in generic acidic media in a well-controlled way under hydrothermal conditions, forming well-defined hollow/rattle-type silica spheres. Furthermore, some salts such as NaCl and Na(2)SO(4) were found to be favorable for the formation of hollow/rattle-type silica spheres.