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ETNA Electronic Transactions on Numerical Analysis. Volume 30, pp. 278-290, 2008. Copyright  2008, Kent State University. ISSN 1068-9613. Kent State University http://etna.math.kent.edu NEW QUADRATURE RULES FOR BERNSTEIN MEASURES ON THE INTERVAL ELÍAS BERRIOCHOA , ALICIA CACHAFEIRO , JOSÉ M. GARCÍA-AMOR , AND FRANCISCO MARCELLÁN Abstract. In the present paper, we obtain quadrature rules for Bernstein measures on , having a fixed number of nodes and weights such that they exactly integrate functions in the linear space of polynomials with real coefficients. Key words. quadrature rules, orthogonal polynomials, measures on the real line, Bernstein measures, Chebyshev polynomials AMS subject classifications. 33C47, 42C05 1. Introduction. In this paper, we present quadrature rules for Bernstein measures on such that they exactly integrate functions in the linear space of polynomials with real coefficients. These types of rules have a fixed number of nodes and quadrature weights for each Bernstein measure. Since we have proved in the case of the unit circle the existence of a quadrature rule for Bernstein-Szegő measures with similar properties from the point of view of the exactness (see [2]), the use of the transformations between measures supported on the & interval and symmetric measures supported on the unit circle !#" $"% , the so-called Szegő transformation (see [1, 8, 15, 16]), allows us to obtain the results. This ' approach of relating quadrature rules on the unit circle and the interval has been used successfully to investigate Gauss-Szegő quadrature rules (see [3]). Indeed, for each Bernstein measure, we give the nodes and the weights. In order to apply our method in the computation of the integral of any polynomial with respect to the Bernstein measure, we only need to know the coefficients of the expansion of the polynomial in the corresponding Chebyshev basis. The well-known Clenshaw-Curtis quadrature rule (see [5]) uses this type of expansion in terms of the Chebyshev polynomials of the first kind but it exactly integrates polynomials only up to a certain degree depending on the number of nodes. Our method has unlimited exactness on the space of polynomials. The paper has the following structure. In Section 2, we introduce the Bernstein measures on and we recall the Szegő transformations of measures supported on a bounded interval and the unit circle. In Section 3, we deduce a quadrature rule for Bernstein-Szegő measures on the unit circle which is exact in the linear space ( of polynomials with complex coefficients. In Section 4, we prove the main results of the paper concerning quadrature rules for the Bernstein measures supported on the interval . In Section 5, we give some numerical examples, and a technical result about the changes of basis is presented. ' 2. Bernstein measures on the interval and Bernstein-Szegő measures on the unit circle . The aim of this paper is to obtain quadrature rules for the Bernstein measures 6 78 + / corresponding to rational modifications of the Jacobi measures )*,+.- /1024350 273 0 243 )2 9 Received March 26, 2008. Accepted for publication June 28, 2008. Published online on October 16, 2008. Recommended by L. Reichel. ETSI Industriales, Universidad de Vigo, 36200 Vigo, Spain ( : esnaola@, acachafe ; @uvigo.es, garciaamor@edu.xunta.es). The first three authors were partially supported by Ministerio de Educación y Ciencia under grant number MTM2005-01320. Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganes (Madrid), Spain (pacomarc@ing.uc3m.es). This author was partially supported by Ministerio de Educación y Ciencia under grant number MTM2006-13000-C03-02 and project CCG07-UC3M/ESP-3339 with the financial support of Comunidad de Madrid and Universidad Carlos III de Madrid. 278 ETNA Kent State University http://etna.math.kent.edu 279 NEW QUADRATURE RULES FOR BERNSTEIN MEASURES <=?>AB @ and CDE>FB @ , i.e., the so-called Chebyshev measures of the first for the parameters B @ , of the second kind <DJCIKB @ , of the third kind <G B @ , CIKB @ , and of kind <GHCI B @ (see [13, 16]). the fourth kind <I B@ , CL ' S N If MN40O2$3PRQ SUT1VXW S 2 is a positive polynomial on with real coefficients, we consider the following rational modifications of the above measures, i.e., the Bernstein measures: )* @ ] Z\[ (2.1) )*X_0 2Y3P Z ] 2 )*1bc0 2Y3P M'N40 2Y3 )2 MN.0 2Y3 ` )2 2 B 2 0O2$3P MN40O2$3 a8 Z B )* B 2 ` ] ^ [ )2 0 2Y3P ] 2 )2 2 M'N40 2Y3 a8 Z ' which, of course, are positive Borel measures on . Taking into account the Féjer-Riesz representation (see [11, 16]), we know that there o'ooqp S N exists an algebraic polynomial deN40 7& 3fgQ STXVXh S , with h SjiLk for , h Vmr lmgn B such that MxN40 yz{}|3%~" dN.0 x3" . n , without zeros in stuv6" $"7w oo'o , Next we show that, applying suitabletransformations, the measures )* , become the Bernstein-Szegő measure on Z Z , )cX0O|3 B. Xx U . In order to obtain this result, we recall the connection between measures supported on Z Z and , respectively, through the following four transformations (see [1, 8, 15, 16]). to the unit circle. 2.1. Szegő transformations of measures from the interval ' Let * be a nontrivial probability Borel measure on , absolutely continuous with respect to the Lebesgue normalized measure, with weight function #0O2Y3 , i.e., )*%0 2Y3]H#0O2$3)2 . We will transform this measure in the following four different ways. ' transforms the interval Z Z into , First transformation. Since 2 yz{4| Z Z B @ #0 yz{4|c3"{|6" )| then we can define a measure on such that and )7 0 |3L @ @ Z Z 0 3 . It induces another measure on that we also denote by for the sake @ @ of simplicity. Notice that when * is the Chebyshev measure of the first kind, then is the @ Lebesgue normalized measure. 'ooo¤ & If 0O2$3P QH T1V W O¡$ 0 2Y3 , with W i¢k , £ Jn , where ¡ 0O2Y3 is the sequence 7¥§¦ of Chebyshev polynomials of the first kind and ¨0©c3e QH T1V\W , then it is easy to relate the inner products induced by both measures * and as follows: @ ª 0 2Y3 ¡ 0O2$3q«'¬v ^ ª 8 ¨0©c3 ¨F®.¯ @° o «±q² Second transformation. The substitution 2³´yz{.| and the multiplication by b7µO¶ @· ¹º ¼ µ defines a finite positive Borel measure B on Z Z by )7 B 0O|3a¸ b µO¶ · )| if » @ ¸ ¼ )2F½ 8¾ ¯ @ @ ¯ . The measure B induces in a natural way another measure on that we also denote by B . Notice that when * is the Chebyshev measure of the second kind, then B is the Lebesgue normalized measure. oo'o¤ & If 0O2$3P QH T1V\¿ ®À, 0 2Y3 , with ¿ i¢k , £XJn , where À 0O2Y3 is the sequence 7¥§¦ of Chebyshev polynomials of the second kind and ¨0©c3 Q T1VÁ¿ , then it is easy to obtain the following relation between the inner products corresponding to both measures: ª 0 2Y3 À, ª 0 2Y3« ¬ ¨0 73 Â ¨A }¯ @° o @ « ± ETNA Kent State University http://etna.math.kent.edu 280 E. BERRIOCHOA, A. CACHAFEIRO, J. GARCIA-AMOR, AND F. MARCELL ÁN Third transformation. The substitution and the multiplication by 2ÃÄyz{4| Z Z b ¹º @ µ defines a finite positive Borel measure _ on by )7§_c0O|c3Å B@ #0©yzc{4|3"ÆUÇ B " )| if ¼ 8¾ @ Â ¼ )2?