Bases in some spaces of Whitney functions
This paper is devoted to the construction of bases in spaces of Whitney jets on certain Cantor-type sets. The Cantor-type set \(K(\gamma)\) is defined by a summable sequence of positive numbers \((\gamma_s)_{s=1}^\infty\). More precisely, this sequence is used to define polynomials \(P_s\), which in turn define sets \(E_s:=\{x\in\mathbb R:P_{2^{s+1}}(x)\leq 0\}\) which are a union of disjoint intervals. Finally, the set \(K(\gamma)\) is defined as the intersection of all these sets \(E_s\). The zeros of the polynomials \(P_s\) are ordered in a particular way into a sequence \((x_k)_{k=0}^\infty\). Using this sequence, the authors define the functions \[ e_0(x):=1\quad\text{and}\quad e_N(x) := \prod_{k=1}^N(x-x_k) \] for \(x\in K(\gamma)\). The authors exhibit a condition on the sequence \((\gamma_s)_{s=0}^\infty\) which ensures that \((e_N)_{N=0}^\infty\) forms a Schauder basis of the space \(\mathcal E(K(\gamma))\) of Whitney jets on \(K(\gamma)\). For a general sequence \((\gamma_s)_{s=0}^\infty\), however, it may happen that this sequence no longer is a basis. For this case, the authors provide a construction to produce a Schauder basis of \(\mathcal E(K(\gamma))\) from the sequence \((e_N)_{N=0}^\infty\). The paper concludes with a discussion of the possibility of defining a continuous extension operator \(\mathcal E(K(\gamma))\to\mathcal C^\infty(\mathbb R)\) by extending the basis elements.
scientific article; zbMATH DE number 6841341