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 Report No. 373 
 
 COO-IU69-OI5I+ 
 
 ORTHOGONALITY RELATIONS FOR CHEBYSHEV POLYNOMIALS 
 
 by 
 
 M. K. Jain THE UBRARY OF THE 
 
 M. M. Chawla 
 
 January 19 70 
 
 FEB 3 1975 
 
 UNIVERSITY OF ILLINOIS 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/orthogonalityrel373jain 
 
Report No. 373 
 
 ORTHOGONALITY RELATIONS FOR CHEBYSHEV POLYNOMIALS* 
 
 M. K. Jain 
 M. M. Chawla 
 
 January 1970 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 618OI 
 
 This report is supported in part by contract U. S. AEG AT(ll-l)li|69, 
 and in part by the National Science Foundation under grant NSF-GJ-217. 
 
TABLE OF CONTENTS 
 
 Page 
 
 1. INTRODUCTION 1 
 
 2. TWO NEW ORTHOGONALITY RELATIONS FOR THE 
 
 CHEBYSHEV POLYNOMIALS 5 
 
 2 .1 Gauss-Che"byshev Quadrature with 
 
 Fixed Abscissa x = +1 5 
 
 2 .2 Gauss-Chebyshev Quadrature Formula 
 
 with Fixed Abscissa x = -1 7 
 
 2 . 3 I^ew Discrete Point Orthogonality Relations 10 
 
 3. NEW SCHEMES FOR THE APPROXIMATE COMPUTATION OF 
 CHEBYSHEV-FOURIER COEFFICIENTS 11 
 
 h. NUMERICAL INTEGRATION BY TRUNCATED CHEBYSHEV- 
 FOURIER EXPANSION lU 
 
 5. EXAMPLES l6 
 
 REFERENCES l8 
 
ABSTRACT 
 
 Two new discrete point orthogonality relations for the Chebyshev 
 polynomials of the first kind have "been obtained. Semi-closed Gaiiss- 
 Chebyshev quadrature formulas, required for the purpose of obtaining these 
 relations, have also been developed. These orthogonality relations lead 
 to better approximations to the coefficients in the Chebyshev-Fourier 
 expansion of a function and consequently to better values for an integral 
 when the integrand is approximated by a truncated Chebyshev-Fourier series, 
 
1. INTRODUCTION 
 
 Let T (x) = cos(n arc cos x) , n = 0, 1, 2,..., be the 
 n 
 
 Chebyshev polynomials of the first kind defined over [-1,1]. These 
 polynomials are known to be orthogonal in two ways (see Lanczos [l], 
 pp. vii, xvii); 
 
 (l) Tit if r = s = 
 
 1 T (x) T (x) I 
 
 ^ -^— TT72-^^=U^^ r = s ^ 
 
 -1 {1-^^r'^ y 
 
 1 if r 7^ s 
 
 (2) Tn if r = s = or n 
 
 E" T (x.) T (x ) =<^ifr= s i- orn 
 
 lo if r 7^ s 
 
 In (2), X. = cos(jiT/n); n > 0; r, s <_n. The double prime on sigma 
 indicates that the first and the last terms are to be halved. 
 
 The discrete point orthogonality relations (2) have been used 
 by Clenshaw [2] for the approximate computation of the coefficients in 
 the Chebyshev-Fourier expansion of a function and for the approximate 
 integration of a function by means of truncated Chebyshev series. 
 
 Our first result establishes that the discrete point orthogonality 
 relations (2) can be obtained from the orthogonality relations (l) by 
 replacing the integral by an appropriate Gauss -Chebyshev quadrature formula. 
 