½ . The measure È_ induces a measure on that we also denote by _ . » ¸ ¯ @ @ Notice that when * is the Chebyshev measure of the third kind, then c_ is the Lebesgue normalized measure. o'oo¤ & 0O2$3 If 0O2$3= Q TXV É ÊL 0O2$3 , with É iËk , £LÌn , where Ê is the 4¥§¦ sequence of Chebyshev polynomials of the third kind and ¨0©c3= QH T1V É , then the inner products corresponding to both measures are related as follows: ª Ê 0O2Y3 ² ª 0 2Y3 « ¬ ² 8 ¨0©c3 ¨ @° ¯ ² Â ¯ o « ±Í Fourth transformation. The substitution 2fÅyz{.| and the multiplication by Z Z b4µO¶ · @ allows us to define a finite positive Borel measure Èb on by )7xb70|3 B@ #0yz{4|x3" yzÈÆ B " )| ¼' 8¾ if » @ ¸ ¼ )2Î½ . The measure §b induces a measure on that we also denote by Èb . ¯ @ @ ¯ Notice that when * is the Chebyshev measure of the fourth kind, then b is the Lebesgue normalized measure. 'oo'o¤ & If 0O2$3P Q T1V ) Ï} 0 2Y3 , with ) i¢k , £ Hn , where Ï 0O 2$3 is the sequence 4¥§¦ of Chebyshev polynomials of the fourth kind and ¨0 73aEQJ T1V ) , then we can relate the inner products corresponding to both measures as follows: ª 0O2Y3 Ï} ² ª 0 2Y3 « ¬ ² ¨0©c3 }¯ Â @ ° }¯ ¨ ² o « ±Ð Now, if we consider the Bernstein measures )* , )* B ,o')oo* _ , and )* b defined in (2.1) and we @ transform each )* by the th transformation, , then it is easy to prove that we obtain the same Bernstein-Szegő measure )c10 |3P B} X . 3. Quadrature rules for Bernstein-Szegő measures. Since our aim in this work is to 'oo'o obtain quadrature rules for the Bernstein measures )* , m , defined in (2.1), we are going to recall a recent result about quadrature rules for Bernstein-Szegő measures on the unit circle . p s Let d N 0©c3 be an algebraic polynomial of degree with zeros outside and d N 0 nc3 r n Z Z and let us consider the Bernstein-Szegő measure )7 defined on by (3.1) )710O|c3a )| ^ Z]Ñ Ñ dN Ò ° B Ñ o Ñ & If we denote by ÈÓ 0 73 the monic orthogonal polynomial sequence with respect to the Ø ÙU ¤IÚÎp 4¥§¦ N measure )7 ( Ô5ÕÖa×$0®.3 ), then Ó 0©c3%H ¯ V , for (see [16]). In [2] we have obtained the following quadrature rule for Bernstein-Szegő measures. & T HEOREM 3.1. Let )7 be a Bernstein-Szegő measure like (3.1) and let± ÈÓ 0 7¤Þ 3 Úß 7¥§¦ p N 3 be the corresponding Ô5ÕÖa×X0©.3 . Assume that Ó 0©c3ÛÜ ¯ (Ý T 0© for p @ with'o o'o Ë& â if Þã and Q Ý T . Then there exists a quadrature rule with nodes áà à @ which uses the values of the function and its derivatives in these nodes and Ý @ such that it exactly integrates functions in the linear space ( of polynomials with complex coefficients, i.e., for every i ( we get äcå (3.2) ± æ ¯ æ Ý @ 0©c3)*%0©c3 Moreover, ç -± ¯ @ Jn à for Á 'oo'oUè . T @ â TXVç -â â o 0 3 ETNA Kent State University http://etna.math.kent.edu 281 NEW QUADRATURE RULES FOR BERNSTEIN MEASURES Proof. See [2]. Notice that the above result can be completed with the following proposition where the explicit computation of the weights - â in the quadrature rule (3.2) is given. ± p ç P ROPOSITION 3.2. Let us assume that d N 0 73é É Nx(Ý T 0 3 , with Q Ý T , @ @ and let d - â be the numerators of the following partial fraction decomposition: Ñ B Ñ Ñ É N 0 ( Ñ Ý T ± @ N 3 N ( Ý T 0© Then the weights @ 3 ± ( Ý T Ù@ ° @ " dN40©c3'"E" d T â T @ B Ñ â @ æ Ý 8 -â d Â -â ° ± æ T â T @ @ o -â Ù@ â ê ° o @ ã}ë i , thus we can write Ñ d for 0 73" N -â ç Proof. Since ± æ of the quadrature rule (3.2) satisfy -â ç ± æ Ý @ ¯ d Ñ dN40©c3 Ñ 0 73 d N ¯ @ ± ° ( ±q² æ Ý T @ d 3 Ý T N ( Ý T @ N ® É N§( @ ( æ d Ý T ° ± -â â @ Ù@ ° ® @ Ý â T @ ± Ù@ ° ± ° Ý T ± ±Uí ° ° Ý ±Uí ±U² æ 8 ° ¯ 8Jìì'ìx8 @ @ â @ @ -â @ © ] ( ± Ý T @ 0 Ñ â T B Ñ Ñ É N Ý T ( Ñ Ñ N Ñ Ñ É N É Nx( B Ñ Ù@ ° N ± @ â T @ â Ù@² ° ê æ 8tì'ììx8 -â â T @ o -â Ý â Ù@í ° ê Taking into account the previous decomposition and the Cauchy’s theorem, we get ä )| F© ° ^ Z]Ñ ä7å ^ Z )c Ñ Ñ Ñ d æ Ý ±q² æ d â T æ ± d 0 73jîï B N 0©c3 Ñ ä å ^ Z Ñ 0 73 B Ñ Ñ dN1® Ò ° ¯ @ -â 0ã â T @ â T @ ¯ 0 3 @ @ ³ 3ë ) â}ðñ 8Jìì'ì8 ° æ â T Oó ô®õ ² æ Ý Ý -â 0ã â ¯ ° @ Ý ! 