An explicit representation for the Cotes numbers of Turán quadrature formulas based on the zeros of the Chebyshev polynomials
of the second kind and its asymptotic behavior are given. The asymptotic formula for the corresponding Christoffel type functions
is also provided.
Electrospray ionization mass spectrometry (ESI-MS) was used for the study of cyclization of organic chelating compounds (chelators). Four chelating compounds were studed: Symmetrical ethylenediaminediacetic acid (s-EDDA), Unsymmetrical ethylenediaminediacetic acid (u-EDDA), N-(2-hydroxyethyl) ethylenediaminetriacetic acid (HEDTA), and N-(2-hydroxyethyl)iminodiacetic acid (HEIDA). The chelators were cyclized with treatments of acids and heating. The open and cyclized form of the chelators were semi-quantified by both positive and negative ion modes ESI-MS. The kinetics of chelator cyclization was studied as a function of reaction temperature and the pH of the matrix. The cyclization of s-EDDA was found to be a pseudo-first order reaction in s-EDDA and overall second order. The cyclizations of HEIDA and HEDTA are reversible reactions. Higher temperature and lower pH favors cyclization.
The generalized Christoffel function λp,q,n(dμ;x) (0<p<∞, 0≦q<∞) with respect to a measure dμ on R is defined by
The novelty of our definition is that it contains the factor |t−x|q, which is of particular interest. Its properties are discussed and estimates are given. In particular, upper and lower bounds for generalized Christoffel functions with respect to generalized Jacobi weights are also provided.