 Theorem 1 . For r, s _^n, (l) =^(2) 
 
 Proof . The (n+l) - point Gauss -Chebyshev quadrature formula of the closed 
 
 type over [-1,1] with fixed abscissas x = +1 (see Chawla [3]) is 
 
/^Ht7p = ^ ^ f(cos ^) +E (f) 
 -1 (l-x^)^/^ ^ k=0 
 
 where E (f) denotes the error of the quadrature formula and is given by 
 
 n^^^ " "22n-l (2n)! 
 
 for some ne[-l,l]. The quadrature formula (3) is therefore exact for all 
 polynomials of degree <_ 2n-l. Replacing the integral is (l) by the 
 quadrature formula (3), together vith the error, we obtain 
 
 (5) Ttt if r = s = 
 
 n I 
 
 - E"T(x.)T(x.)+E(TT)=(Jifr = s^O 
 h.qTj sj nrs 12 
 
 (^0 if r 7^ s 
 
 If r + s < 2n-l, then E ( T T ) = and (5 ) gives 
 — nrs 
 
 (6) Tn if r = s = 
 E" T (x.) T (x.) =( g- if r = s y 
 
 1^0 if r 7^ s 
 
 Next, let r = s = n. Since T^(x) = ■^(T (x) + l) and T (x) = 2'^"^x^ plus 
 
 n 2 2n n 
 
 lower degree terms, therefore, T (x) = 2 x plus lower degree terms, 
 
 and from (k) we obtain E (T T ) = - — . Thus, for r = s = n we have 
 
 n n n 2 
 
 from (5) 
 
 Z" T (x.) T (x.) = ^J- E (T^)) 
 = n 
 
Combining (6) and (T), the orthogonality relation (2) follows. Clearly 
 the argument holds for the converse also. 
 
 Let f(x) be continuous and of bounded variation on [-1,1], 
 then f(x) has a imiformly convergent Chebyshev-Fourier expansion: 
 
 (8) 
 
 f(x) = E' a. T.(x] 
 j=0 ' ' 
 
 the prime on the sigma means that the first term is to be halved. The 
 Chebyshev coefficients are given by 
 
 (9) 2 ^ ^^^) T.(x)dx 
 
 ^1=7-^ p'\/9 ' J = 0, 1, 2,... 
 
 J ^ _1 (l-x^)^/^ 
 
 Replacing the integral in (9) by the Gauss -Chebyshev quadrature 
 formula of the closed type (3) and neglecting the error we obtain 
 
 ( 10 ) „ n 
 
 a ~ - r' f(x.) T (x.) 
 r n .^Q J r J 
 
 where x. = cos(j7T/n). This is the scheme for the approximate computation 
 J 
 
 of the Chebyshev coefficients described by Clenshaw [2], 
 
 If we define the numbers a^ , a , . . . , a as 
 
 1 n 
 
 (11) 2 "^ 
 
 a = - E" f(x.) T (x.) 
 
 then it is easy to show [2] that 
 
 (12) 
 
 a=a+E(a- +a^^) 
 
 r r _ 2np-r 2np+r 
 
This shows that a^ can be taken as an approximation for a^ with an 
 
 error of 2(a +...); a can be taken as an approximation for a with 
 
 an error of a^ ^ +.... However, 
 2n-l 
 
 a -, = a ^ + a ^^ + a_ -,+... 
 n-1 n-1 n+1 3n-l 
 
 so that unless the coefficients a converge very rapidly, a ^ will not 
 be a good approximation to a . But 
 
 a = 2(a + a^ +. . .) 
 
 n n 3n 
 
 so that — a gives an approximation for a with an error of a^ +.... 
 2 n ° ^-^ n 3n 
 
2. TWO NEW ORTHOGONALITY RELATIONS FOR THE 
 CHEBYSHEV POLYNOMIALS 
 
 We have seen that the approximation (10) for the calculation 
 of the Chebyshev coefficients results from replacing the integral in 
 (9) by the two point closed Gaioss -Chebyshev quadrature formula. This 
 raises another possibility of using semi-closed Gauss -Chebyshev 
 quadrature formulas with a fixed abscissas either at x = +1 or at x = -1 
 for replacing the integral in (9). Since the semi-closed formulas are 
 more accurate (one degree of precision more) than the fully closed 
 formula, it is expected that the resulting approximation will be better 
 than that given by (lO). 
 
