Abstract

We give a formula for the logarithmic dimension of the generalized Cantor-type set $K$. In the case when the logarithmic dimension of $K$ is smaller than $1$, we construct a Faber basis in the space of Whitney functions $\mathcal{E}(K)$.

Highlights

  • This paper is the extension of [2] and [12]

  • In [2], the logarithmic dimension λ0 was suggested as the Hausdorff dimension corresponding to the function ψ(r)

  • We extend this result to the general case

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Summary

Introduction

This paper is the extension of [2] and [12]. In [2], the logarithmic dimension λ0 was suggested as the Hausdorff dimension corresponding to the function ψ(r). 1 r that defines the logarithmic measure. Some applications of the logarithmic dimension to the isomorphic classification of Whitney spaces were presented. In [12], the first author constructed bases in the spaces E(K2(αn)), where the set K2(αn) is obtained by the Cantor procedure with replacing each interval by two adjacent subintervals of equal length. As in [2], we consider more general Cantor-type sets K((Nαnn)) , see the definition below.

Logarithmic dimension for the generalized Cantor-type sets
Relation to potential theory and the extension property
Polynomial bases for small Cantor-type sets

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