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xv- Report No. 369 
 
 COO-1U69-0150 
 
 LINE AND AREA ORTHOGONALITY OF JACOBI POLYNOMIALS 
 
 by 
 
 M. M. Chawla and M. K. Jain 
 
 fH 
 
 \* 
 
 December 1969 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
Report No. 369 
 
 LINE AND AREA ORTHOGONALITY OF JACOBI POLYNOMIALS* 
 
 by 
 
 M. K. Jain 
 M. M. Chawla 
 
 December 1969 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 This report is supported in part by contract U. S. AEC AT(ll-l)lU69, 
 and in part by the National Science Foundation under grant NSF-GJ-217. 
 
LINE AND AREA ORTHOGONALITY OP JACOBI POLYNOMIALS 
 
 M. M. CHAWLA and M. K. JAIN 
 
 Department of Computer Science, University of Illinois 
 Urbana, Illinois 61801 
 
 1. In t re duct ion. 
 
 j->e n 
 
 n 
 
 P '' (x) , n-0,1,2, ... , be the set of Jacobi polynomials 
 which form an orthogonal system on [-1,1] with respect to the weight 
 function (l-x) (l+x; , <x > -1, ys > -1; that is, 
 
 (1) 
 
 (fc^ ?"'$ -j O-*?O+xf?f'>\x)T>r%Ux=0, "+* 
 
 These polynomials are standardized, as usual, such that P '' (l) = ( J . 
 
 Let p"' ; "(x) designate the Jacobi polynomial P ' * (x) normalized according 
 n n 
 
 to (1) so that p*(x) = P^^^xVUP^'^ll , where IIP^'^II = (P ( *'f°, P (jC '^)* 
 n n ' n n N n T n 
 
 Let £f> designate the ellipse z^(^ +$"') 1 ^ = pe lfi , Oi0 4 2Tf , P > 1, 
 in the z-plane (x=Re(z)) with foci at z = i 1 and semiaxes ■§•( P + P ) and 
 "KP-J J ~')» By A(£\) we shall designate the class of functions analytic on 
 "the closed interval [-1,]] (A [-1,1]) and which can be continued analytically 
 into the z-plane so as to be single valued and regular in £- , the interior 
 of £ , and continuous in £« , the closed ellipse £« . 
 
 For f,geA(A) we introduce the line integral inner product 
 
 wnich induces the line integral norm 
 
 (3) If II = J Kcx)| iKz^d, 
 
 I 
 
 ^ "ft 
 
-2- 
 
 ar.d the area integral inner product 
 
 ( 4 ) (M) = Jj jW6oJ *M 900^4 
 
 «p - 
 
 which induces the area integral norm 
 
 (5) If II* = j/lW(z)| lf<>/</*4, 
 
 £ p 
 
 In (2) and (4), )w(z)| and Jw(z)j are weight functions such that 1/ 1 )| <. 00 
 
 and li 1 U D < oo. 
 
 L? 
 
 It has been shown by Walsh [1] that the Chebyshev polynomials of the 
 first kind, T (z) = cos (n arc cos z) = P 2 ' (z), form an orthogonal 
 system on £n with respect to the inner product (2) with weight |w(z)| = 
 |1— z I a , and that these polynomials are also orthogonal on £„ with respect 
 to the inner product (4) with weight |w(z)j = |l-z j . Szego [_2] has 
 shown that the Chebyshev polynomials of the second kind, U (z) = sin[(n+l) arc 
 
 cos zj/(l-z ) 2 = P ' (z) , form an orthogonal system with respect to 
 
 2 - 1 - 
 (2) with weight |l-z j a ; it is also known that these polynomials are 
 
 orthogonal on cp with respect to (4) with weight unity (See, e.g., Davis [3J ) • 
 
 The purpose of this paper is to show that the (complex) Jacobi 
 
 polynomials P '' (z) form an orthogonal system on £p with respect to 
 
 the line integral inner product (2) with weight Jw(z)| = |l-zf |l+z( , 
 
 •*>-l, p >— 1; further, for & > -1 , p > -\ , these polynomials also form 
 
 an orthogonal system on £\ with respect to the area integral inner product 
 
 (4) with weight |w(z)| = I 1 "" 2 ! ~"" 2 ll+zr"* 2 . 
 
