S. R. Nasyrov's problem of approximation by simple partial fractions on an interval (Q6569629)

Let \(C\) be the unit circle and let \(SF(C)\) denote the set of simple partial fractions with poles in \(C\). This interesting paper deals with the questions of finding conditions on a Borel measure \(\mu\) on \([-1,1]\) so that \(SF(C)\) is dense in the complex space \(L^p([-1,1], d\mu)\) for \(1<p<\infty\). It is known that this fails for the Lebesgue measure, see [\textit{M. A. Komarov}, Constr. Approx. 58, No. 3, 551--563 (2023; Zbl 1534.41008)]. Theorem 1 here gives a positive answer, when \(\frac{1}{x\pm 1} \in L^p([-1,1],d\mu)\). The proof uses the uniform convexity of \(L^p\) spaces and an interesting result from [\textit{P. A. Borodin}, Izv. Math. 78, No. 6, 1079--1104 (2014; Zbl 1316.46015); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, no. 6, 21--48 (2014)], that for a closed additive group \(G\) in a uniformly convex space and for any Lipschitz function \(\phi:[-1,1] \rightarrow G\), a closed real linear span of \(\{\phi(a)-\phi(b): a,b \in [-1,1]\}\) is contained in \(G\).