Basis in the space of \(C^\infty\)-functions on a graduated sharp cusp
Let \((x_k)_{k\in\mathbb{N}}\), \((\delta_k)_{k\in\mathbb{N}}\), and \((\psi_k)_{k\in\mathbb{N}}\) be null-sequences of positive real numbers. Assume that for some \(C> 0\) they satisfy \(x_k\leq C\delta_k\) and \(\delta_k\geq C\delta_{k+1}\), \(k\in\mathbb{N}\). Define \(a_k:= x_k- \delta_k\), \(b_k:= x_k+ \delta_k\), \(I_k:= [a_k, b_k]\), \(a_0:= b_1\), \(K:= \{0\}\cup\bigcup^\infty_{k=1} I_k\times [-\psi_k, \psi_k]\), \(\psi: ]0,1[\to \mathbb{N}\), \(\psi(x):= \psi_k\) for \(x\in [a_{k+1}, a_k[\), and \[ \Omega_\psi:= \{(x, y)\in \mathbb{R}^2: 0< x< 1,\,| y|\leq \psi(x)\}. \] Assume that \(I_k\subset [0,1]\) and \(C\text{\,dist}(I_k, I_{k+1})\geq \delta_k\), \(k\in\mathbb{N}\). Denote by \({\mathcal E}(K)\) the space of all Whitney functions on \(K\) and let \[ C^\infty(\overline\Omega_\psi):= \{f\in C^\infty(\Omega_\psi): f^{(\alpha)}\text{ is uniformly continuous on }\Omega_\psi\text{ for each }\alpha\in\mathbb{N}^2_0\}. \] The set \(\overline\Omega_\psi\) has a ``cusp'' at the origin and varying the sequence \((\psi_k)_{k\in\mathbb{N}}\) obtains a continuum of pairwise nonisomorphic nuclear Fréchet spaces \(C^\infty(\overline\Omega_\psi)\). The authors use scaled Chebyshev polynomials to construct a basis in the space \({\mathcal E}(K)\). An argument of Mityagin is then applied to construct a continuous projection \(Q\) on \(C^\infty(\overline\Omega_\psi)\) with range \({\mathcal E}(K)\) and it is shown that this basis of \({\mathcal E}(K)\) can be enlarged to a basis of \(C^\infty(\overline\Omega_\psi)\). Moreover, it is proved that \({\mathcal E}(K)\) is isomorphic to the space \(s\) iff there is \(M> 0\) such that \(\psi_k> \delta^M_k\) for all \(k\in\mathbb{N}\).
scientific article; zbMATH DE number 2013178
scientific article