 For this purpose, in the following, we develop the two semi- 
 closed Gauss -Chebyshev quadrature formulas and use them to obtain two 
 new discrete point orthogonal relations for the Chebyshev polynomials 
 similar to (2) . 
 
 2.1 Gauss -Chebyshev Quadrature with Fixed Abscissa x = +1 
 
 (^ -^) (i -i) 
 
 Let w(x) = (x-l) P (x) where P is the Jacobi 
 n n 
 
 1— x 1/2 
 polynomial on [-1,1] corresponding to the weight (tt~) . Let x , x , ..., x 
 
 (- -h 
 2' 2 
 
 be the n zeros of P (x), and let L (x) be the Lagrange interpolation 
 
 polynomial of degree n for f corresponding to the abscissas x = 1, x , ..., x , 
 
 ^n(^) = „-T(ffe ^(1) ^ X .'i^^nl^) ^(^' 
 
2 -1/2 
 Multiplying both sides by the weight ( 1-x ) and integrating over [-1,1] 
 
 (13) 1 ^/ N^ 1 L (x)dx n 
 
 •^ 2%= /~Vl72 = Af(l) -H Z A^f(^) 
 
 -1 (1-x^)^/^ -1 {1-^r'^ k=i ^ ^ 
 
 where the weights are given by 
 
 A = 1 r w(x)dx _ 1 r w(x)dx 
 
 " "'<1) -1 (x-l)(l-x2)l/2' \ " ^^ .i (,.^)(i.x2)l/2- 
 
 k = 1, . . . , n. 
 The precision of the quadrature formula (13) is clearly 2n. Since 
 
 P (x) = c 
 
 n n 1 - X 
 
 where c is a positive constant, the n free abscissas of the formiila (13) 
 
 n 
 
 (^ -^) 
 2' 2 
 are the zeros of P (x). 
 
 2kTT ^ 
 
 Therefore , 
 
 ^ 1 T (x) - T A^) „ 
 
 2n+l ^ ,^ s,^ 2^1/2 2n+l 
 
 -1 (l-xj(l-x ) 
 
 ^k 
 
 and A = jrj r where we have put 
 
 ■\' 
 
2' 9 
 
 But y , k = 1,..., n are the weights in the Gauss-Jacobi quadrature 
 formula 
 
 (IW 1 1 X 1/2 ° 
 
 -1 k=l 
 
 and are given (see [U] by 
 
 \ = 2^^ ""^ ^2^^^ 
 
 and thus 
 
 2lT 
 
 The Gauss-Chebyshev quadrature form-ula with fixed abscissa at x = +1 is, 
 therefore, 
 
 \ f(x)dx 2^ ^, ^. 2kTT . _^ „+, . 
 
 where E ( f ) denotes the error of the formula. 
 
 n 
 
 2 .2 Gauss-Chebyshev Quadrature Formula with Fixed Abscissa x = -1 
 
 (-- -) (-- -) 
 
 Let v(x) = (x+l) P (x) where P (x) is the Jacobi 
 
 1+x 1/2 
 polynomial of degree n over [-1,1] corresponding to the weight {- ) 
 
Let the zeros of P (x) be denoted by x^ , . . . , x , and let M (x) be 
 
 n In n 
 
 the Lagrange interpolation polynomial of degree n for f corresponding to 
 
 the abscissas x= -1, x, , . . . , x : 
 
 1 n 
 
 n v'(-l)(x+lj , _, v'(x,)(x-x,) k 
 
 ^^ V^x^Mx-x^, 
 
 2 —1/2 
 Integrating over [-1,1] with weight (l-x ) , 
 
 (16) 1 , . 1 M (x)dx 
 
 / ^ l^/^ ~ / -^^-77J= Bf(-l) + E Bf(x,) 
 -1 (l-x2)l/2 (i_x2)l/2 k=l ^ ^ 
 
 where 
 
 ■n _ 1 f v(x)dx _ 1 . v(x)dx 
 
 " ^'^^-l (x.l)(l-x2)l/2' 'k - ^^T^i (_ )(l-x2)l/2' 
 
 ■\' 
 
 k = 1,. .., n. 
 
