Chebyshev Quadrature Formula Research Articles

The zeros of Faber polynomials generated by an 𝑚-star

It is shown that the zeros of the Faber polynomials generated by a regular m m -star are located on the m m -star. This proves a recent conjecture of J. Bartolomeo and M. He. The proof uses the connection between zeros of Faber polynomials and Chebyshev quadrature formulas.

On Gauss-Kronrod quadrature formulae of Chebyshev type

We prove that there is no positive measure $d\sigma$ on (a, b) such that the corresponding Gauss-Kronrod quadrature formula is also a Chebyshev quadrature formula. The same is true if we consider measures of the form $d\sigma (t) = \omega (t)dt$, where $\omega (t)$ is even, on a symmetric interval $( - a,a)$, and the Gauss-Kronrod formula is required to have equal weights only for n even. We also show that the only positive and even measure $d\sigma (t) = d\sigma ( - t)$ on $( - 1,1)$ for which the Gauss-Kronrod formula has all weights equal if $n = 1$, or has the form $\smallint _{ - 1}^1f(t)d\sigma (t) = w\Sigma _{\nu = 1}^nf({\tau _\nu }) + {w_1}f(1) + w\Sigma _{\mu = 2}^nf(\tau _\mu ^\ast ) + {w_1}f( - 1) + R_n^K(f)$ for all $n \geq 2$, is the Chebyshev measure of the first kind $d{\sigma _C}(t) = {(1 - {t^2})^{ - 1/2}}dt$.

On Chebyshev quadrature for ultraspherical weight functions

It is well-known that for the ultraspherical weight function (1-x2)λ-1/2 there exist no Chebyshev quadrature formulae in the strict sense having n nodes, where n is sufficiently large and λ>0, whereas on the other hand for λ=0 every Gaussian quadrature formulae is a Chebyshev formula in the strict sense. In this paper we study the open question of Chebyshev quadrature for λ <0. It is shown that there exists no Chebyshev quadrature formula in the strict sense having more than two nodes for λ≤λ0=-.30056... (for definition of λ0 see (1.8) below). Moreover, results are obtained for Chebyshev-type formulae and Chebyshev formulae of closed type. For the remaining values of λ (λ0<λ<0) numerical results are given.

A Model for Studying the Stress-Field around Simple Fractures in Human Patella

SummaryA mathematical model for estimating the stress-field in the vicinity of cracks in human patella has been studied. In conformity with experimental observations with regard to the mechanical properties of osseous tissues, elasticity and anisotropy of the patella have been paid due consideration. The present study being analytical, the problem is first formulated mathematically, and posed as a boundary value problem. Using Mellin transforms technique, the problem is reduced to solving a Fredholm integral equation which is treated numerically by employing Chebyshev quadrature formula. Numerical results are presented. It is suggested that the techniques used may also find application to engineering structures.

On Characterization of Functions by Their Gauss–Chebyshev Quadratures

If p is a polynomial, then all but a finite number of the Gauss–Chebyshev quadrature formulae for p are exact. The purpose of this paper is to establish a converse as well as some related results. In particular, a formula is given to recapture a function from its Gauss–Chebyshev quadratures.

On optimal Chebyshev-type quadratures

A well-known result of Bernstein states that a Chebyshev quadrature formula of the form $$\int\limits_{ - 1}^1 {f(x) dx = \frac{2}{n}\sum\limits_{k - 1}^n {f(t_k ) + R_n (f), t_k real,} }$$ can have algebraic degree of exactnessp=n only if 1?n?7 orn=9. The nodest k are necessarily symmetric with respect to the origin, so that in factp=2[n/2]+1. If symmetry of the nodes is imposed, it is known from work of Gautschi and Yanagiwara and others that next-to-highest algebraic degreep=2[n/2]?1, beyond the classical cases above, can be attained only whenn=8, 10, 11, and 13. For these values ofn, "optimal" formulas have been obtained which minimize |R n (x p+1)| among all symmetric Chebyschev quadratures of degreep=2[n/2]?1. We show here that these same formulas in fact minimize |R n (x i )| for eachi?p+1.

On weight functions for Chebyshev quadrature

It is shown that weight functions or moment sequences admitting Chebyshev quadrature for alln are scarce in a probabilistic sense. The method involves relating the quadrature problem to the moment spaces of Karlinet al. (1953, 1966), and obtaining the volume of that part of moment space which corresponds to weight functions for which Chebyshev quadrature formulas can be constructed. The result is that the probability of a moment sequence admitting Chebyshev quadrature of orders 1,2, ...,N approaches zero asN??. This is generalized to the case when arbitrary, rather than equal, positive weights are preassigned.

