Abstract
We consider a finite class of weighted quadratures with the weight function x − 2 a ( 1 + x 2 ) − b on (−∞,∞), which is valid only for finite values of n (the number of nodes). This means that classical Gauss-Jacobi quadrature rules cannot be considered for this class, because some restrictions such as {maxn}≤a+b−1/2, a<1/2, b>0 and ( − 1 ) 2 a =1 must be satisfied for its orthogonality relation. Some analytic examples are given in this sense.MSC:41A55, 65D30, 65D32.
Highlights
The differential equation x px + q n(x) + x rx + s n(x)– n r + (n – )p x + – (– )n s/ n(x) = ( )was introduced in [ ], and it was established that the symmetric polynomials rs n(x) = Sn p x q [n/ ] [n/ ] = k k=[n/ ]–(k+ ) ( i + (– )n+ + [n/ ])p + r ( i + (– )n+ + )q + s xn– k i=are a basis solution of it
Was introduced in [ ], and it was established that the symmetric polynomials rs n(x) = Sn p x q
If this equation is written in a self-adjoint form, the firstorder equation d x px + q W (x) = rx + s W (x)
Summary
By solving equation ( ), the polynomial solution of monic type is derived According to [ ], these polynomials are finitely orthogonal with respect to the second kind of beta weight function x– a( + x )–b on (–∞, ∞) if and only if {max n} ≤ a + b – / , i.e., we have x– a ( + x )b The orthogonality property ( ) shows that the polynomials Sn( , , – a – b + , – a; x) are a suitable tool to finitely approximate the functions that satisfy the Dirichlet conditions [ – ].
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