3ë 'oo'oUè T æ Ý â T @ ± æ ¯ T @ 'ooo @ -â Ù@ ° ê @ d @ ± æ 0©c3òîï d @ ä å ^ Z ±Uí 8 -â â T1V ! -â Â @ â )c â}ðñ ° o ã6ë Thus using (3.2) we have - â for ; ãmHn . âqö In the next section, weç obtain quadrature rules for Bernstein measures on applying the previous results and the transformations of measures introduced in Section 2. ETNA Kent State University http://etna.math.kent.edu 282 E. BERRIOCHOA, A. CACHAFEIRO, J. GARCIA-AMOR, AND F. MARCELL ÁN ' . Let us consider the Bernstein 4. Quadrature rules for Bernstein measures in measures given in (2.1) and the Bernstein-Szegő measure (3.1) obtained using the above transformations. Our aim is to prove that for each of these Bernstein measures there exists a quadrature rule with a fixed number of nodes and weights, which is exact in the linear space ÷ of polynomials with real coefficients. ¼ ¼ù§¼ . There exists a quadrature rule using the T HEOREM 4.1. Let )* 0O2Y3 cø @ & & nodes x Ý T and the weights @ exactly integrates polynomials, i.e.,ç ý T @ ¯ -ûúûúûú @ â T1V -ûúûúûúû- ± Ýü æ ÷ i -â ¿ 0 2Y3Pÿþ TXV ¡$ with ¿ 0O2Y3 we get ä @ 0 2Y3)* @ îï @ ¯ T -â â T1Veç @ æ If 0O2$3Å 0 2Y3 ¬² ä with ¡$ O 1 T V\¿ 0 2Y3 8 þ ¨0 73 ¨ 0 ¯ @ 3 Q @ 0 2Y3 @ ¯ ä ± ] þ ¿ )| ^ TXV ^ Z ¯ Ñ Ñ dN @ â o ^ Ù T ðñ Ù becomes the measure )c given by @ Ò ° Ñ B ä 8 æ ¿ þ Ñ , such that it p Ý T 'oo'o i?k MN40O2$3 æ , £m n , and . Therefore, we can write B 2 Q iFk T1V )2 Z\[ ¿ ¿ þ Proof. Using the first transformation, the measure )* BU. X U . )710O|3a @ ^ 0O2$3P ± æ ¯ æ Ý , with @ ¯ ¨0©c3Û ^ T1V ¯ 1 T V þ Q )| ^ Z Ñ , then o B Ñ ° Ñ dN B Ñ for òHxÒ . 'oo'o In order to compute these integrals, we apply Theorem 4.1. Indeed if are the o'oo Ý @ Ý T Q zeros of are their multiplicities with d N 0©c3 , which are located in s , and p & Ý @ @ , then there exist weights - â T -ûúûúûú - â T1V -ûúûúûúû- ± such that for every i ( we get ä ç ^ Z]Ñ ¯ ± æ ¯ æ Ý B Ñ Ò ° Ñ dN @ ¯ )| ¢® ° Ýü @ Ñ T â -â ç â TXV @ @ © ° and ä ¢© Ò ° ä æ ¯ þ T1V ¿ X T V þ Q 0 73% 8 © ^ ° Ñ dN ¯ Therefore, if ¨ ^ Z]Ñ ° B with ¿ iák )| ^ Z Ñ ± æ ¯ æ Ý )| Ñ dN1© Ò ° B Ñ Ñ and, as a consequence, the statement holds. B Ñ Ñ T â T1V @ @ ç -â , then ^ ± æ ¯ æ Ý T @ â T1V o ® @ îï â ç -â ° æ þ T1V ¿ ^ â Ù T Ù ðñ ETNA Kent State University http://etna.math.kent.edu 283 NEW QUADRATURE RULES FOR BERNSTEIN MEASURES o'oo R EMARK 4.2. When d N 0©c3 has simple zeros @ and the quadrature rule means that if 0O2Y3auQ T1VÁ¿ þ ä @ )2 0O2$3 ] Z\[ B 2 @ ¯ ^ ¡$ 0 2Y3 N æ M'N40 2Y3 B ø T HEOREM 4.3. Let )* B & Ý T and the weights @ integrates polynomials, i.e.,ç ý 0O2Y3a & T -â -ûúûúûú @ Ýü ä @ 0 2Y3)* @ æ 0O2$3P @ ¯ ± æ ¯ æ Ý B T â T1V @ -â ç ) þ â T1V 1 T V þ ®À ) and ¨0©c3RQ 0O2Y3 ª ª ' « ¬ 0 2Y3 Since for ä i @ B 0O2$3)* @ ¯ ä we have ¨ ¨ 1 T V þ 0 73 0 73, Â ) 1 T V þ ) þ ) T1V ¯ )| ^ Z Ñ Ò ° Ñ dN B Ñ ä ç B â ° T1V ° o ÷ . Using the results about the becomes )710 |3P BU. X U . If i B c3P þ X T V þ Q Â ) c¨ B TXV ¯ ¨0©c3 æ Ñ Â ) þ and c¨0 -â ª }« ± i¢k , then @ 0 2Y3 æ o æ ° ý ¨0 3 Q ° ¬ , 0O2Y3 Proof. We write » @ 0O2$3)* B 0 2Y3 ¯ @ second transformation given in Section 2, the measure )* 0O2$3PHQ ^ ç @ with ) 0 2Y3 T1V we get T1V À ) þ @ ¯ æ 0O2$3% ç @ þ ¿ N @ . There exists a quadrature rule using the nodes p , with Q Ý T , such that it exactly â T1V -ûúûúûú - ± ÷ i ¼' ¼ @ ù¯ §¼' æ T o'oo' in s , the weights are , with ¿ i¢k , then ç N o « ± 0 3 Â ) B , then )| ^ Z Ñ Ò ° Ñ dN B Ñ mJ Ò Ñ Proceeding like in the proof of Theorem 4.3, we can apply Theorem 4.1 and get ä ¯ ä æ þ ) ) þ 1 T V as well as æ Â B T1V ¯ ä @ ¯ 0O2Y3)* B æ þ 