Second-Order Linear Differential Equations with Solutions in Analytic Function Spaces

This research is concerned with second-order linear differential equation f′′+A(z)f=0, where A(z) is an analytic function in the unit disc. On the one hand, some sufficient conditions for the solutions to be in α-Bloch (little α-Bloch) space are found by using exponential type weighted Bergman reproducing kernel formula. On the other hand, we find also some sufficient conditions for the solutions to be in analytic Morrey (little analytic Morrey) space by using the representation formula.

Reproducing kernels of Sobolev spaces on ℝd and applications to embedding constants and tractability

The standard Sobolev space [Formula: see text], with arbitrary positive integers [Formula: see text] and [Formula: see text] for which [Formula: see text], has the reproducing kernel [Formula: see text] for all [Formula: see text], where [Formula: see text] are components of [Formula: see text]-variate [Formula: see text], and [Formula: see text] with non-negative integers [Formula: see text]. We obtain a more explicit form for the reproducing kernel [Formula: see text] and find a closed form for the kernel [Formula: see text]. Knowing the form of [Formula: see text], we present applications on the best embedding constants between the Sobolev space [Formula: see text] and [Formula: see text], and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in [Formula: see text], whereas worst case integration errors of algorithms using [Formula: see text] function values are also exponentially small in [Formula: see text] and decay at least like [Formula: see text]. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.

Inhomogeneous shearlet coorbit spaces

In this paper, we establish inhomogeneous coorbit spaces related to the continuous shearlet transform and the weighted Lebesgue spaces [Formula: see text] for certain weights [Formula: see text]. We present an inhomogeneous shearlet frame for [Formula: see text] which gives rise to a reproducing kernel [Formula: see text] that is not contained in the space [Formula: see text]. We show that the inhomogeneous shearlet coorbit spaces are Banach spaces by introducing a generalization of the approach of Fornasier, Rauhut and Ullrich.