Norms of projectors onto spaces with Riesz bases (Q1576458)

Let \(\Omega\subseteq \mathbb R^d, d=1,2,\dots{}\) be a closed connected set, let \(X\) be a linear space of bounded, real-valued functions on \(\Omega\) and \(\langle\cdot,\cdot\rangle :X\times X\rightarrow\mathbb R\) be a semi-definite inner product on \(X.\) Let \(\mathcal S\subset C(\Omega)\cap X\) be a Hilbert subspace of \(X\) with respect to \(\langle\cdot,\cdot\rangle.\) Then for each \(f\in X\) there exists a unique best approximant \(u_f\in \mathcal S.\) Let \(P:X\rightarrow\mathcal S\) be the metric projection defined by \(Pf=u_f\) and let \(\|P\|_\infty:= \sup_{f\neq 0}\|Pf\|_{L_\infty(\Omega)}/\|f\|_{L_\infty(\Omega)}.\) Under certain conditions the author obtains upper bounds for \(\|P\|_\infty,\) in terms of the properties of a Riesz bases in \(\mathcal S.\) Better upper bounds are obtained in the particular case when \(X=C[a,b], \mathcal S= \mathcal S_r(\Delta)\subset C^{r-2}[a,b]\) is the space of polynomial splines of order \(r\) with knots \(\Delta: a=x_0<\dots{}<x_{n+1}=b\) and \(\langle f,g\rangle=\int_a^b f(x)g(x) dx.\) The case when \(X=L_\infty[a,b],\mathcal S=\mathcal S_r(\Delta)\) and \(\langle f,g\rangle=\sum_{i=1}^nf(t_i)g(t_i),\) is also treated.