Computers Math. Applic. Vol. 30, No. 3-6, pp. 155-168, 1995 Pergamon CopyrightQ1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221/95 $9.50 + 0.00 0898-1221(95)00093-3 Interlacing Properties of the Zeros of the Orthogonal Polynomials and Approximation of the Hilbert Transform G . MASTROIANNI AND D. OCCORSIO Dipartimento di Matematica, Universitg della Basilicata Via N. Sauro 85, 85100 Potenza, Italy Mastroianni©pzvx85. c ineca, it Abstract--In this paper, starting from interlacing properties of the zeros of the orthogonai polynomials, the authors propose a new method to approximate the finite Hilbert transform. For this method they give error estimates in uniform norm. Keywords--Orthogonal polynomials, Lagrange interpolation, Hilbert transform. 1. I N T R O D U C T I O N The finite Hilbert transform of the function f is defined as H(f;t) = f l f(x) dx = i x -- t lim ~0+ / f(x) x dx, (1.1) -- t where the integral is in the Cauchy principal value sense. A condition for the existence of H(f) is that the function f is continuous in [-1,1] ( f e C(°)([-1,1])) and f~w(f;u)u-ldu < oc, (f e DT), where w(f,.) is the ordinary modulus of continuity. In numerical analysis and in approximation theory, it is very useful to approximate the finite Hilbert transform, because it appears in several problems of various nature. Furthermore, many physics and engineering problems involve Cauchy integral equations and sometime their solutions have zeros or singularities in i l . For this reason, the solution is expressed as a product of a smooth function for a Jacobi weight v o'z, a,/~ > - 1 . Then we are interested in approximating H(vO,Zf). Actually, by global approximation of the function f, two kinds of quadrature rules have been proposed to approximate H(va'Zf): "product rules" and "Gaussian rules." About the product rules, there exist efficient methods [1,2] founded on recurrence formulae for a large class of weight functions, but the computation of the coefficient requires some effort. The Gaussian rules are simpler than the previous one, but they present numerical instability and generally diverge whenever the function f is only Hhlder continuous. In this paper, we propose a new method to approximate H(v~'Zf). This procedure employs Gaussian rules, using interlacing properties of the zeros of orthogonal polynomials [3] and recent results about the Lagrange interpolation [4]. We also prove that this method is numerically stable and convergent. Typeset by ~4.AdS-TEX 155 156 G. MASTROIANNI AND D. OCCORSIO In the following section, we state the above-mentioned method. Section 3 contains results about the convergence, while in Section 4, there are their proofs. Finally in Section 5, some numerical examples are given. 