1 T V @ ± æ ¯ æ Ý T @ â T1V À ) ç T 0O2Y3 B @ ± æ ¯ -â þ T1V ç â T1V Ñ T B )* ®) -â þ ) Â â ® B T1V ° â o © ° 0O2$33 ç æ ) þ 1 T V â T1V @ æ -â @ æ @ Ñ Ñ @ æ Ý Ò ° Ñ dN 0 2Y3P ¯ B Ñ Ò ° Ñ dN )| @ @ Ñ ^ Z]Ñ Therefore, we have ä ^ Z ± æ ¯ æ Ý )| ° â æ ® ° ç -â þ T1V ©) Â B ° â ® ° o o ETNA Kent State University http://etna.math.kent.edu 284 E. BERRIOCHOA, A. CACHAFEIRO, J. GARCIA-AMOR, AND F. MARCELL ÁN o'oo R EMARK 4.4. If the zeros §N , of @ N and the quadrature rule becomes 'ooo @ ç ç ä @ æ þ 0O2$3 T1V @ ¯ ®À ) B )* æ 0 2Y3P T are simple, then the weights are 0©c3 N N æ d ç @ þ ) æ T1V þ ç Â ) B T1V o Â ¼ ¼ @ ¼ ù ¼ T HEOREM 4.5. Let us consider the positive measure )* _ 0O2$3 @ ' & @ ¯ supported on . Then there exists a quadrature rule with nodes Ý T and weights & p @ T â T1V -ûúûúûúû- ± -â -ûúûúûú , with Q Ý T , such that it exactly integrates polynomials, i.e., ç Ýü @ @ ¯ ý æ ÷ i 0O2Y3a we get ä ± æ ¯ æ Ý @ @ @ æ 0 2Y3)*X_c0O2$3P @ ¯ T â T1V @ -â ç Q Ï. and ¨0 73% 0 2Y3 ª ² ª « ¬ Í 0 2Y3 Taking into account that, for ä i 0O2$3)*X_c0 2Y3P Â â @ ° T1V ° o . Then if 0 2Y3 B. Xx Â @ þ ¨0©c3 æ T1V ¯ « ± 1 T V þ þ þ ª Q æ ç ² 0 3 0 3% @ ¯ ä @ ¨ -â , we get i¢k , c¨ ² ¨0 73 æ 3 i¢k becomes )cX0O|3P _ , with X T V þ Q 0© TXV Proof. Using the third transformation, )* 1 T V þ â ° with 0O2Y3 þ Ï. þ 1 T V ¨ o « ± 0 73 , then Â @ TXV )| ^ Z Ñ ° Ñ dN o B Ñ Ñ Proceeding like in the proof of Theorem 4.3, we can apply Theorem 4.1 and deduce ä þ ä ¯ T1V ¯ as well as æ æ þ Â @ T1V ^ Z ^ Z ± æ ¯ æ Ý @ Ñ d Ñ Ñ d @ @ T @ â T1V ç T B @ ± æ ¯ Ñ -â þ TXV T ° â -â þ Â æ ° ç -â þ T1V â ° o @ ° TXV â ° T1V æ ç þ @ â T1V @ -â â TXVç æ 0 2Y3)*X_70O2$3P ¯ Ñ Ñ æ @ æ Ý ® ° N B Ñ Ò ° N )| Hence, ä Ñ ± æ ¯ æ Ý )| Â @ ° â ° 0 3 o ETNA Kent State University http://etna.math.kent.edu 285 NEW QUADRATURE RULES FOR BERNSTEIN MEASURES 'ooo' R EMARK 4.6. If the zeros and the quadrature rule is ä @ æ þ 0 2Y3 TXV @ ¯ Ï} of d N @ æ )*X_0O2Y3a ç T þ T1V @ æ þ ç Â @ T1V N @ ç N æ o'oo are simple, then the weights are 0 73 N ç o ¼ ¼ @ ¼ @ Â ¯ ¼ ù T HEOREM 4.7. Let us consider the positive measure )* b 0O2Y3Ü & @ supported on . Then there exists a quadrature rule with nodes Ý T and weights & p @ T â T1V -ûúûúûú - ± -â -ûúûúûú , with Q Ý T , such that it exactly integrates polynomials, i.e., ç Ýü @ @ ¯ ý æ ÷ i 0O2$3)* ¯ ± æ ¯ æ Ý @ @ æ b 0 2Y3P @ T ç â T1V @ -â þ T1V 0O2$3P Q 1 T V þ ª ©ÊA and ¨0©c3 0 2Y3 «¬xÐ Q ² ª ' 0 2Y3 1 T V þ 8 ¨ 0 73 ä ä @ 0 2Y3)*1b40O2$3P )* 8 for â @ ° © ° o . Then if B. Xx q @ T1V o «± ¨0 3 i , then Â Â 8 ¨0©c3 @ þ TXV ¯ ª æ TXV , we get þ )710 |3Å iák Â -â ç «± 1 T V þ þ ² ¨0 3 æ iFk æ 3 becomes , with 8 0© b 73%uQ @ ¯ Therefore, taking into account that c¨0 ² â ° with 0 2Y3 Proof. Using the fourth transformation, ÊA 0O2$3Pÿþ T1V we get ä @ )| ^ Z Ñ ° Ñ dN o B Ñ Ñ Proceeding like in the proof of Theorem 4.3, from Theorem 4.1 we get ä ä ¯ þ æ þ T1V Â @ ^ Z]Ñ Hence, ä 0O2$3)* ¯ ± æ ¯ æ Ý @ @ @ Ñ Ñ d T @ â T1V ç ± æ ¯ Ñ -â þ T1V T ° â T1V æ ç â T1V @ -â þ T1V ° ç -â þ T1V æ 8 ® þ @ æ b 0 2Y3P @ -â ç â T1V æ Ý ° N T B Ñ æ @ Ñ )| ^ Z B Ñ Ñ dN © Ò ° ± æ ¯ æ Ý )| T1V ¯ and æ â ° ® Â â o @ ° © Â ° @ ° â ° ® ° o ETNA Kent State University http://etna.math.kent.edu 286 E. BERRIOCHOA, A. CACHAFEIRO, J. GARCIA-AMOR, AND F. MARCELL ÁN 'ooo' R EMARK 4.8. If the zeros and the quadrature rule is ä @ æ þ ¯ Ê 0 2Y3 T1V @ of d N @ N æ ç æ )*1bc0O2Y3a T o'oo' are simple, then the weights are 0 73 N ç @ þ æ 8 T1V ç þ Â @ T1V N @ ç o R EMARK 4.9. In order to compute the nodes of the quadrature rules, we propose to ± W 3 0 2 use the next property. If is a factor of the polynomial M N 0O2$3 of arbitrary degree p B ³ [ W W W > , then the Joukowski transformation of with modulus less than one ( with B ! [ W "W > ".