 2 * 
 
 Let L (£*; |W(z)| ) denote the Hilbert space resulting from the 
 
 completion of A(£\) with respect to the area integral norm (5), and let 
 H (£\; |w(z)J ) denote the Hilbert space resulting from the completion 
 of A(£p) with respect to the line integral norm (3). The Hilbert space 
 
-3- 
 
 5, N 
 
 L (tp? 1) has been used extensively for obtaining estimates for the errors 
 of numerical approximation for functions analytic on E— 1,1] (the method 
 Suggested by Davis; see, e.g., j^4] ) • As a consequence of our results we 
 show that the area integral inner product (2) is unsuitable for obtaining 
 error estimates for the class of functions A[— 1,1] ; this has already been 
 observed in Chawla £6], As a side result we obtain relationships between 
 the real integral, with Jacobi weight function, and the (complex) line 
 and area integrals for an fC-A(£«). As an important application of the 
 line integral orthogonality of the Jacobi polynomials on £p , we show 
 that estimates for the errors of rules of numerical approximation for 
 analytic functions can be obtain directly from Cauchy's integral formula. 
 2. Line Integral Orthogonality of Jacobi Polynomials 
 
 The Jacobi polynomials form a line integral orthogonal system on £« 
 rith the Jacobi weight function. This is contained in 
 ?HSOREM 1 . Let 
 
 ; 6) 'ti — '" ' ■■■' c 
 
 p 
 
 Len, for <* > -1, /?>-!. 
 
 \1) 
 
 ||.-z|*|i^7>^)lC<>)^ ~ <k.,h > »Vi«M,V' 
 
 £p 
 
 Proof . We follow Walsh Lit Theorem 12j. For « > -1, ^>-l, we have 
 
 
 
 f l.-.l'Wl' £«£(»> W 
 
-4- 
 
 V;hcrC °<n,vS^ iG a normalizing factor. Since for m=n, 
 
 *°S„« - IKH 
 
 X 
 
 e t 
 
 the line integral orthogonality ( 7 ) of the Jacobi polynomials (6) follows. 
 
 follov; 
 
 «nce for P->1, o^J P ) -, ! and » p£ || .» (2)* for all „> 0, there 
 
 Co 
 
 JO^lLjPy 1 . On -l 4!i l, P*( z ) = ( 2 )~* p »( a ) # 
 
 £-1210. let *„J, ,_fc then p * (x) . (2/7ty*1J x) , n>0 , pJ W . (*)"* 
 Since on ^ , T n (z) = -|( ^ n + £~ n ) , 
 
 (3) «/-;ii = (P** +?**)*, »Z0 
 
 The corresponding line integral orthonormal system of Chebyshev polynomials 
 
 is, therefore, 
 
 3 * A^ea Integral Orthogonality of JaooM Po1 vrmni , a1 „ 
 
 lie next show that the Jacobi polynomials ?£">(,) form an orthogonal 
 system on the area of £^ with weight |l- Z j*^ |l +a |'-* . 
 
 THEOREM 2 . Let 
 
 10) £>)- /CC*)/C««)*, »-o>i,$.>.~, 
 
 iiere 
 
 in, for c<>-j f p y _ if 
 
 Proof. Let A designate the annulus: £ =Re i0 , 1< R < p , ^ 6 4 ZV , 
 
-5- 
 
 in the ^ -plane, which is the map of £p in the z-planc. Then, with 
 
 -1 
 
 <=■*($ + 5 ). 
 
 £=u+iv, 
 
 (13) jj 1 W(,)j fc* W £(.) J^ « J{ , W (,), £(,) ££) j. ,,_ ,-^JJ, 
 
 P,. 
 
 c K 
 
 Now, if |'.i(z)| = |l-z| " |l+zj^\ th 
 
 en 
 
 
 Prom (7), (8) and (14) we obtain 
 
 (15) Si i«r-* 1 - -i''~ 1; t« K>) -4 - J^ f i J ««fi ■«!' p>)^w/^0 
 
 c 
 
 c 
 
 »fl£PJ£fi4 
 
 P.... A 
 
 = Vj>fii f 
 
 
 and (12) now follows from (15). We note an alternative expression for 
 
 (16) 
 
 C 
 
 Prom (11) we observe that as />-*l, p^{ p )-* 0, and since p*(z) 
 uniformly bounded on -1 £ z £l, there follows 
 
 COROLLARY 2. The polynomials ?*(z) are unbounded on -Ui*l f 
 and hence unbounded on every £\ , p> 1. 
 
 See also Sewell [7 J , Theorem 5.3.1 and remarks on p. 154. 
 