2. T H E METHOD We assume f E DT. We have H(v~'Zf; t) = /1 f(x) va,Z(x ) dx = f l f(x) - f(t) va,~(x ) dx 1 x-t + f(t) 1 X t (2.1) dz, lx--t : = F(f; t) + f ( t ) g (v~'~; t ) . Since it is possible to use an efficient procedure due to Gautschi and Wimp [5] to evaluate H(va'~; t), we have to compute the function F(f). The idea is to approximate the function F ( f ) by a Lagrange polynomial M LM(F(f); t) = ~ lj(t)F(f; tj) j=l on suitable interpolation knots t l , . . . , t M . Since we cannot evaluate exactly F(f; tj), making use of a Gaussian rule, we replace F ( f ; t j) by N FN(f; tj) = ~ Ag,k(V ~'~) f(Xg,k(V~'~)) -- f(tj) Xy,k(V ~'B) -- tj ' j = 1 .... , M, (2.2) k=l where {XN,k(V"'/3)}N= I are the zeros of the Jacobi polynomial p y ( v ~'~) the Christoffel numbers related to the weight v a'~. Therefore, we approximate H(va'~f) by ~I (va'Z f; t) := LM(FN(f); t) + f ( t ) H (v~'Z; t ) . and {/~N,k(Va'/~)}~= 1 are (2.3) Now we want to select the knots {tj}M1 and N to obtain an optimal interpolation process LM, i.e., such that the Lebesgue constants [[LM[[ = supllfll= 1 [[Lm(f)ll ([[.[[ denotes the s u p - n o r m ) , diverge as log M. Moreover, to preserve numerical stability, we intend to choose the zeros {XN,k(Va'~)}N=I sufficiently far from the knots {tj}/= M 1. In this aim, we need some results. For any weight u, we denote by {pk(u)}~°=o the corresponding sequence of orthonormal polynomials with leading positive coefficient. In the following, we will denote by Lip A the class of continuous functions in [-1, 1], such that w(f; ~) _< Kfi ~, 0 < A _< 1 where k is a positive constant. The following two theorems concern interlacing properties of the zeros of orthogonal polynomials. THEOREM 2.1 [3]. If w is an arbitrary weight function and @(x) = (1 - x 2) w(x), then the zeros ofpn+l(w) interlace the zeros ofpn(~). Moreover, ifw(x) = ~(x)(1 - x)~(1 + x) ~, a,/3 > - 1 , 6 LipA, 0 < A _< 1, denoted by y2n+l,i = COS?72n+1,i, i = 1 . . . . , 2 n + 1 the zeros of pn+l(w)p,~(@), in increasing order, we have C ~2nq-l,z -- 772n,i+1 ~ --~ n with C positive constant independent of i and n. i ----1 , . . . , 2n Hilbe~Trans~rm 157 THEOREM 2.2 [3]. If W is an arbitrary weight function and Wl(X) = (1 - x)w(x), w2(x) = (1 + x)w(x), then the zeros of p,(w2) interlace the zeros of pn(wx). Moreover, ff w(x) = ~(x) (1 - x)a(1 + x) ~, a, 13 > - 1 , ~ 6 Lip A, 0 < A _< 1, denoted by z2,,, = cos a2n,i, i = 1 , . . . , 2n the zeros of Pn (w 1)Pn (w2), in increasing order, we have ~2n,i -- O'2n,i+l c ~ --, i = 1 , . . . , 2n -- 1 n with C positive constant independent of i and n. T h e next theorem concerns the construction of Lagrange interpolation processes on the zeros of Jacobi polynomials. THEOREM 2.3 [4]. Let {xn#(v~'~) }'~=l be the zeros of the Jacobi polynomial pn(v ~,~) and let Xn,l-s+k (v ~'e) = -I + k 1 + xn,1 (ve'6), S 1- Xn,n+j (V~'~) = 1 - ( r - j) xn, n (v 7,6) r k=0,...,s- 1, j -- 1 . . . . ,r. For a given bounded function g, we denote by L~,~,s(v~'e; g) the Lagrange polynomial interpolatn+r ing 9 on the knots {Xn#(V 7 '6 )}i=l-s" If the parameters r, s 6 N and 7, ~ satisfy the conditions ~+1 ~/ ~<~<~+~, 5 6 1 6 ~+~<s<~+~, 5 then ]lL,~,r,~ (v'~"~) H _< C logn, where C is a positive constant independent of n. Making use of the previous theorems, we are able to select the knots tj, j = 1 , . . . , M. In the following, we assume a, t3 < 1. Whenever, for example, a _> 1, then we assume (1 - x) a-[a] as the weight ([a] denotes the integer part of a), and f ( x ) ( 1 - x)[ ~1 as the function. Similarly, we behave whenever/3 >_ 1. Now we consider some cases t h a t more frequently happen. At first, we suppose - 1 < a,/3 _< 0. In this eventuality, the zeros of pm(v~+l'~+l)pm+i(v~,~ ) verify the assumptions and the conclusions of Theorem 2.1. Therefore, we choose as Gaussian knots Xm+la (v c~'z) , i = 1 .... , m + 1. B y using T h e o r e m 2.3 for 7 = a + 1,6 = / 3 + 1, we choose as interpolation knots {tj}Ma --1 ~ Xm, 0 (Va+l'/3+l) ,Xm, 1 (va+l'/3-kl) , ... , Xm,m (Va+lJ3+l) , Xm,rn+l (V°~'bl'/3+l) =- 1. Then, (2.2) becomes m+l Fro+, (f; xm,, = k=l (v s (x +i,k - s zm+l,k (v~,~) - xm,i (v~+1,~+1) i =0,...,re+l, and finally, we obtain ~I (va'B f ; t ) = Lm,l,1 (Fm+l(f);t) + f ( t ) H ( v a ' ~ ; t ) . 158 G. MASTROIANNI AND D. OccoRslo Now we suppose 0 < a,/3 < 1/2. By using Theorem 2.1, the polynomial pm+l(Va-l'~-l)pm(Va'" ) has zeros sufficiently far apart; therefore, the Gaussian knots are xm,i (va'"), i = 1,...,m. Moreover, from Theorem 2.3 used for 7 = ~ - 1, ~ = / ~ - 1, it follows tha~ the interpolation knots {tj }jM=I are Xm+l,~ ( v a - l ' " - l ) , i = 1 , . . . , m + 1. Let be 1/2 < a, ~ < 1. By Theorems 2.1 and 2.3, we have that Xm, ~ (Ca'"), i = 1 , . . . , m, are the Gaussian knots, while the interpolation knots {tj}M=l are Xm+l,i (va-l'"-l) , i = O,...,m + 2. Still now, we have assumed that a and /3 have equal sign. It is useful to consider the case 0 < a < 1/2 and - 1 < f~ <_ 0. Theorem 2.2 assures a "good" distance between two consecutive zeros ofpm(va'")pm(va-l'"+l), as long as by Theorem 2.3 for ~ = a - l , 5 = f l + l it follows r = 0, s = l . So that xm,~ (va'~), i = 1,...,m, are the Gaussian knots and Xm, i (va-l'B+l), i ----0 , . . . , m , are the interpolation knots {tj}~= 1. Other cases can be considered, but we omit details. As we have previously observed, our method is very simple, since it reduces to the construction of a Gaussian rule and to the evaluation of a Lagrange polynomial, both of them with respect to Jacobi weights. It is well known that for such weights there exist efficient methods [6,7] to calculate zeros and Christoffel constants. A matter of more general interest is that this method can be applied when we replace the Jacobi weight v a'z in (2.1) by w = ~v a'~, ~ E Lip A, 0 < A _ 1, provided that zeros and Christoffel constants are really computable, for instance ~ is a rational function (see [8,9]). Finally the previous procedure can be used to approximate weakly singular integrals, with singularities inside ( - 1 , 1), or Hadamard integrals. 