Approximation and existence of Schauder bases in Müntz spaces of \(L_{1}\) functions (Q276809)

The author considers Müntz spaces defined on the finite interval \([0,1]\) and denoted by \(M_{\Lambda,p}\) as the completion (in the \(L^p([0,1])\)-norm) of the linear span of the monomials \(t^\lambda\), where \(\lambda\in \Lambda\) and \(\Lambda\) is an increasing sequence in \((0,\infty)\) satisfying the gap condition \(\inf_k (\lambda_{k+1}-\lambda_k)>0\) and the Müntz condition \(\sum_{k=1}^\infty \lambda_k^{-1}<\infty\). He shows that \(M_{\Lambda,p}\) and \(M_{\Gamma,p}\) are isomorphic for \(1\leq p<\infty\) whenever \(\Lambda=\{(\lambda_k)\}\), \(\Gamma=\{(\gamma_k)\}\) with \(\lambda_k\leq \gamma_k\) for all \(k\in \mathbb N\) and \(\sup_k (\beta_k-\lambda_k)<\big(\sum_{k=1}^\infty \lambda_k^{-1}\big)^{-1}\). The author then analyzes the case \(p=1\) showing two types of results. First it is shown that for each \(0<\gamma<1\) there exists \(\beta>0\) such that the space \(M_{\Lambda,1}\) is contained in the Weil-Nagy class \(W^\gamma_\beta L^1\) consisting of the functions in \(L^1([0,1])\) such that the \((\gamma, \beta)\)-Weil derivative \(f^\gamma_\beta\) belongs to \(L^1([0,1])\), where NEWLINE\[NEWLINEf^\gamma_\beta(x)=\sum_{k=1}^\infty k^\gamma\big(a_k(f)\cos (2\pi k x+ \beta\pi/2)+ b_k(f)\sin (2\pi k x+ \beta\pi/2)\big)NEWLINE\]NEWLINE and \(a_k(f)\) and \(b_k(f)\) are the Fourier coefficients of \(f\). Second it is proved that \(M_{\Lambda,1}\) has a Schauder basis.