½ . Therefore, if all the ) is a node of the quadrature rule withmultiplicity factors are linear with real zeros, then the nodes are in . Otherwise if the factors are quadratic with complex zeros, we have a pair of complex conjugated nodes. Notice that this fact does not play a relevant role in the method. With this method for determining nodes and the previous one given in Proposition 5.1 for determining the weights, we prepare the method in order to be applied without errors in the calculus. 8Ã 2 , which is 5. Numerical examples. Let us consider the polynomial M '0 o2Yo'3F o @ positive on 8³ as well as the Bernstein measuresB )* for , , given in (2.1) with 8 ^ M 0O2$3 2 . Notice that M 0 yz{4|c3#K" d 0 xÒ'3'" , with d 0©c3E ; see Remark 4.9 @ @ @ @ for the details. In order to see how our method works and using the theorems of the previous B ³ 2 2 section, for the polynomial 0 2Y3a , we compute the integrals » @ 0O2$3)* 0O2Y3 'oo'o ¯ @ for a as follows. First we write 0 2Y3 in terms of the four Chebyshev polynomial bases ^ 0O2$3P 0O2$3P Ï ¡ B B 8 ¡ 0O2$3 _ Ï B 0O2$3 @ V 0O2Y3 8 _ Ï B 0O2Y3 ¡ 0O2$3 @ À 0 2Y3% V 0 2Y3 B Ê 0 2Y3% 8 B B@ À 0 2Y3 0 2Y3 @ 8 B@ Ê 0O2$3 B@ Ê 0 2Y3 @ V 0O2Y3 o Now we apply the result concerning the Bernstein-Szegő measure in order to get the nodes 8u ^ B @ and we compute and the weights. Since d 0 73 , then the node is as @ @ @ ç follows: ä ç @ )| 8 ^ Z " ¯ B ^ " äcå ^ Z BÙ @ Â 8 B B@ o Therefore, ä ä @ 0O2$3)* @ @ 0 2Y3P @ ¯ ä ^ ä @ 0O2$3)* B ä @ @ ¯ B B @ B @ ^ @ 0O2Y3 @ 0O2$3 ^ 8 @ @ @ 0O2$3 ! 2$3 )2 2Y3 _ ^ " @ _ @ V 0 2Y3 Z B ^ @ ` Ï ^ # ^ 8 0 B 2 8D 0 8 Ï 2 ] Z b )2 B ] Z\[ 0 2Y3 @ ^ ^ [ À ^ ^ Ï V 0 2Y3 ° 8J @ ¡ 0O2Y3 @ ^ @ @ À 8 ¡ 0O2Y3 B ^ B 0O2$3)*X_0 2Y3P ¯ B @ ¯ ¡ @ 0 2Y3P @ ¯ ä @ ¯ ^ 8 ] 2 a8 ^ 2 @ 8 0 )2 2Y3 ETNA Kent State University http://etna.math.kent.edu 287 NEW QUADRATURE RULES FOR BERNSTEIN MEASURES and ä ä @ @ 0 2Y3)*1b70O2Y3a @ ¯ ¯ Ê 0 2Y3 @ B 8 B ^ 8 0 2Y3 @ $ 8 ^ @ 8 ^ V 0O2Y3 8 _ ` Ê ^ @ 8 Ê 2 8 ^ 2$3 o ^ @ )2 8 0 @ 2 ] Z B ^ @ a8 In the previous computations, a key step was the explicit expression of the polynomials in the appropriate basis. In order to simplify these computations, next we present a result about the change of basis. Â B Â B S @ W S 2 @ ¿ S ¡ S 0O2$3 , with ¿ S PÂ ROPOSITION 5.1. If 0 2Y3 Q SUT1 , then 0O2$3e Q SUT1 VE VÞ S S @ - âWâ , where we denote by h - â the entry corresponding to the l -row and ã -column Q VÞh â TX ÷ of the matrix of change of basis in between the canonical basis and the basis of the first kind Chebyshev polynomials. These entries are B V -â ° B â T1V -ûúûúûú - h n n ^ n n &" ^ % &" ^ n oo'o n ^ ¤Å 0 ^%^ B ¯ n Â @ * +, oo'o ì'ìì¤ 3 ¤Å³ ^ _ 0 3ë n ^ B S n ^ n ¯ B S @ 8 l ^ B S Â ^ ^ 0 n 8D l ^ ë @ 8 ^ 30 l ^ B S Â _ 3 ^ 0 n .% 8 ^ l ^ B S Â 'oo'o¤ @ Â * +, o'oo n B S Â ( 8D l 3 ' ^ ¤ ^ ¤Û l @ , - â ° â T1V -ûúûúûú - B n ^ 30 ^ B ¯ B S Â ¤ 0 / 30 ë n h ^ ¤ ^ B 8 l 'oo'o and for ljtn ')( n , B S - â ° â T1V -ûúûúûúû- B h " % @ oo'o¤ for l Â @ ^ B S n ^ 8 l ^ B S Â B n 0 ^ 8 l ^ ë 8 ^ 30 l ^ B S Â b 3 n ^ 0 21 8 l 3% 04 8 ^ 30 l ë @ oo'o n ^ B ^ B S Â ^ ¤ 30 l 8t ¤á 8 ^ l 3 n o Notice that if the degree of the polynomial 0 2Y3 is even, then some analogous formulas can be obtained. The other changes of bases can be done using the previous ones. Taking into account the expressions of the different Chebyshev polynomials in terms of the Chebyshev polynomials of ETNA Kent State University http://etna.math.kent.edu 288 E. BERRIOCHOA, A. CACHAFEIRO, J. GARCIA-AMOR, AND F. MARCELL ÁN the first kind, if 0O2$3P ^ ¿ V ^ ^ ¿ V À B ¿ ° æ ¯ V 0O2Y3 8 ^ ¿ V 8 Ê @ ¿ @ @ ° Ï