3. C O N V E R G E N C E Denoted by IIn the class of polynomials of degree n at most, for any function g E C(°)([-1, 1]), En(g) = min [ [ g - P[[ PCII,~ is the best uniform approximation error by algebraic polynomials. The following theorems evaluate the difference between the exact function H(va'~f) and the approximant H(va'~f). Since (va'" f; t) = LM(FN(f); t) + f ( t ) g (va'"; t ) , /1 the error committed by our method is due to the approximation of F ( I ; t) = 1 f ( x ) - f ( t ) v a , . ( x ) dx X t by LM(FN(f); t) (see (2.1) and (2.3)). Then we have to estimate ~M(f; t) := F(f; t) - LM(FN(f); t). Ifc~, ~ >_ 0, f • C(°)([-1,1]), and f ~ w ( f ; u ) u - l d u for It] < 1, and we can state the following theorem. (3.1) < oc, (i.e., f • DT), then F(f;t) exists HAlbert Transform 159 THEOREM 3.1. Let v~'Z,a,/3 >_ O, and let f E DT, then II u(S)ll <- CEM-I(S)log2 M, where C is a positive constant independent of f and M. Whenever a,/3 < 0 and we assume only f E DT, then generally F ( f ) is not bounded at the endpoints :kl. However when f is a more regular function, for example f E LipA, with m a x ( - 2 a , -2/3) < A _< 1, the above method works on [-1, 1] and the error estimate is given by the following theorem. THEOREM 3.2. Let v a'~, a,/3 < 0 and assume f(k) E Lip A, with k > O, and m a x { - 2 a , -2/~} < A <_ 1, then log 2 M [[~u(f)[[ --< C U;~+ k , k >_ O, where C is a positive constant independent of f and M. As we have previously observed, H(va'~f), in general, is not bounded in ±1 for a, /3 < 0. Then it is natural to give a weighted estimate of the error. THEOREM 3.3. Let v ~'~, a, /3 < O, and let f E DT, then IlCM(f)v-"'- ll <_Clog2 M f l/M (3.2) w(f;u)u-ldu, J0 moreover, if $ E C(1)([-1, 1]), then log 2 M M EM-I(f'), (3.3) where C is a positive constant independent of f and M. REMARK. We suspect that log2M that appears in our estimate can be replaced by logM, as well as sometime happens in the estimate of the product rules. In any case, we can pay for this price in exchange for the simplicity of the method. 