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nodes are not equally spaced and tend to cluster toward the end points ofthe interval.3.2Connection to Best Uniform Approxima-tionGiven a continuous functionfin [a, b], its best uniform approximationp*ninPnis characterized by an error,en=f-p*n, which equioscillates, as definedin (2.40), at leastn+ 2 times. Thereforeenhas a minimum ofn+ 1 zerosand consequently, there existsx0, . . . , xnsuch thatp*n(x0) =f(x0),p*n(x1) =f(x1),...p*n(xn) =f(xn),(3.15)In other words,p*nis the polynomial of degree at mostnthat interpolatesthe functionfatn+ 1 zeros ofen. Of course, we do not constructp*nbyfinding these particularn+1 interpolation nodes. A more practical questionis: given (x0, f(x0)),(x1, f(x1)), . . . ,(xn, f(xn)), wherex0, . . . , xnare distinctinterpolation nodes in [a, b], how close ispn, the interpolating polynomial ofdegree at mostnoffat the given nodes, to the best uniform approximationp*noffinPn?To obtain a bound forkpn-p*nk∞we note thatpn-p*nis a polynomial ofdegree at mostnwhich interpolatesf-p*n. Therefore, we can use Lagrangeformula to represent itpn(x)-p*n(x) =nXj=0lj(x)(f(xj)-p*n(xj)).(3.16)It then follows thatkpn-p*nk∞≤Λnkf-p*nk∞,(3.17)
42CHAPTER 3.INTERPOLATIONwhereΛn= maxa≤x≤bnXj=0|lj(x)|(3.18)is called theLebesgue Constantand depends only on the interpolation nodes,not onf. On the other hand, we have thatkf-pnk∞=kf-p*n-pn+p*nk∞≤ kf-p*nk∞+kpn-p*nk∞.(3.19)Using (3.17) we obtainkf-pnk∞≤(1 + Λn)kf-p*nk∞.(3.20)This inequality connects the interpolation errorkf-pnk∞with the bestapproximation errorkf-p*nk∞. What happens to these errors as we increasen? To make it more concrete, suppose we have a triangular array of nodesas follows:x(0)0x(1)0x(1)1x(2)0x(2)1x(2)2...x(n)0x(n)1. . .x(n)n...(3.21)wherea≤x(n)0< x(n)1<· · ·< x(n)n≤bforn= 0,1, . . ..Letpnbe theinterpolating polynomial of degree at mostnoffat the nodes correspondingto then+ 1 row of (3.21).By the Weierstrass Approximation Theorem (p*nis a better approxima-tion or at least as good as that provided by the Bernstein polynomial),kf-p*nk∞→0asn→ ∞.(3.22)However, it can be proved thatΛn>2π2logn-1(3.23)
3.3.BARYCENTRIC FORMULA43and hence the Lebesgue constant is not bounded inn. Therefore, we cannotconclude from (3.20) and (3.22) thatkf-pnk∞asn→ ∞, i.e.that theinterpolating polynomial, as we add more and more nodes, converges uni-formly tof. That depends on the regularity offand on the distribution ofthe nodes. In fact, if we are given the triangular array of interpolation nodes(3.21) in advance, it is possible to construct a continuous functionfsuchthatpnwill not converge uniformly tofasn→ ∞.3.3Barycentric FormulaThe Lagrange form of the interpolating polynomial is not convenient for com-putations.If we want to increase the degree of the polynomial we cannotreuse the work done in getting and evaluating a lower degree one.How-ever, we can obtain a very efficient formula by rewriting the interpolatingpolynomial in the following way. Let