 ^--■^■r.les. 1. Let <*=£, £ =£, then p*(x) - (2/'}f)~ S U^x) , n^O. 
 Since on $ , U^z) = ( ? n+1 - jf*" 1 VCg-^" 1 ), 
 
 are 
 
d7) IKi = (V"^ p""-^' 
 
 ^ 
 
 so that 
 
 (is) ?;<=) a±Y(p^ z +^ )-*uw 
 
 are line integral orthonormal over Cp . Now, 
 
 (19) /up) = ['Cr^^r^-*)^ 
 
 therefore the polynomials 
 
 are area integral orthonormal over L with weight |w(z)( = 1. 
 2. Let *— £■, £=-|; then from (8) and (ll) 
 
 
 tfhile 
 
 W- *.( P f- 
 
 = ^ 4 p , 0>0 
 
 nerefore, the Chebyshev polynomials of the first kind, 
 
 2l) ?>) - *(t)*(P u, -P"* , )"*"E(0> »>° 
 
-7- 
 
 c 2 -1 
 
 are area integral orthonormal over en with weight |l-z j 
 
 4- delations between Real, Line and Area Integrals of an Analytic Function 
 Assume that fe A(L), PP" 1 * Then f possesses the uniformly convergent 
 Fourier expansions: On -l<ix^l f 
 
 .'-3 * 
 
 (22) f oo - f. <C />*<*; = £ (jjTw%) Th "''"OO 
 
 *"l - C M — V ** 
 
 where 
 
 (23) 
 
 « 
 
 * « f' (i-x^O*/ fOO|>*(x>kc 
 
 (24) 
 
 •p 
 
 «■=£/ £, 
 
 wnere 
 
 
 
 
 
 
 (25) 
 
 
 
 
 wr 
 
 = f 
 
 4 
 
 and also 
 
 the 
 
 Fo 
 
 urier 
 
 expansion 
 
 (26) 
 
 
 i 
 
 fo = 
 
 X 
 
 
 t * _ j , i,.*ri, + »i^wT*ww 
 
 t v ! c * ^ T>^/%) 
 
 where 
 
 (27) c; - JT 1,.^^ |lt2 f-'A ^ ?>) J*J a 
 
 Since all these Fourier expansions (22), (24) and (26) of the same f«=A(£V) 
 
 arc identical on -I/-2U, the coefficients of P^ '^(z) in these 
 
 n 
 
 expansions must be the same. Equating the coefficients in (22) and (24), 
 
 
 (26) 1£h a* = y h , "to 
 
 Substituting for a* and b* from (23) and (25), we have 
 
-8- 
 
 2 
 
 For n=0, (29) gives 
 
 i r' 
 
 (30) j ■ 
 
 
 since p" x " is a constant, 
 o 
 
 Next, equating the coefficients in the expansions (24) and (26), 
 we obtain 
 
 Substituting for b* and c* from (25) and (27), we obtain 
 
 n n ' 1 
 
 (32) p B (flj i.-.r»'«i'.^ m\M =110* [fi-r^W 7 ^ fto^ 
 
 For n=0, (32) gives 
 
 (33) %(?) j" i.*ri!«f#o 1^1 = > rf $m~V^"V^ 
 
 Finally, comparing (30) and (33) we obtain 
 
 (34) ?oCF) ( O-sfO-^/fOo ^x - Jf 1-4"^ li^T^f C-)^ 
 
 $ 
 
 We note the following special cases of (34). 
 (i) *=■!, p=|- : 
 
 (35) ±^-p-*){ 1 o-x^f^J*- = jJfojM 
 
 This particular relation was derived by Davis f5j in connection v;ith simple 
 Quadratures. 
 
-9- 
 
 (ii) oc —fr, ^=-1- : 
 
 
 (iii) ix=0, £ =0 : 
 
 (37) htn j ft*w* = j| l-^r y V^M 
 
 -J •s J 
 
 v;here r 
 
 and L(£J = length of A . 
 
 5# ^ rrors of Numerical Approximation for Analytic Functions thrmi^h 
 Cauchy's Integral Formula 
 For f €A(£^), p > 1, we have from Cauchy's integral formula 
 
 (33) f (x> = -L f f(^ 
 
 $ 
 
 x ^L-lflJi and, on £^ , f( z ) possesses the uniformly convergent Fourier 
 expansion (24) in terms of the line integral orthonormal Jacobi polynomials 
 P*(z). Substituting (24) in (38), we have 
 
 (39) i Cx; = £ C J^- 
 
 since P*(z) = p*( 2 )/ || p* Jj . Also, from the equation of closure, 
 
 (40) wit « [ i,-,.ri..^if^^ = z /tf i* 
 
 If E denotes the error of a linear process of numerical approximation 
 for an ft" (^), p > 1, then from (39) 
 