4. T H E PROOFS We need the following lemma. LEMMA 4.1 [10]. I [ f c c ( r ) ( [ - 1 , 1 ] ) , r > O, then [or each n E N there exists a polynomial qn of degree at most n >_ 4(r + 1) such that f(k)(x)--q(a)(X) < c o n s t [ n - l ~ ] r - k w ( f ( ~ ) ; n - X ~ ) , 0<k<r, q(k)(x) < c o n s t [ A n ( x ) ] r - k w ( f ( r ) ; A n ( x ) ) , k > r, -l<x<l, -1<x<1, where An(x) = n-1 x/ri- - x 2 + n -2. Setting E ( f ; t) - FN(f; t) := R N ( f ; t), by (3.1), we get I O u ( f ; t ) l <_d l F ( / ; t ) - L i ( F ( f ) ; t ) l + I L u ( R N ( f ) ; t ) l . Let qM be the polynomial of Lemma 4.1 related to the function f . Setting rM = f -- qM and taking into account that d2M(P) = O, VP E HM, we have • M(rM; t) ~ CIF(rM; t)l + ILM(F(rM); t) I + ILM(RN(rM); t)l. (4.1) 160 G. MASTROIANNI AND D. OCCORSlO PROOF OF THEOREM 3.1. Using [11, (3.22) of Lemma 3.6, p. 457], we obtain -l<t<l, ,F(rM;t)l<¢/~_ ~ rM(~--rM(t)_t va'a(x)dx<Cw(f;~) l°gM' a, fl> 0. On the other hand, IF(rM;±l)l <-Ctlrmll<-w (f; ~ ) . so that IIF(rM)II<Cw(f; ~)logM. (4.2) Consequently, / 1 \ IILM(F(rM))It ~ ClILMII" IIF(rM)II ~ ClILMII~ ~f; Mfl log M. Since we have chosen the interpolation knots tj, j = 1 , . . . , M in such a way that we get ,,LM(F(rM))II< Cw (f; ~ ) log2M. IILMII < c log M, (4.3) Now, we recall that I N IRN(rM;t)I <_C IF(rM;t)I+ ~AN# (v~'~) i=1 (4.4) zN,i - xg,i(v~'~) Setting = COS/gN#(Va'~) and t = cos/9, by using [11, (3.17) in Lemma 3.5, p. 455], we have, uniformly with respect to N IO-ON,,(v°.~)l>~ (v rM (XN,i(Va'~)) --rM(t) Zg,i(V ~'1~) -- t ~ C03 Combining the last inequality with (4.4) and (4.2), we get "RN(rM)" < Cw(f; ~ ) logM, uniformly with respect to N, and ,,LM(RN(rM))H< Cw( f ; ~ ) ,,LM,IlogM, so that IILM(RN(rM))II <-Cw( f; ~ ) log2M. Combining (4.2), (4.3) and the last inequality with (4.1), we obtain 'lgPM(f)ll< Cw(f; ~ ) log2M. Now for a polynomial P E HM, we have "~M(f)'I = 'I~M(f -- P)" < Cw ('f - P';-~-M) log2 M _< 2Cllf - PII log2 M. Making the infimum on P E HM, Theorem 3.1 follows. II Hilbert Transform 161 First we suppose k = 0 and assume f E Lip A, with m a x { - 2 c ~ , - 2 B } < A < 1. We start by (4.1). Making use of [11, (3.23) of Lemma 3.6, p. 457], we have P R O O F OF T H E O R E M 3 . 2 . IF(rM;t)l ~_ C l oM~ gM , max(-2a,-2~)<A< 1, -1 <t<l. (4.5) Now we observe that F(rM; +1) make sense for the restriction on A. For t = - 1 , we have IF(rM;--1)[ <_C f - l + ~ [rM(xL____r_Za(_l)] (1 + x) ~ dx d--1 + i_ 1 IrM(x) -- rM(--1)] Vc~,~(X)dx =: Cl(t) + C2(t) I+M_ ~ X+ 1 If(z) - f ( - 1 ) l ( 1 + x) ~-1 dx Cl(t) _< C f-l+ J--1 + (4.6) IqM(X) -- qM(--1)l (1 + X) ~ dx s-1 x+l Iq~(~)l(1 + x) n dx, (1 + x) n-l+~ dx + < C J -1 J -1 C - 1 < ~ < - 1 + M----~, since (1 - x) > 1. By applying Lemma 4.1, c c Cl(t) <_ M2n+2----------~ +~ c c 1 (1 + x) n+~-I dx <_ M2n+2-------~ + M~_~ M~+1+2 ~ . Taking into account that 2A + 2/7 > A, we have C Cl(t) <_ M---~. (4.7) Now we consider C2(t). Making use of Lemma 4.1, C2(t) < C [ J_-I+M-~ _ ~ v~'n-l(x) dx _ l+ (l+x)~+~-ldX+-M-~ dx C C < M2~+2n <_ M; ~. (4.8) The same estimate holds for t = 1. Combining (4.5)-(4.8), we can conclude NF(M)H < c logM M~ , max(-2a,-2~) < A_< 1, (4.9) and consequently, M [[LM(F(rM))H < C "Og2 ] - M ~ , (4.10) since ][RN(rM)II < C IIf(rM)[I N q- E ~N,i (Va'~) rM (XN# (V°'~)) -- rM(t) - ~ N . 