100%(2)2 out of 2 people found this document helpful
This preview shows page 41 - 44 out of 213 pages.
Butqis a nonzero polynomial of degree 2n+ 2 exactly. Therefore,x1andxn+2have to be simple zeros and sox1=-1 andxn+1= 1. Note that thepolynomialp(x) = (1-x2)[e0n(x)]2∈P2n+2has the same zeros asqand sop=cq, for some constantc. Comparing the coefficient of the leading orderterm ofpandqit follows thatc= (n+ 1)2.Therefore,ensatisfies theordinary differential equation(1-x2)[e0n(x)]2= (n+ 1)2kenk2∞-e2n(x).(2.61)We knowe0n∈Pnand its n zeros are the interior pointsx2, . . . , xn+1. There-fore,e0ncannot change sign in [-1, x2].Suppose it is nonnegative forx∈[-1, x2] (we reach the same conclusion if we assumee0n(x)≤0) then, takingsquare roots in (2.61) we gete0n(x)pkenk2∞-e2n(x)=n+ 1√1-x2,forx∈[-1, x2].(2.62)Using the trigonometric substitutionx= cosθ, we can integrate to obtainen(x) =kenk∞cos[(n+ 1)θ],(2.63)forx= cosθ∈[-1, x2] with 0< θ≤π, where we have chosen the constant ofintegration to be zero so thaten(1) =kenk∞. Recall thatenis a polynomialof degreen+ 1 then so is cos[(n+ 1) cos-1x]. Since these two polynomialsagree in [-1, x2], (2.63) must also hold for allxin [-1,1].
2.4.CHEBYSHEV POLYNOMIALS33Definition 5.The Chebyshev polynomial (of the first kind) of degreen,Tnis defined byTn(x) = cosnθ,x= cosθ,0≤θ≤π.(2.64)Note that (2.64) only definesTnforx∈[-1,1].However, once thecoefficients of this polynomial are determined we can define it for any real(or complex)x.Using the trigonometry identitycos[(n+ 1)θ] + cos[(n-1)θ] = 2 cosnθcosθ,(2.65)we immediately getTn+1(cosθ) +Tn-1(cosθ) = 2Tn(cosθ)·cosθ(2.66)and going back to thexvariable we obtain the recursion formulaT0(x) = 1,T1(x) =x,Tn+1(x) = 2xTn(x)-Tn-1(x),n≥1,(2.67)which makes it more evident theTnforn= 0,1, . . .are indeed polynomialsof exactly degreen. Let us generate a few of them.T0(x) = 1,T1(x) =x,T2(x) = 2x·x-1 = 2x2-1,T3(x) = 2x·(2x2-1)-x= 4x3-3x,T4(x) = 2x(4x3-3x)-(2x2-1) = 8x4-8x2+ 1T5(x) = 2x(8x4-8x2+ 1)-(4x3-3x) = 16x5-20x3+ 5x.(2.68)From these few Chebyshev polynomials, and from (2.67), we see thatTn(x) = 2n-1xn+ lower order terms(2.69)and thatTnis an even (odd) function ofxifnis even (odd), i.e.Tn(-x) = (-1)nTn(x).(2.70)
34CHAPTER 2.FUNCTION APPROXIMATIONGoing back to (2.63), since the leading order coefficient ofenis 1 and thatofTn+1is 2n, it follows thatkenk∞= 2-n. Thereforep*n(x) =xn+1-12nTn+1(x)(2.71)is the best uniform approximation ofxn+1in [-1,1] by polynomials of degreeat mostn. Equivalently, as noted in the beginning of this section, the monicpolynomial of degreenwith smallest infinity norm in [-1,1] is˜Tn(x) =12n-1Tn(x).(2.72)Hence, for any other monic polynomialpof degreenmaxx∈[-1,1]|p(x)|>12n-1.(2.73)The zeros and extrema ofTnare easy to find. BecauseTn(x) = cosnθand 0≤θ≤π, the zeros occur whenθis an odd multiple ofπ/2. Therefore,¯xj= cos(2j+ 1)nπ2j= 0, . . . , n-1.(2.74)The extrema ofTn(the pointsxwhereTn(x) =±1) correspond tonθ=jπforj= 0,1, . . . , n, that isxj= cosjπn,j= 0,1, . . . , n.(2.75)