Chapter 3 Orthogonal Polynomials and Related Approximation Results The Fourier spectral method is only appropriate for problems with periodic boundary conditions. If a Fourier method is applied to a non-periodic problem, it inevitably induces the so-called Gibbs phenomenon, and reduces the global 1 convergence rate toO(N − ) (cf. Gottlieb and Ors ag (1!""##. $onse%uently, one should not apply a Fourier method to problems with non-periodic boundary conditions. Instead, one should use orthogonal polynomials which are eigenfunc- tions of some singular &turm-'iouville problems. The commonly used orthogonal polynomials include the 'egendre, $hebyshev, (ermite and 'aguerre polynomials. The aim of this chapter is to present essential properties and fundamental appro)imation results related to orthogonal polynomials. These results serve as preparations for polynomial-based spectral methods in the forthcoming chapters. This chapter is organi ed as follows. In the *rst section, we present relevant prop- erties of general orthogonal polynomials, and set up a general framewor+ for the study of orthogonal polynomials. ,e then study the -acobi polynomials in &ect. ../, 'egendre polynomials in &ect. ... and $hebyshev polynomials in &ect. ..0. In Sect. ..1, we present some general appro)imation results related to these families of orthogonal polynomials. ,e refer to & ego2 (1!"1#, 3avis and 4abinowit (1!50# and Gautschi (/660# for other aspects of orthogonal polynomials. 3.1 Orthogonal Polynomials Orthogonal polynomials play the most important role in spectral methods, so it is necessary to have a thorough study of their relevant properties. Our starting point is the generation of orthogonal polynomials by a three-term recurrence relation, which leads to some very useful formulas such as the $hristoffel-3arbou) formula. ,e then review some results on eros of orthogonal polynomials, and present ef*cient algorithms for their computations. ,e also devote several sections to discussing some important topics such as Gauss-type %uadrature formulas, polynomial inter- polations, discrete transforms, and spectral differentiation techni%ues. -. &hen et al., Spectral Methods: Algorithms, Analysis and Applications, 0" &pringer &eries in $omputational 7athematics 01, 3OI 16.166"8!"5-.-106-"1601-" ., �c &pringer-9erlag :erlin (eidelberg /611 05 . Orthogonal ;olynomials and 4elated <ppro)imation 4esults 3.1.1 Existence and Uniqueness Given an open intervalI== (a,b)( ∞ a<b +∞), and a generic weight func- tion ω such that − ≤ ≤ ω(x)>6, x I and ω L 1(I), (..1# ∀ ∈ ∈ / two functionsf andg are said to be orthogonal to each other inL ω (a,b) or orthog- onal with respect to ω if b (f,g) ω == f(x)g(x) ω(x)dx=6. �a <n algebraic polynomial of degreen is denoted by n n 1 pn(x)=k nx +k n 1x − +...+k 1x+k 6,k n =6, (../# − � where k i are real constants, andk n is the leading coef*cient ofp n. { } ∞ < se%uence of polynomials p n n=6 with deg(pn) =n is said to be orthogonal / { } inL ω (a,b) if b (pn,p m)ω = pn(x)pm(x)ω(x)dx= γnδmn, (...# �a / where the constant γn = p n is non ero, and δmn is the >ronec+er delta. � �ω Throughout this section, p n denotes a se%uence of polynomials orthogonal { } with respect to the weight function ω, andp n is of degreen. 3enote byP n the set of all algebraic polynomials of degree n, namely, ≤ / n Pn == span 1,x,x ,...,x . (..0# :y a dimension argument, � � Pn = span p 6,p 1,...,p n . (..1# { } < direct conse%uence is the following. Lemma 3.1.p n 1 is orthogonal to any polynomial q P n. + ∈ Proof Than+sto(..1#, for anyq P n, we can write ∈ q=b n pn +b n 1 pn 1 +...+b 6 p6. − − (ence, (pn+1,q) ω =b n(pn+1,p n)ω +b n 1(pn+1,p n 1)ω +...+b 6(pn+1,p 6)ω =6, − − where we have used the orthogonality (...#. �� (ereafter, we denote the monic polynomial corresponding top n by n (n) n 1 (n) ?pn(x)==p n(x)/kn =x +a n 1x − +...+a 6 . (..@# − ..1 Orthogonal ;olynomials 0! ,e show below that for any given weight function ω(x) de*ned in(a,b), there e)ists a uni%ue family of monic orthogonal polynomials generated by a three-term recurrence formula. Theorem 3.1. !or any given positive #eight function ω L 1(I), there exists a ∈ $niq$e seq$ence of monic orthogonal polynomials ?pn #ith deg(?pn) = n, #hich can be constr$cted as follo#s { } ?p6 =1,?p 1 =x α6, − (.."