7 ~ _~~ =: IIF(rM){I + E(t). /=1 30:3/6-L (4.11) 162 G. MASTROIANNI AND D. OoooI¢.SIO Setting XN,~(v~'~) = cosON,i(Va'a) and t = cos0, by using [11, (3.18) in Lemma 3.5, p. 455], we have r~(t) _< c ~ ~ , , (v °,~) r~ (x~., (v"'~)) - r~(t) z~,~ (v~,~) - t 1o-o.,.(.o,~)l_>~ < c log M v~+.},~+ j (t), -Ma (4.12) uniformly with respect to N. We observe that in this case t E { t l , . . . , ~:M}and ON,k(va,~) --Oj > C/M. Then by (4.11), (4.9), and (4.12), taking into account that a + A/2,/3 + )~/2 > 0, we get IIRN(rM)II < C l o g M - M ~ ' uniformly with respect to N, and rlLM(RN(rM))II<_CIILMtllo~M, so that log 2 M [[LM(RN(rM))H < C M------Z- (4.13) Combining (4.9),(4.10),(4.13) with (4.1), the theorem is proved for k = 0. To prove the theorem for k > 0, we start from (4.1) again. For It[ < 1, we have := Bl(t) + B2(t) + B3(t). (4.14) .1(,) ,.. Applying Lemma 4.1, Bl(t) <C {, r'-'v''o,''',., ~a_l Ix-t[ dx+ a-1 -~--~ dx } . Taking into account [11, (3.12) in Lemma 3.3, p. 453], we get Bl(t) < C{ (1-- Mt)-~-~+" f;-'~,~ (l +IXz)~+'?+~ v~' +"'~?+o(t) } ~ -- tl dx + MX+k logM . -- 1 Since (k + .~)/2 +/3 > O, making still use of [11, (3.12) in Lemma 3.3, p. 453], we get Bx(t) < C v~?+.,~-~+~(t) M),+ k logM. (4.15) By similar developments, we obtain B3(t) < C v~'Z~+~'~f-~+a(t) logM. MX+k (4.16) Hilbert Transform 163 Now we estimate B2. Applying Lemma 4.1, we have for t - (1 + t)/M 2 < ~z < t + (1 - t)/M 2, 1--t v 'a(x) dx B2(t) <_ C Jt--~M-~ C [t+M---'z k-l#x- k-l+x-~ <_ M,~+k_l jt_lM_+_~ V 2 ~-~, 2 + ~ ( z ) d x < C M~+k+l, (4.17) since (k + A - 1)/2 + a, (k + A - 1)/2 + fl > 0. Then combining (4.15)-(4.17) with (4.14), we get log M IF(rm;t)l < CM~+-----g, - 1 < t < 1. (4.18) Assume t = -1. By (4.14), since B I ( - 1 ) = 0, we have IF(rm;-1)l < C [J--I--I+M----C-~IrM(X~(_~M (--1)1 < C [_1+-% Ir~(G)l(x J-1 - [1 ~M- irM(x)_rM(_l)lv~,~(x)d (l + x)~ dx+j-l+ + x) ~ dx + /l_ - ]rM(X)]v~,~(x)dx ' X+I I+M~ C - 1 < ¢= < - 1 + M----7. Then, proceeding as in the case k = 0, we obtain IF(rm;-1)1 C <_ MX+k. The same estimate holds for t -- 1. Thus, IF(rm; 4-1)1 < C (4.19) M~+------~. Combining (4.18)-(4.19), we get C IIF(rM)[I <_ ~ Easily we obtain also IILM(F(rM))II <_~ Setting XN,i(v a'~) we get = cosON#(V~'~) logM. C and t = cosO, since (4.20) log 2 M. t C {tl,... ,tu} (4.21) and ON,k(v~'~) -- Oj >_ C/M, ,RN(rM;t)I<_C{IF(rM;t),+ IO-°N.,(v~'~)l>~)~N.i(Va.13)rM(XN,i(V~))~_._NN.