V 0O2$3 @ ¿ T B À, ° 8 8 À ¿ 0 2Y3 @ ¯ @ ¯ Â ¿ ÊL ° @ 8 Ê ¿ 0 2Y3 ¿ 0O2$3 0O2$3 À 0O2$3 @ æ ¯ 8 8 ¿ T Â ¿ @ æ ¯ 8 V 0O2$3 ° ¿ ¿ T 0 2Y3P ^ ^ 0 2Y3P , then S 0O2$3 B 0 2Y3P ¡ QHST1VÁ¿ S ¿ Â Ï ° @ 8 Ï ¿ 0 2Y3 0O2$3 o @ R EMARK 5.2. Taking into account these changes of bases, we can calculate the number of operations (sums and products) that we ¤ need for applying our method for the computation of the integral of a polynomial of degree with respect to the polynomial modifications of Â BU Â ^ B @ , the number the Bernstein measures )* 0 f 3 . The number of sums is Â ¤ of products is B @ , and a prefixed number of evaluations of a polynomial of degree exist. Taking into account the high number of zeros of the matrix of change of basis, a more efficient algorithm could be developed. If we want to apply the quadrature rule to a function which is not a polynomial, usof the function or the expansion of an ing, e.g., the measure )* , then either the expansion & @ approximation of the function in the basis ¡ 0O2$3 is needed. 4¥§¦ ' Let us consider, e.g., an analytic function on . In order to compute @ W N ¡ N70 2Y3 and we compute exactly the integral ¯ Va by QJN T1 » @ 0 2Y3)* 0O2$3 , we approximate @ ¯ @ #5 ¯ V@ W N . The idea is to approximate » @ 0 2Y3)* 0O2$3 by » @ 0 Q N T1 @ ¯ @ , and next to evaluate the error; see [6, 7]. ¯ @ A well-known method based on the FFT (fast Fourier transform) can be used to estimate & ¡ 0 2Y3 the expansion in the basis of the interpolatory polynomial of 0O2$3 defined in 7¥§¦ . It can be applied when the evaluations of the function are determined in fixed nodes. & ^ ¡ T h We will choose h & Ì and the zeros '2 -ûúûúûú of the polynomial m0 2Y3 as & Ì @ Z Z nodes. Let us consider '| such that yzc{4| 2 , Â and such n n ^ that yzc{ , as well as the FFT associated with these consider B 2 @ nodes, i.e., we ÙU Â Ùq N B @ ¿ N§ Q such² that 0O| 3m 0 73ò 0 3 0 2 3 . Then 0O2Y3 ¯ N T1V ¯ ¯ V@ W N ¡ » @ 0 Q N T1 ¡ ¯ @ 0O2$3 N 0O2$33)* @ > 7 =: 7 N70 2Y33)* 0 2Y3 @ 7 96 8 Ù Â B > ? > : 7 ý ;6 'oo'o 7 ;: <8 ? A@ @ B h with 2Î satisfies 0O2 3ò 0 2 3 , . From thep symme try of the nodes, the problem has a solution using only functions of the form yzc{0 |3 with B h 0 h z 3 operations and using a well-known algorithm. The solution is the expansion of & . the minimax interpolation polynomial in terms of the basis ¡ 0 2Y3 7¥§¦ Next we present three numerical experiments based on the ideas developed above. We ¼ 8j Â ø ¼' ¼ consider the measure )* used in Section 5 with M 0O2$3P 2 , i.e., )* 0O2Y3 b Q N T1V W N C 9D FE ¡ N70O2$3 ^ / G / nodes, i.e., a few number of nodes in order to have a low computational cost. @ @ @ @ ¯ and we take The FFT algorithm is well developed in all the standard calculus programs. Although we can use any of them, we apply the FFT algorithm using the Numerical Math “Trig Fit” of Mathematica. The discretization has an effect on the accuracy of the quadrature which is the same as we have when we use a polynomial interpolatory quadrature for a function, i.e., we have an error given by error of the interpolatory process. For the three analytic functions, we obtain the function 0©c3 which gives the solution of the complex interpolatory problem and we present a table with the approximation value of the > ETNA Kent State University http://etna.math.kent.edu 289 NEW QUADRATURE RULES FOR BERNSTEIN MEASURES integral using our method, the approximate value of the integral obtained using the command NIntegrate of Mathematica and the difference between both approximations. > &H . " &% IFI KJ KJ L % " I I KM I "&" % &" %&% / N1;I "F I OM 4 F I % I KM M I "&" FP N1 & 1 KM I " I " &F" IQ1 KM " F "&" J 1 IF% KM I Function Approx. Approx. NIntegrate Bound error % F % % F % N 1 5 LH R . 