 (41) £ ^£, "* 1W 
 
 From Schwarz inequality the error E(f) can now be bounded by 
 
-10- 
 
 (42) \ui)\ 4»n ( x 
 
 |f(tf(=9)J*V* 
 
 If ||f|| is estimated from 
 
 (43) ? If t * ^(^((i-ri^i^s)' 
 
 where M(£p) = max f(z)j on £ , then we can estimate the error from: 
 THEOREM 3 . Let ffA(^), p>l. Then, 
 
 (44) |£ft>l - ( IMW*) ( g. Tj^T ; M(W 
 
 For various error functionals of numerical analysis one can use 
 a fixed Jacobi weight function and the corresponding orthonormal system 
 of polynomials P w (z) in the estimate (44); however, for certain particular 
 error functionals it might be more appropriate to use the related Jacobi 
 weight function and the orthonormal system of polynomials. A striking 
 example is the following. 
 
 Let E (f) denote the error of an n-point Gauss-Jacobi quadrature 
 formula with weight (l-x) (l+x)" : 
 
 (45) 
 
 
 Here x , , k=l, ... , n, are the n real and distinct abscissas which are 
 
 n,k' ' ' 
 
 the zeros of the corresponding Jacobi polynomial p*(x) and J| = 
 
 (l/p*'(x ) ) -1 I (l-xf (l+x/ (p*(x)/(x - x ))dx , k=l, ... , n, are 
 
 the corresoonding positive Christoffel numbers. Since E (f) = whenever 
 
 n 
 
 f is a polynomial of degree 4 2n-l, we have from (44) the following estimate 
 for the error of the Gauss-Jacobi quadrature formula. 
 
-II- 
 
 Tj \ :: ^r. . 
 
 £ i> 
 
 6. ^.symTuoiic Estimates 
 
 The second factor on the right side of (4-6) can be recognized as 
 the line integral norm of E over II ( £. ; |l-z| |l+z)' ): 
 
 n 
 
 *\l* 
 
 (47) »Ejf - I \E*CV)\ 
 
 l i 
 
 Similarly, the area integral norm of E over |_ (£,; |l-z| z |l+z)^ a ) can 
 
 be obtained from the Riesz representation theorem (Davis £3] f (9*3.13)) in 
 
 terms of the area orthonormal polynomials y (z) : 
 
 (48) ie,, i* = 1 kodi 1 
 
 Since for z e £ p , P^ 1 * and - n sufficiently large (Szego L8] , (8.21.9)), 
 
 (49) ?„ c ^) « ^ a-irvo'-v^' 
 
 it follows that 
 
 and since for large n, iJP^'^jj cs 2° <+ ^/n 1 therefore 
 
 tp 
 
 Prom (ll) and (51), 
 
 ( 52) M'> « a,)- 1 p a "—^' . 
 
 therefore from (10) we have for large n and P>1, 
 
 (53) ?„*0) * Cai)*?*^ 
 

 
 oj a ;a ^53; , ior large n, 
 
 i _ v. 2 
 
 ; -mce 
 
 !F ,* A _,_ £ k K,aoi 
 
 
 Combining (54) and (55) it follows that for large n, 
 
 (56) iiHJL ^ too* >EJI, 
 
 -p 
 
 EhUE, the area integral nor, will overestimate the errors in Davis' method(M) 
 -03 also Chawla [6]. 
 
 J.L. WALSH, Interpolation and Approximation by Rational Functions in 
 the Complex Plane, American Mathematical Society Colloquium Publication 
 No. XX, Providence, ft. I., 1935. 
 
 G. ^ZGO, A Problem Concerning Orthogonal Polynomials, Transactions 
 of the American Mathematical Society, vol. 37, 1935, PP. 195-205. 
 P.J. DAVIS, Interpolation and Approximation, Blaisdell, New York, 1963. 
 P.J. DAVIS, Errors of Numerical Approximation for Analytic Functions, 
 in Survey of numerical Analysis, J. Todd (Editor), McGraw-Hill, 
 New York, 1952. 
 
 ?.J. DAVIS, Simple Quadratures in the Complex Plane, Pacific Journal 
 of Mathematics, vol. 1$ , 1965, pp.813-8^. 
 
 A.M. CHAWLA, On Davis' Method for the Estimation of Errors of Gauss- 
 Chebyshcv Quadratures, SIAM Journal on Numerical Analysis, vol. 6, 
 I959 ; pp. 103-117. 
 
 -AE. SEWELL, Degree of Approximation by Polynomials in the Complex 
 -lane, Annals of Mathematics Studies, Mo. 9, Princeton University 
 Press, Princeton, New Jersey, 1942. 
 
 G. iaEGO, Orthogonal Polynomials, American Mathematical Society 
 Colloquium Publications No. XXIII, New York, 1959. 
 
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