Sampling Theory with Average Values on the Sierpinski Gasket

Abstract

In the case of some fractals, sampling with average values on cells is more natural than sampling on points. In this paper we investigate this method of sampling on SG and SG 3 . In the former, we show that the cell graph approximations have the spectral decimation property and prove an analog of the Shannon sampling theorem. We also investigate the numerical properties of these sampling functions and make conjectures which allow us to look at sampling on infinite blowups of SG. In the case of SG 3 , we show that the cell graphs have the spectral decimation property, but show that it is not useful for proving an analogous sampling theorem.2010 Mathematics Subject Classification. 28A80.if the limit exists. Since E m (u, u) is nondecreasing and nonnegative, this limit always exists in the case u = v. Definition 2.3. We say that a real-valued function u on SG has finite energy ifThe set of such functions is denoted dom E.We can use this to define a metric on SG.Definition 2.4. The resistance metric on SG is the function R :An important property of R(x, y) is that c|x−y| β ≤ R(x, y) ≤ C|x−y| β for some constants c and C, where β = log(5/3)/ log(2).We also note the following fact that will be useful in the subsequent section Theorem 2.5. The space dom E modulo the constant functions is a Hilbert space with inner product E(u, v).

Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions

“…In [28], Strichartz introduced the notions of cell graphs and cell graph energies for the Sierpinski gasket SG, and provided an equivalent definition of Laplacians on SG, instead of using vertex graphs and vertex graph energies as we usually did in fractal analysis. See [25,30] for some interesting works using related considerations.…”

Quantitative estimates for nonlinear sampling Kantorovich operators

“…In particular, sampling type operators (in their multivariate version) can be used in order to reconstruct and approximate images, see e.g. [37,29,18,19].…”

Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression

“…In order to achieve good approximations, one needs a smooth signal f (see [9]). Since in the real world applications signals are not very regular it has been natural to weaken the assumptions on f ; for this reason the sinc-type kernels where later replaced by the generalized kernels that could provide good approximations for signals that do not need to be smooth (see, e. g., [1,2,8,10,35,36,37] and see also [4,23] for the case of multivariate signals). Furthermore, recent studies considered Kantorovich variants of sampling operator acting on functions defined on Orlicz spaces a Borel probability measure µ.…”