Embeddings of Müntz spaces: the Hilbertian case (Q2838958)

Given a strictly increasing sequence \(\Lambda = (\lambda_n)_n\) of positive real numbers with \(\sum_n {1 \over \lambda_n} < \infty\), the \textit{Müntz space} \(M^p_\Lambda\) is the closed linear span in \(L^p ([0, 1])\) of the monomials \(x^{\lambda_n}\). The authors investigate whether \(M^p_\Lambda\) can be embedded in \(L^p (\mu)\) for some finite positive Borel measure \(\mu\) on \([0, 1]\). Their results concern mainly the case \(p = 2\) and use the following function \(\psi\): let \(\hat{\mathcal P}_n\) be the linear span of the \(x^{\lambda_m}\) for \(m \neq n\), and \(d_n = \text{dist}\, (x^{\lambda_n}, \hat{\mathcal P}_n)\); then \(\psi\) is defined as \(\psi (x) = \sum_{n \geq 1} d_n^{- 1} x^{\lambda_n}\). The authors prove that if \(\psi \in L^2 (\mu)\), one has an embedding \(i_\mu^2 : M^2_\Lambda \hookrightarrow L^2 (\mu)\) and \(\| i_\mu^2 \| \leq \| \psi\|_{L^2 (\mu)}\). Using that, they obtain that if \(\mu\) is supported by a compact subset of \([0, 1[\), then \(i_\Lambda^2\) is in all Schatten classes \(S_q\), \(q > 0\). It follows that if \(\Psi (x) = \psi (x^{1/4}) \, \psi ' (x^{1/4})\) and \(\Psi \in L^2 (\mu)\), then \(i_\mu^2\) is Hilbert-Schmidt.NEWLINENEWLINEFor lacunary sequences \(\Lambda\), they prove that \(M^2_\Lambda\) embeds into \(L^2 (\mu)\) if (and only if, with an additional technical assumption on \(\Lambda\)) \(\mu\) is \textit{sublinear}. They say that a measure \(\mu\) on \([0, 1]\) is sublinear if, for some \(C > 0\), \(\mu ([1 - \varepsilon, 1]) \leq C \varepsilon\) for all \(0 < \varepsilon < 1\). When \(\mu ([1 - \varepsilon, 1]) = o\, (\varepsilon)\), \(i_\mu^2\) is compact. If \(\mu ([1 - \varepsilon, 1]) = O\, (\varepsilon^\alpha)\) for some \(\alpha > 1\) (\(\Lambda\) may be assumed to be a finite union of lacunary sequence), then \(i_\mu^2\) is in all Schatten classes \(S_q\), \(q > 0\).NEWLINENEWLINEEmbeddings of \(M^p_\Lambda\) into \(L^p (\mu)\) for \(1 < p < 2\) are obtained via a suitable version of the Riesz-Thorin theorem.NEWLINENEWLINEExamples are given such that: 1) \(M_\Lambda^2 \hookrightarrow L^2 (\mu)\) but \(M_\Lambda^1 \not \hookrightarrow L^1 (\mu)\), and such that: 2) given any \(0 < r < q\), \(i_\Lambda^2 \in S_q\) but \(i_\mu^2 \notin S_r\).