in this article, by using chebyshev’s polynomials and chebyshev’s expansion, we obtain the best uniform polynomial approximation out of p2n to a class of rational functions of the form (ax2+c)-1 on any non symmetric interval [d,e]. using the obtained approximation, we provide the best uniform polynomial approximation to a class of rational functions of the form (ax2+bx+c)-1 for both cases b2-4a...

Let S ⊂ R be compact with #S = ∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (Pn) ∞ n=0 with deg Pn = n such that every f ∈ C(S) has a unique representation f = ∑∞ i=0 αiPi and call such a basis Faber basis. In the special case of S = Sq = {qk; k ∈ N0} ∪ {0}, 0 < q < 1, we prove the existence of such a basis. A special orthonormal F...

Bounds are proved for the Stieltjes polynomial En+1, and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials Pn. This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials Gn. Applying these results, convergence theorems are proved for the Lagrange interpolation process...

A new approach to the numerical solution of systems of first-order ordinary differential equations is given by finding local Galerkin approximations on each subinterval of a given mesh of size h. One step at a time, a piecewise polynomial, of degree n and class C°, is constructed, which yields an approximation of order 0(A*") at the mesh points and 0(A"+1) between mesh points. In addition, the ...

Recently, the continuous Jacobi transform and its inverse are defined and studied in [i] and [2]. In the present work, the transform is used to derive a series representation for the Jacobi functions Pe’) (x) -% -1/2. The case e = =0 yields a representation for the Legendre functions and has been dealt with zn [3]. When I is a positive integer n, the representation redu...

The flux reconstruction (FR) methodology has proved to be an attractive approach to obtaining high-order solutions to hyperbolic partial differential equations. However, the utilization of somewhat arbitrarily defined correction polynomials in the application of these schemes, while adding some flexibility, detracts from their ease of implementation and computational efficiency. This paper desc...

We consider the multifractal structure of the Bernoulli convolution νλ, where λ−1 is a Salem number in (1, 2). Let τ(q) denote the L spectrum of νλ. We show that if α ∈ [τ ′(+∞), τ ′(0+)], then the level set E(α) := { x ∈ R : lim r→0 log νλ([x− r, x+ r]) log r = α } is non-empty and dimH E(α) = τ∗(α), where τ∗ denotes the Legendre transform of τ . This result extends to all self-conformal measu...

We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function ei!x on the interval [ 1, 1]. We show that such a rule attains high asymptotic order, in the sense that the quadrature error quickly decreases as a function of the frequency !. However, accuracy is maintained for all values of ! and in particular the rule elegantly reduces ...

The use of polynomial functions to describe the average growth trajectory and covariance functions of Nellore and MA (21/32 Charolais+11/32 Nellore) young bulls in performance tests was studied. The average growth trajectories and additive genetic and permanent environmental covariance functions were fit with Legendre (linear through quintic) and quadratic B-spline (with two to four intervals) ...