~ ~ _~/ := M(t) } F(rM;t) + Z(t). (4.22) Then we have AN,, (v IO-ON.dV~. )l>xr r._ (xN,, ZN# (v ~'~) -- t + IrM(t)i AN# (v a'~) I. 164 G. MASTROIANNIAND D. OCCORSZO Making use of [11, (3.16) in Lemma 3.4, p. 454], we get E(t) <__C 1 M~+k ~-" IO-Omdv~',~)l>~ MIx, N,~ (v'~,:~)- tl "XN'dV'~'~) + :+~':~+a'~ (t) log M } M~+k <_c v~+aF"e+~'P Ma+k (t) log M + r(1 + x.,v,, (va't~)) B+ -k+~ C t<0 t>O Thus, taking into account that a + (~ + k)/2 > 0, /3 + (A + k)/2 > 0, and making use of [12, Lemmas 5.7 and 5.8, pp. 164-165], we have log M E(t) <_ C MX+k, Ill ~_ 1. (4.23) Combining (4.20) and (4.23) with (4.22), we get log M IlRN(rM)II < C M~+k, and consequently, (4.24) log M [[LM(RN(rM))t[ <_CIILM[IM~+k, so that log 2 M IILM(RN(rM))H <_C Ma+k. (4.25) Combining (4.20),(4.21),(4.25) with (4.1), the assertion is completely proved. | PROOF OF THEOREM 3.3. By (4.1), it follows OM(rM; t)v-~'-~(t) <_C{tF(rM; t)l + ILM(F(rM); t)l + ILM(RN(rM); t)]} v-a'-~(t). (4.26) Setting I = [-1,1], IM = It - (1 + t)/(2M2),t + (i - t)/(2M2)], I~M = I - IM, IF(rM;t)lv-~'-:3(t) --<CV-a'-~(t){IflrM(X)--rM(t)]v~'~(x)dx+ t I~i flrM(X)--'M(t)I X va'o(x)dx := Al(t) + A2(t). - ( J~- ~-~ .h-~ t ~(~i ~7~ -tl) }, dx+l:+¢---~IP~(~=)ldx _<c ~[~+'~'~ l+t 1-t t - 2M--~ < ~= < t + 2M-~. Hilbert Transform 165 By Lemma 4.1, we get Al(t)<_¢ {z <_ C To give a bound to ~w(f;u) u-ldu+~ co(f; U) U-1 < z w(f;u)u-tdu+w (f; du. (4.27) A2(t), we make use of [11, (3.12) in Lemma 3.3, p. 453], A2(t) <_CIIrMIIv-~'-a(t) f v"'Z(x)v~-_Ti dx. I'M Since [11] [v-~'-~(t)H (v~'a; t) l < C log M, a, f3 < 0, Itl _< 1, then A2(t) <_w (f; -~-) log M. (4.28) Combining (4.27) and (4.28), we obtain IIF(rM)v-O'-~ll (4.29) w(f;u) u-' du, < ClogM and consequently, 1 I[LM(F(rM))[v -~'-~ < Clog2 M f0 w(f;u)u-l du. (4.30) Taking into account IRN(rM; t)l v-~'-a(t) < c IIF(rM),-°,-~II + v-°,-~(t) Z ~N,~ (V°'~) i=1 rM (XN,~ (V~'~)) -- rM(t) ] x--;,,(v~,~--~-t (4.31) --: IIF(,-~)v-°,-~ll + r~(t). Now we set xN,i(v a'~) = COSON,i(V~'~) and t = cosO. Since t E {tt,... ,tM} and ON,k(V~';3) -Oj >_C/M, we have ON.k(V'~.~)--Oj ~N,~(,"'~) jxZCv~,~ - tl } Since AN,~(v a'o) ON.k(v°.a)--O~ IxN,, (v",~) - tJ by applying Lemma 4.1, we have <ClogM, a,~<0, It I_<1, 166 G. MASTROIANNIAND D. OccoRsio C o m b i n i n g t h e l a s t i n e q u a l i t y w i t h (4.29) a n d (4.31), we g e t '[RN(rM)V-a'-~H <_C{logMfol/Mw(f;u)u-lduq-w(f;1)logM} 1/M w(f;u)u-ldu. _< C l o g M Jo Thus, by Theorem 2.3, we get 1 IILM(RN(rM))V-"'-ZII w(f;u)u-ldu. < Clog2 M (4.31) Then (3.2) follows combining (4.29)-(4.31) with (4.26). Now assume f E C(1)([-1, 1]). Then by (3.2), we have N(I)M(f) v-a'-f~][ < C log 2 i Ilf'll - M (4.32) ' Let qM-1 the best uniform approximation polynomial of f~ and QM E HM such that Q ~ = qM-1. Then, by (4.32), we get IlCM(f) v-~'-~[I = [I~M(f --QM) v-='-~[I _< C l o g 2 5. N U M E R I C A L M I[(f -QM)'II)~ <_C_...._.~__ . l2 Mo EM_I(f,) g | EXAMPLES In this section, we present some numerical results obtained by using our method, showing that the numerical errors agree with the estimate of the previous theorems. EXAMPLE 1. This example can be found in [1]. .