5.2. Function > " %S1 .% / L 1 &" IQ1 %4 " F "F" % " % % % I FI " I I " % " 1 " 1;J %Q1 S%&M IF% P / 5.1. Function Ç o c^ 0 73% 8 UyÆUÇÈé2 ] 8 @ n ¯ o ^ n n ] o f @ n @ n o Â Í B¼ Â 8D6o n 8 o n @ @ nnn o @ ³o b @ o ^ n ¯ ] } 8 8 o o } È ^ n 8 Èn o @U@ 8 o o } n B n n ¯ 8 o o n ^ n . ^ n @ 8 _ o n ^ n n R 'I Bound error 5 M / 5.3. Function . 1 % "&" T1LI % 1 1 % > / I 4 1 J "&" %S1 M &" 1;IF%S1 P I I&I 1 I&IFI&% 1 F% % I " F " & / Function Approx. Approx. NIntegrate Bound error R % %" % %" % 5 4 LF" Function Í Â Â B¼ 0©c3% Approx. NIntegrate o ^ ^ o ^ ^ È]8D o È 8 B o } 8 o } 8J o n . ^ n 8 o n ¼ Â nnn ^ È BU¼ Í Â _ 8 o } 8 o @ o n @ }o ^ b 8 n o n ^ @U@ 8 nnnn ^ n }o ^^ n 8J _ @ % È @ b n o ^ n n 8Jo ^ 8 V 4o ^ 8 ¯ _ o &&" Approx. _ ¼ Â ^ J / @@ @ ¼ # V @ ¯ 4J b 8 @ o ^ ^ n n 8 _ n x n 8 n V @ n ¯ 7o n ¯ o B ^ 8Jo o ^^ ^ n V ^ _ .o ^^ 4 8 n 8 @ _ o é ^ _ ^ n n ¯ o @ o n ] o ^ } 8Jo o @ n ¯ ^ nnnn B n ¼ n ¯ }%8 ^ nn 8 o ^ ^ @ Ç qyÆUÇé2 0©c3% 8 o B n ¯ }o o 8 n ¯ 8 8 b @ n ¯ o o }' ^ ' n n nn o B n n n 8 @ b 8 o ^ n o x nnn n @ _ % @ n }o ^ c^ o ^ n n ¯ Acknowledgments. The authors are very grateful to the referees whose suggestions contributed to the improvement of the manuscript. In particular, they thank them for pointing out references [3, 6, 7]. REFERENCES [1] [2] [3] [4] [5] [6] E. B ERRIOCHOA , A. C ACHAFEIRO , AND J. G ARC ÍA -A MOR , Connection between orthogonal polynomials on the unit circle and bounded interval, J. Comput. Appl. Math., 177 (2005), pp. 205–223. E. B ERRIOCHOA , A. C ACHAFEIRO , AND F. M ARCELL ÁN , A new numerical quadrature formula on the unit circle, Numer. Algorithms, 44 (2007), pp. 391–401. A. B ULTHEEL , L. D ARUIS , AND P. G ONZ ÁLEZ -V ERA , A connection between quadrature formulas on the unit circle and the interval , J. Comput. Appl. Math., 132 (2001), pp. 1–14. T. S. C HIHARA , An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. C. W. C LENSHAW AND A. R. C URTIS , A method for numerical integration on an automatic computer, Numer. Math., 2 (1960), pp. 197–205. D. E LLIOT , The evaluation and estimation of coefficients in Chebyshev series expansion of a function, Math. Comp., 18 (1964), pp. 274–284. ETNA Kent State University http://etna.math.kent.edu 290 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] E. BERRIOCHOA, A. CACHAFEIRO, J. GARCIA-AMOR, AND F. MARCELL ÁN D. E LLIOT AND G. S ZEKERES , Some estimates of the coefficients in the Chebyshev series expansion of a function, Math. Comp., 19 (1965), pp. 25–32. J. G ARC ÍA -A MOR , Ortogonalidad Bernstein-Chebyshev en la recta real, Doctoral Dissertation, Departamento de Matematica Aplicada I, Universidad de Vigo, 2003, in Spanish. W. G AUTSCHI , Orthogonal Polynomials: Computation and Approximation, Oxford University Press, Oxford, 2004. YA . L. G ERONIMUS , Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval, Consultants Bureau, New York, 1961. U. G RENANDER AND G. S ZEG Ő , Toeplitz Forms and Their Applications, University of California Press, Berkeley, 1958. Z. S. G RINSHPUN , On oscillatory properties of the Bernstein-Szegő orthogonal polynomials, J. Math. Anal. Appl., 272 (2002), pp. 349–361. , Special linear combinations of orthogonal polynomials, J. Math. Anal. Appl., 299 (2004), pp. 1–18. T. R IVLIN , The Chebyshev Polynomials. John Wiley and Sons, New York, 1974. B. S IMON , Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI, 2005. G. S ZEG Ő , Orthogonal Polynomials, Fourth ed., American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, RI, 1975.

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