On three‐dimensional rotational averages

On Asymptotic Differentiability of Averages

Exploiting Nanoroughness on Holographically Patterned ThreeDimensional Photonic Crystals

The fabrication of three-dimensional (3D) diamond photonic crystals with controllable nanoroughness (≤120 nm) on the surface from epoxy-functionalized cyclohexyl polyhedral oligomeric silsesquioxanes (POSS) is reported. The nanoroughness is generated on the 3D network due to microphase separation of the polymer chain segments in a nonsolvent during the rinsing step in holographic lithography pr...

Rademacher Averages on Noncommutative Symmetric Spaces

Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let (εk)k≥1 be a Rademacher sequence, on some probability space Ω. For finite sequences (xk)k≥1 of E(M), we consider the Rademacher averages ∑ k εk ⊗ xk as elements of the noncommutative function space E(L(Ω)⊗M) and s...

On Spherical Averages of Radial Basis Functions

A radial basis function (RBF) has the general form

Problems on Averages and Lacunary Maximal Functions

We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H to L bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an Lp regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L boundedness of lacunary maximal ...