~_1 ex H(f; t) = -lx-t dx. In this case, the interpolation nodes are m + 2 and they are {Xm,i \V(1 ' 1m]ji=l ~ U {=J:=l}, while the knots of the Gaussian rule are the zeros of Pm+l(V°'°). In the following, we give some tables of the absolute error obtained through comparison with the analytical solution for various value of t. Table 1. m = 8. Table 2. m = 16. t IH(vC~'~f;t)-ffI(va'~f;t)l t ]H (v~'Df;t) - ~I (v~'~f;t) ] 0.01 0.146540E-09 0.01 0.422770E-013 0.05 0.131077E-09 0.05 0.265931E-012 0.1 0.8641595E-10 0.1 0.964777E-013 0.2 0.482977E-10 0.2 0.236846E-012 0.3 0.142488E-09 0.3 0.918579E-013 0.6 0.135784E-09 0.6 0.466931E-013 0.8 0.958530E-10 0.8 0.295877E-012 0.9 0.452853E-10 0.9 0.458954E-012 0.95 0.648729E-10 0.95 0.112702E-012 0.99 0.468135E-10 0.99 0.478533E-012 Hilbert T r a n s f o r m EXAMPLE 167 2. H(f;t) : 1 dx - - dx. = 1 "~(1.1) 2 - x 2 x - t This example can be found in [13]. In this case, we choose the same knots used in the previous one, but taking into account t h a t the singularities of the function 1/x/(1.1)2 - x 2 are "close" to the interval [ - 1 , 1], a larger number of knots is required. Table 4. m --- 32. Table 3. m = 16. t [H(va'#f;t)-ffI(va'#f;t)l t [H (v",Of; t) - ~" ( v " , ~ f ; t) l 0.25 0.295195E-04 0.25 0.348525E-07 0.99 0.700020E-03 0.99 0.340218F-,-06 Table 5. m = 64. t EXAMPLE IH ( v ~ ' ~ f ; $) - / ~ (va'f~f; t) ] 0.25 0.351559E-09 0.99 0.143051E-08 3. H(f;t) : f: zlzl dz t dx. 1 ~/1 - z 2 x - In this example, a = /3 = - 1 / 2 , then we choose [Sx rn,ik:ou 1~2,1~2"vim }Ji=l (.J {2:1}, as interpolation knots, while the zeros of Pm+l(V -1/2'-1/2) as the knots of the Gaussian rule. Table 7. m = 64. Table 6. m = 32. t ]H(va'Bf;t) - ~ I ( v a ' ~ f ; t ) l t [H (va'fTf; t) -- ~I (v ~'~ f; t) [ -0.99 0.572003E-06 --0.99 0.267368E-07 0.01 0.257048E-02 0.01 0.468045F-,-03 0.05 0.418544E-03 0.05 0.280239E-03 0.25 0. I35369E-03 0.25 0.156793E-04 0.8 0.655437E-05 0.8 0.595368E-06 0.9 0.282223E-05 0.9 0.225838F_,-06 0.999999 0.159008E-08 0.999999 0.221591E-09 Table 8. m = 128. t IH (v~'~f;t) - ~I (va,f~f;t) l -0.99 0.318389E-06 0.01 0.174744E-04 0.05 0.232971F-,-04 0.25 0.534911E-06 0.8 0.16786IE-06 0.9 0.115249E-06 0.999999 0.331633E-06 168 G. MASTROIANNIAND D. OCCORSIO REFERENCES 1. G. Monegato, The numerical evaluation of one-dimensional Cauchy principal value integrals, Computing 29, 337-354, (1982). 2. G. Monegato, Convergence of product formulas for the numerical evaluation of certain two-dimensional Cauchy principal integrals, Numer. Math. 43, 161-173, (1984). 3. G. Criscuolo, G. Mastroianni and D. Occorsio, Convergence of extended Lagrange interpolation, Math. Comp. 55, 197-212, (1990). 4. G. Mastroianni, Uniform convergence of derivatives of Lagrange interpolation, J. Comput. and Appl. Math. 43, 1-15, (1992). 5. W. 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