# ?pn+1 = (x αn)?pn βn ?pn 1,n 1, − − − ≥ #here (x?pn,?pn)ω αn = / ,n 6, (..5a# ?pn ω ≥ � �/ ?pn ω βn = � � / ,n 1. (..5b# ?pn 1 ω ≥ � − � Proof It is clear that the *rst two polynomials are ?p6(x) 1,?p 1(x)=x α6. ≡ − To determine α6, we see that(?p6,?p1)ω =6 ifandonlyif b b (x?p6,?p6)ω α6 = ω(x)xdx ω(x)dx= / , a a ?p6 ω � �� � � where the denominator is positive due to (..1#. ,e proceed with the proof by using an induction argument. <ssuming that by a similar construction, we have derived a se%uence of monic orthogonal polynomials n ?pk . Ae)t,wesee+ ?pn 1 of the form { }k=6 + n / − (n) ?pn+1 =x?pn αn ?pn βn ?pn 1 ∑ γk ?pk, (..!# − − − − k=6 with αn and βn given by (..5#, and we re%uire ?pn 1,?pk =6,6 k n. (..16# + ω ≤ ≤ � � Ta+ing the inner product with ?pk on both sides of (..!#, and using the orthogo- n nality of ?pk ,we *nd that(..16# is ful*lled if and only if { }k=6 (?pn 1,?pn)ω = (x?pn,?pn)ω αn(?pn,?pn)ω =6, + − (?pn+1,?pn 1)ω = (x?pn,?pn 1)ω βn(?pn 1,?pn 1)ω =6, − − − − − n / − (n) (..11# (?pn+1,?p%)ω = (x?pn,?p%)ω ∑ γk (?pk,?p%)ω − k=6 (...# (n) / = (x?pn,?p%)ω γ ?p% =6,6 % n /. − % � �ω ≤ ≤ − 16 . Orthogonal ;olynomials and 4elated <ppro)imation 4esults Hence, the *rst e%uality implies (..5a#, and by the second one, / (x?pn,?pn 1)ω (?pn,x?pn 1)ω ?pn ω βn = −/ = −/ = � � / , ?pn 1 ω ?pn 1 ω ?pn 1 ω � − � � − � � − � where we have used the fact n 1 − (n) x?pn 1 =?pn + ∑ δk ?pk, − k=6 n and the orthogonality of ?pk k=6 to deduce the last identity. It remains to show that (n) n {/ } the coef*cients γk k=−6 in (..!# are all ero. Indeed, we derive from 'emma ..1 that { } x?pn,?p% = (?pn,x?p%)ω =6,6 % n /, ω ≤ ≤ − � � (n) which, together with the last e%uation of (..11#, implies γk 6for6 k n /, in (..!#. This completes the induction. ≡ ≤ ≤ − Ae)t, we show that the polynomial se%uence generated by (.."#B(..5# is uni%ue. ∞ For this purpose, we assume that ?qn is another se%uence of monic orthogonal { }n=6 polynomials. &ince ?pn, given by (.."#,is also monic,we have deg ?pn+1 ?qn+1 n. :y 'emma ..1, − ≤ � � (?pn 1,?pn 1 ?qn 1)ω =6,(?q n 1,?pn 1 ?qn 1)ω =6, + + − + + + − + which implies ?pn 1 ?qn 1,?pn 1 ?qn 1 =6 ?p n 1(x) ?qn 1(x) 6. + − + + − + ω ⇒ + − + ≡ This proves� the uni%ueness. � �� The above theorem provides a general three-term recurrence relation to construct orthogonal polynomials, and the constants αn and βn can be evaluated e)plicitly for the commonly used families. The three-term recurrence relation (.."# is essential for deriving other properties of orthogonal polynomials. For convenience, we *rst e)tend it to the orthogonal polynomials p n , which are not necessarily monic. { } Corollary 3.1. Let p n be a seq$ence of orthogonal polynomials #ith the leading { } coef&cient kn =6. 'hen #e ha"e � pn+1 = (anx b n)pn c n pn 1,n 6, (..1/# − − − ≥ #ith p 1 ==6, p 6 =k 6 and − kn+1 an = , (..1.a# kn kn+1 (xpn,p n)ω bn = / , (..1.b# kn pn ω � � / kn 1kn+1 pn ω cn = − / � � / . (..1.c# kn pn 1 ω � − � ..1 Orthogonal ;olynomials 11 Proof This result follows directly from Theorem ..1 by inserting ?pl =p l/kl with l=n 1,n,n+ 1 into ( .."# and (..5#. − �� The orthogonal polynomials p n with leading coef*cients k n are uni%uely de- termined by (..1/#B(..1.#. Interestingly,{ } the following result,{ which} can be viewed as the converse of $orollary ..1, also holds. ,e leave its proof as an e)ercise (see ;roblem ..1#. Corollary 3.2. Let k n =6 be a seq$ence of real n$mbers 'he three(term re( c$rrence relation( {..1/#�)(..1.} # generates a seq$ence of polynomials satisfying the properties: the leading coef&cientof pn is kn and deg(pn) =nC • pn are orthogonal #ith respect to the #eight f$nction ω(x)C •{ }/ theL (norm of pn is gi"en by • ω / γn = p n = (a6/an)c1c/ ...c nγ6,n 6, (..10# � �ω ≥ / b #here γ6 =k 6 a ω(x)dx. <n important conse%uence� of the three-term recurrence formula (..1/#B(..1.# is the well-+nown *hristoff(+arbo$x form$la. Corollary 3.3. Let p n be a seq$ence of orthogonalpolynomials #ith deg(pn) =n Then, { } n / pn+1(x)pn(y) p n(x)pn+1(y) kn+1 pn ω − = ∑ � �/ p %(x)p %(y), (..11# x y kn p % − %=6 � �ω and n / kn+1 pn ω / pn� +1(x)pn(x) p n� (x)pn+1(x)= ∑ � �/ p % (x).