 Since 
 
 (-!» |) T (x) + T _^ fx) 
 
 P 2 2 (^) ^ ^ n n+1 
 
 n n 1 + X 
 
 the abscissas x, = cos ( (2k-l) TT/(2n+l) ) , k = 1,..., n. Now, 
 
 f ^.n 1 T (x) + T _^^(x) 
 
 2^-^ -1 (l.x)(l-x2)l/2 2n.l 
 
 and 
 
B =-i-X 
 
 k 1+x k 
 
 where we have put 
 
 (-i -) 
 
 " (4 i)' -1 ^-'^ ""-^ 
 
 But X are the weights in the Gauss-Jacobi quadrature formula 
 
 K. 
 
 ^1^) ^ l+x 1/2 "" 
 
 -1 k=l 
 
 and are given by 
 
 \ " mT cos^((k-|)7r/(2n+l)), k = 1,..., n. 
 Therefore , 
 
 \ - 2^;;i> ^ = l--" ^ 
 
 The Gauss -Chehyshev quadrature formula with fixed abscissa x = -1 is, 
 therefore. 
 
 (18) 
 
 ^ f(x)dx _ _2tt_ ''^^, , (2k-.l) V , p-, ->, 
 
 ( , ' 2.1/2 - i^^,^ ^^"°" 2n+l ^^ \^^^ 
 -1 (1-x j k=l 
 
 where E (f ) denotes the error of the formula. The lower prime on the sigma 
 n 
 
 indicates that the last term is to be halved. The formula (l8) is of 
 precision 2n. 
 
10 
 
 An error analysis of v.he serai-closed Gaviss-Chebyshev quadrature 
 formulas (15) and (l8) with fixed abscissa at x = 1 and -1, respectively, 
 will be given in Section 3. 
 
 2.3 New Discrete Point Orthogonality Relations 
 
 From the orthogonality relations (l) for the Chebyshev 
 
 polynomials we deduce the following two discrete point orthogonality 
 
 relations similar to (2). 
 
 Replacing the integral in (l) by the quadrature formula (15) 
 
 of precision 2n, we obtain: 
 
 Theorem 2 . If x^ = cos ( ■ -, ) » ^ = 0, 1,..., n, then for r, s <_ n, 
 
 (19) r(2n+l)/2 if r = s = 
 
 "" ) 
 
 Z' T (x, ) T (x ) =< (2n+l)/4 if r = s ^^ 
 
 L if r 7^ s 
 
 Again, replacing the integral in (l) by the quadrature formula 
 (l8) of precision 2n, we obtain 
 
 Theorem 
 
 _3.. If x^ = cos ((2k-l)TT/(2n+l)), k = 1,..., n+1, then for r, s <_n. 
 
 (20) r(2n+l)/2 if r = s = 
 
 n+1 I 
 
 Z. T^(x^) T^(xj^) =/ (2n+l)/l| if r = s ?^ 
 k=l ^ ^ I 
 
 V if r 7^ s 
 
11 
 
 3. NEW SCHEMES FOR THE APPROXIMATE COMPUTATION OF 
 CHEBYSHEV -FOURIER COEFFICIENTS 
 
 We have seen that the Clenshaw scheme for calculating 
 
 approximately the Chebyshev-Foiirier coefficients a. consists of 
 
 J 
 
 replacing the integral in (9) "by the fully closed Gauss -Chebyshev 
 quadrature formula (3). Alternative "better schemes can be obtained 
 by replacing the integral by either of the semi-closed formulas; 
 however, these formulas lack in symmetry. 
 