On Chebyshev-type quadratures

According to a result of S. N. Bernstein, n-point Chebyshev quadrature formulas, with all nodes real, do not exist when n = 8 n = 8 or n ≧ 10 n \geqq 10 . Modifications of such quadrature formulas have recently been suggested by R. E. Barnhill, J. E. Dennis, Jr. and G. M. Nielson, and by D. Kahaner. We establish here certain empirical observations made by these authors, notably the presence of multiple nodes. We also show how some of the quadrature rules proposed can be constructed by solving single algebraic equations, and we compute the respective nodes to 25 decimal digits. The same formulas also arise in recent work of P. Rabinowitz and N. Richter as limiting cases of optimal Chebyshev-type quadrature rules in a Hilbert space setting.

Almost-Interpolatory Chebyshev Quadrature

The requirement that a Chebyshev quadrature formula have distinct real nodes is not always compatible with the requirement that the degree of precision of an n-point formula be at least equal to n. This condition may be expressed as ${\left \| d \right \|_\nu } = 0,1 \leqq p$, where $d = ({d_1}, \cdots ,{d_n})$ with \[ {d_j} = \frac {{{\mu _0}(\omega )}}{n}\sum \limits _{i = 1}^n {x_i^j - {\mu _j}(\omega ),\quad j = 1,2, \cdots ,n,} \] ${\mu _j}(\omega ),j = 0,1, \cdots$, are the moments of the weight function $\omega$ used in the quadrature, and ${x_1}, \cdots ,{x_n}$ are the nodes. In those cases when ${\left \| d \right \|_2}$ does not vanish for a real choice of nodes, it has been proposed that a real minimizer of ${\left \| d \right \|_2}$ be used to supply the nodes. It is shown in this paper that, in such cases, minimizers of ${\left \| d \right \|_p},1 \leqq p < \infty$, always lead to formulae that are degenerate in the sense that the nodes are not all distinct. The results are valid for a large class of weight functions.

Almost-interpolatory Chebyshev quadrature

The requirement that a Chebyshev quadrature formula have distinct real nodes is not always compatible with the requirement that the degree of precision of an n-point formula be at least equal to n. This condition may be expressed as ‖ d ‖ ν = 0 , 1 ≦ p {\left \| d \right \|_\nu } = 0,1 \leqq p , where d = ( d 1 , ⋯ , d n ) d = ({d_1}, \cdots ,{d_n}) with \[ d j = μ 0 ( ω ) n ∑ i = 1 n x i j − μ j ( ω ) , j = 1 , 2 , ⋯ , n , {d_j} = \frac {{{\mu _0}(\omega )}}{n}\sum \limits _{i = 1}^n {x_i^j - {\mu _j}(\omega ),\quad j = 1,2, \cdots ,n,} \] μ j ( ω ) , j = 0 , 1 , ⋯ {\mu _j}(\omega ),j = 0,1, \cdots , are the moments of the weight function ω \omega used in the quadrature, and x 1 , ⋯ , x n {x_1}, \cdots ,{x_n} are the nodes. In those cases when ‖ d ‖ 2 {\left \| d \right \|_2} does not vanish for a real choice of nodes, it has been proposed that a real minimizer of ‖ d ‖ 2 {\left \| d \right \|_2} be used to supply the nodes. It is shown in this paper that, in such cases, minimizers of ‖ d ‖ p , 1 ≦ p &gt; ∞ {\left \| d \right \|_p},1 \leqq p &gt; \infty , always lead to formulae that are degenerate in the sense that the nodes are not all distinct. The results are valid for a large class of weight functions.

Concerning Gaussian–Chebyshev Quadrature Errors

We introduce a method for estimating errors in the Gaussian–Chebyshev quadrature formulas for functions of low order continuity. In particular, we improve an error bound obtained by Johnson and Riess.

Chebyshev interpolation and quadrature formulas of very high degree

All the zeros x 2 m , i , i = 1(1)2 m , of the Chebyshev polynomials T 2 m ( x ), m = 0(1) n , are found recursively just by taking n 2 n -1 real square roots. For interpolation or integration of ƒ( x ), given ƒ( x 2 m , i ), only x 2 m , i is needed to calculate (a) the (2 m - 1)-th degree Lagrange interpolation polynomial, and (b) the definite integral over [-1, 1], either with or without the weight function (1 - x 2 ) -1/2 , the former being exact for ƒ( x ) of degree 2 m +1 - 1.

Construction of Chebyshev Quadrature Formulae

Construction of Chebyshev Quadrature Formulae