Cantor type functions in non-integer bases

Bases in Banach spaces of smooth functions on Cantor-type sets

We suggest a Schauder basis in Banach spaces of smooth functions and traces of smooth functions on Cantor-type sets. In the construction, local Taylor expansions of functions are used. c ⃝ 2011 Elsevier Inc. All rights reserved.

Non-differentiability points of Cantor functions

Hausdorff Dimension and Hausdorff Measure for Non-integer based Cantor-type Sets

We consider digits-deleted sets or Cantor-type sets with β-expansions. We calculate the Hausdorff dimension d of these sets and show that d is continuous with respect to β. The d-dimentional Hausdorff measure of these sets is finite and positive.

Greedy and Quasi-greedy Expansions in Non-integer Bases

We generalize several theorems of Rényi, Parry, Daróczy and Kátai by characterizing the greedy and quasi-greedy expansions in non-integer bases.

Expansions in Non-integer Bases: Lower, Middle and Top Order

Let q ∈ (1, 2); it is known that each x ∈ [0, 1/(q− 1)] has an expansion of the form x = ∑ ∞ n=1 anq −n with an ∈ {0, 1}. It was shown in [3] that if q < ( √ 5 + 1)/2, then each x ∈ (0, 1/(q − 1)) has a continuum of such expansions; however, if q > ( √ 5 + 1)/2, then there exist infinitely many x having a unique expansion [4]. In the present paper we begin the study of parameters q for which th...