 Replacing the integral in (9) by the semi-closed Gauss- 
 Chebyshev quadrature formula ( 15 ) and neglecting the error, we obtain 
 the approximation 
 
 (21) ^ n 
 
 a = ^— - E' f(x.) T (x.) 
 r 2n+l ^^Q J r J 
 
 where x. = cos (2JTT/(2n+l) ) , j = 0, l,...,n. If we define for 
 
 J 
 
 r = 0, 1,..., n, the numbers 
 
 (22) n 
 
 B = ~- Z- f(x.) T (x.) 
 r 2n+l ^^Q J r J 
 
 then substituting (8) in (22) and using the orthogonality relations (19) 
 
 we obtain 
 
 (23) 
 
 B = a + Z (a/^ ^^ ^ + a.,^ ^^ . ^ ), r = 0, 1,..., n 
 
 r r _ (2n+ljp-r (2n+l}p+r ' > > » 
 
 Jr 
 
 Note that for r = 0, (23) reduces to the quadrature formula (l5) so that 
 the error is given by 
 
12 
 
 {2k) ^ 
 
 E (f) = -TT Z a,. .. 
 n -, (2n+l)p 
 
 p=l 
 
 and for r = 0, 1, . . . , n, 3 ~ a with an error of Z (a/o , -, \ + 
 
 ' ' ' ' r r -, (2n+l)p-r 
 
 p=l 
 
 ^(2n+l)p-r^* 
 
 Similarly, replacing the integral in (9) hy the semi-closed 
 
 Gauss -Chebyshev quadrature formula (l8) and neglecting the error, we 
 
 obtain the approximation 
 
 (25) ^ n 
 
 J 
 
 , (2k-l)TT\ 
 where x. = cos {— — ) j = 1,..., n+1. For r = 0, 1,..., n define 
 
 the numbers 
 
 (26) , n+1 
 
 Y = Trrrr ^' f(x.) T (x.) 
 'r 2n+l J r J 
 
 J -^ 
 
 Substituting (8) in (26) and using the orthogonality relations (20), 
 we obt ain 
 
 (27) 
 
 ^r = \ ■" ^l^ ^-^) ^^2n+l)p-r "^ ^2n+l)p-Hr) 
 
 For r = 0, (27) leads to the quadrature formula (l8), and the error is 
 given by 
 
 (28) 
 
 E (f) = TT 
 n 
 
 p=i 
 
 E-(f) = . E (-l)P^^ a(^^^^)^ 
 
13 
 
 Also for r = 0, 1,..., n, (27) shows that y ~ a with an error of 
 
 oo 
 
 ^ ^"^^ ^^(2n+l)p-r "*■ ^(2n+l)p+r^- 
 p=l '^ '^ 
 
Ik 
 
 h. NUMERICAL INTEGRATION BY TRUNCATED 
 CHEBYSHEV-FOURIER EXPANSION 
 
 Let f(x) have the Chebyshev-Fourier expansion (8). A 
 useful polynomial approximation to f(x) can be fo\md by trioncatin^ 
 the expansion at some j = N: 
 
 (29) N 
 
 f(x) = E- a^ T^(x) 
 
 k=0 
 
 and therefore. 
 
 (30) 1 N 1 
 
 / f(x)dx ~ Z' a, ( / T (x)dx) 
 
 -1 k=0 ^ -1 ^ 
 
 But 
 
 (31) /> 2 . „ , . 
 
