A local discrepancy theorem
This excellent paper continues important work on discrepancy theory with applications on Pollaczek polynomials. The main result gives an estimate using local discrepancy (for a comparable result using global dicrepancy cf. \textit{H.-P. Blatt} and \textit{H. N. Mhaskar}, Ark. Mat. 31, No. 2, 219-246 (1993; Zbl 0797.30032)]. NEWLINENEWLINENEWLINELet \(\sigma\) be a signed Borel measure on \(I=[-1,1]\) and let \(J=[a,b]\subset I\) be a subinterval. The local discrepancy of \(\sigma\) on \(J\) is defined by NEWLINE\[NEWLINE D_J[\sigma]=\sup_{[c,d]\subset J} |\sigma([c,d])|.NEWLINE\]NEWLINE The main result concerns the signed measure \(\sigma=\sigma^{+}-\sigma^{-}\) with \(\sigma^{\pm}\) positive unit Borel measures on \(I\), \(\sigma^{+}(J)\leq M(b-a)^{\gamma}\) for certain \(M>0, 0<\gamma\leq 1\) which are independent of \(a, b\): NEWLINE\[NEWLINE D_J[\sigma]\leq 4\varepsilon\left\{2+\log{\left(1+{b-a\over \delta}\right)}\right\} + 16\delta\log{{1\over \delta}}+2M\delta^{\gamma} NEWLINE\]NEWLINE for all \(\delta\) with \(0<\delta<{1\over 2}\) and \(\varepsilon\) defined by NEWLINE\[NEWLINE \varepsilon=\max_{z\in B(\delta,J)} |U^{\sigma}(z)|. NEWLINE\]NEWLINE Here \(B(\delta,J)=\{z=x+iy\mid a-\delta\leq x\leq b+\delta\), \(|y|\geq \delta^2\}\) and NEWLINE\[NEWLINE U^{\sigma}(z)=\int\log{{1\over |z-t|}}d\sigma(t) NEWLINE\]NEWLINE is the logarithmic potential induced by \(\sigma\). NEWLINENEWLINENEWLINEIntroducing the generalized Pollaczek weight (GP) \(W\) by supp\((W)=I\) and NEWLINE\[NEWLINE W(x)=\exp{\left(-{\sqrt{1-x^2}\over |P(x)|}\right)},\quad -1\leq x\leq 1, NEWLINE\]NEWLINE where \(P\) is a polynomial having all its zeros simple and on \(I\), application of the local discrepancy bound leads to: NEWLINENEWLINENEWLINELet \(\tau\) be a positive Borel measure on \(I\), \(W\) a GP-weight and let the Radon-Nikodým derivative \(\tau'\) satisfy \(\tau'(x)\geq W(x)\) a.e. in \(I\). Then: NEWLINE\[NEWLINE D_{[a,b]}[\nu_n(\tau)-\mu]\leq c{(\log{n})^2\over n} (n\geq 2) NEWLINE\]NEWLINE for any closed interval \([a,b]\subset I\setminus\{z_j\}_{j=1}^m\). Here \(\nu_n(\tau)\) is the zero-counting measure of the orthonormal polynomials w.r.t. \(\tau\), \(\mu\) is the equilibrium measure on \(I\) NEWLINE\[NEWLINE\text{d}\mu(x)={\text{d}x\over \pi\sqrt{1-x^2}} NEWLINE\]NEWLINE and the \(z_j\) are the (real) zeroes of \(P(x)\). NEWLINENEWLINENEWLINEApplication of the global discrepancy estimate leads in the case of the ordinary Pollaczek weight to NEWLINE\[NEWLINE {c\over \sqrt{n}}\leq D[\nu_n(\tau)-\mu]\leq c_1{\log{n}\over \sqrt{n}}\quad (n\geq 2). NEWLINE\]NEWLINE The new result using the local discrepancy estimate, i.e., \(c(\log{n})^2/n\), thus is a considerable improvement in the case that \(\tau'(x)\) is the ordinary Pollaczek weight.
scientific article; zbMATH DE number 1663806
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