Advances in Chebyshev quadrature

A characterization of positive quadrature formulae

“…There are other weight functions that have property T (cf. [4,11] and the references therein). As an example, we mention w0(x) = ( 1 -x2)-1 ( 1+bx), \b\ < 1/4 [11].…”

“…Moreover, it is the only weight function all of whose Gaussian quadratures is equally weighted (cf. [4]). There are other weight functions that have property T (cf.…”

On a theorem of C. Posse concerning Gaussian quadrature of Chebyshev type

“…Chebyshev formulae have been investigated for more than a hundred years (see, e.g., [6] and the references cited therein). They were first considered by Chebyshev [3], who showed that the linear functional Ta>ß>y, (2) W/l:=af /, fion ,dx> a,ß,y£R, Jß y/\x-ß\\x-y\ admit («, 2n-1) Chebyshev formulae for each « £ N, i.e., that each Gaussian quadrature formula is of Chebyshev type.…”

“…Let us additionally note that the linear functionals S" ?i, Recently, using methods of complex analysis and Faber polynomials, Peherstorfer [8] has proved the surprising result, that Tajy and S" ^ also are the only (positive) linear functionals on C[a, b], admitting (1,1) and («, « + 1) Chebyshev formulae for each « £ N\{1} . For other improvements of Posse's result see, e.g., [6,5,9].…”

Weight functions for Chebyshev quadrature

“…The first example other than wo is due to Ullman [8]. He proved that gives an infinite one-parameter family of weight functions having property T. In a more recent paper, Byrd and Stalla [1] provided another class of weight functions with property T, (3) w{x) = Wo{x)2p~+ï+lc, P-L Further examples and related results can be found in [2], [4], [5], and [7]. But the problem of characterizing all weight functions with property T is still open, and an attempt of finding the complete solution to this problem would seem to be too ambitious.…”