 -— — if k IS even 
 
 k -1 
 
 / T (x)dx = 
 
 -1 I if k is odd 
 
 so that 
 
 (32) 1 [N/2] 
 
 / f(x)dx = Z 
 
 -2 
 
 a^ 
 
 -1 k=0 Uk^-1 ^^ 
 
 where [k] denotes the largest integer contained in k. The Chebyshev 
 
 coefficients a can be computed through either (2l) or (25). 
 idk 
 
 Note that as far as approximate integration over [-1,1] is 
 concerned, the lack of symmetry of the approximations (2l) or (25) does 
 not affect as only even coefficients are needed in (32). The improved 
 
15 
 
 accuracy of these algorithms for the approximate computation of the 
 Chebyshev coefficients will therefore produce better approximations to 
 the value of the integral than that resulting from the use of the 
 algorithm (lO) . 
 
16 
 
 ' 5 . EXAMPLES 
 
 2 1/2 -1 
 
 The Chebyshev coefficients for (l-x ) and tan x 
 
 calculated from (22) are compared in Tables 1 and 2 respectively, 
 
 with those calculated from (ll) as also -with their exact values. In 
 
 each case n = 9. In Table 3 we compare the approximate values I and 
 
 ■*" 2 1/2 
 
 I. for 1(f) = / f(x)dx for f(x) = (l-x ) ' and log(l.01+x) as 
 
 ^ -1 
 
 obtained from (32) when the a 's are approximated, respectively, by 
 
 a, 's and 3, 's with the exact values for these integrals. It will be 
 k k ^ 
 
 observed that the 3, 's produce better approximations. 
 
 K. 
 
IT 
 
 Table 1 
 f(x) = (l-x^)l/2 
 
 r 
 
 3 
 r 
 
 a 
 
 r 
 
 (exact) 
 
 
 r 
 
 
 
 +1.2T033T8 
 
 +1.2732395 
 
 +1.2602859 
 
 2 
 
 -0.U2 73309 
 
 -0.U2 1+1+132 
 
 -O.H376913 
 
 k 
 
 -0.0878^+91 
 
 -O.O8U8826 
 
 -0.0992158 
 
 6 
 
 -0.039^289 
 
 -0.0363783 
 
 -0.0527911 
 
 8 
 
 -0.0233851 
 
 -0.0202102 
 
 -o.oUoi+Uoi 
 
 Table 2 
 f(x) = tan'-'-x 
 
 r ! 
 
 ^ 
 
 a 
 r 
 
 (exact) 
 
 a 
 r 
 
 1 
 
 +O.828U2712 
 
 +0.82 81+2 712 
 
 +0.82 81+2 716 
 
 3 
 
 1 
 
 -0.01+737851+ 
 
 -0. 01+73785 1+ 
 
 -0. 01+737878 
 
 5 
 
 +0.00 1+87732 
 
 +O.OOI+87732 
 
 +O.OOI+87895 
 
 7 
 
 \^ 
 
 -0.00059773 
 
 -0.00059773 
 
 -0.00060892 
 
 
 
 
 Table 3 
 
 
 fCx) 
 
 N 
 
 h 
 
 I 
 
 I 
 
 a 
 
 log(l.01+x) 
 
 9 
 5 
 
 1.5699337 
 -0.56131+05 
 
 1.5707963 
 -0.5506975 
 
 1.5696093 
 -O.5662I+77 
 
18 
 
 REFERENCES 
 
 [l] Lanczos, C, "Applied Analysis," Prentice-Hall, Inc., Englewood 
 Cliffs, New Jersey, 1956. 
 
 [2] Clenshaw, C. W. , "Chebyshev Series for Mathematical Functions," 
 National Physical Laboratory Mathematical Tables, Vol. 5, 
 Department of Scientific and Industrial Research, Her Majesty's 
 Stationery Office, London, England, I962 . 
 
 [3] Chawla, M. M. , "Error Bounds for the Gaiiss-Chebyshev Quadrature 
 
 Formula of the Closed I^pe," Math. Comp. , 22 (1968), pp. 889-891. 
 
 [k] Krylov, V. I., "Approximate Calculation of Integrals," Macmillan, 
 New York, New York, I962. 
 
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