Orthogonality of quasi-orthogonal polynomials
aa r X i v : . [ m a t h . C A ] D ec ORTHOGONALITY OF QUASI-ORTHOGONALPOLYNOMIALS
Cleonice F. Bracciali, Francisco Marcell´an, and Serhan Varma
Abstract.
A result of P´olya states that every sequence of quadrature formu-las Q n ( f ) with n nodes and positive numbers converges to the integral I ( f )of a continuous function f provided Q n ( f ) = I ( f ) for a space of algebraicpolynomials of certain degree that depends on n . The classical case when thealgebraic degree of precision is the highest possible is well-known and the quad-rature formulas are the Gaussian ones whose nodes coincide with the zeros ofthe corresponding orthogonal polynomials and the numbers are expressed interms of the so-called kernel polynomials. In many cases it is reasonable torelax the requirement for the highest possible degree of precision in order togain the possibility to either approximate integrals of more specific continuousfunctions that contain a polynomial factor or to include additional fixed nodes.The construction of such quadrature processes is related to quasi-orthogonalpolynomials. Given a sequence { P n } n > of monic orthogonal polynomials anda fixed integer k , we establish necessary and sufficient conditions so that thequasi-orthogonal polynomials { Q n } n > defined by Q n ( x ) = P n ( x ) + k − X i =1 b i,n P n − i ( x ) , n > , with b i,n ∈ R , and b k − ,n = 0 for n > k −
1, also constitute a sequence oforthogonal polynomials. Therefore we solve the inverse problem for linearlyrelated orthogonal polynomials. The characterization turns out to be equiva-lent to some nice recurrence formulas for the coefficients b i,n . We employ theseresults to establish explicit relations between various types of quadrature rulesfrom the above relations. A number of illustrative examples are provided.2010 Mathematics Subject Classification.
Key words and phrases.
Orthogonal polynomials, quasi-orthogonal polynomials, positivequadrature formulas, Gaussian quadrature formulas, Christoffel numbers, inverse problems.Research supported by the Brazilian Science Foundation CAPES under project numberCSF/PVE 107/2012. The work of the author (CFB) has also been supported by Brazilian ScienceFoundation CNPq, grants 305208/2015-2 and 402939/2016-6. The work of the author (FM) hasalso been supported by Ministerio de Econom´ıa y Competitividad of Spain, grant MTM 2015-65888-C4-2-P..
1. Introduction
Some results obtained during the early development of the theory of orthogonalpolynomials were motivated by the desire to build quadrature formulas with positiveChristoffel numbers whose nodes are zeros of known polynomials. Nowadays thesequadratures are succinctly denominated as positive quadrature formulas. The studyof this kind of problems was inspired by the Gauss’ theorem on quadrature withthe highest algebraic degree of precision with nodes at the zeros of the polynomialsorthogonal with respect to the measure of integration as well as by the result ofP´olya [ ] on convergence of quadrature rules. This led Riesz, Fej´er and Shohat tosearch for the properties of certain linear combinations of orthogonal polynomialsand the further developments resulted in deep outcome. The most convincingexample is the Askey and Gasper [
7, 8 ] proof of the positivity of certain sumsof Jacobi polynomials which played a key role in the final stage of de Branges’proof of the Bieberbach conjecture. We refer to the nice survey of Askey [ ] forthe motivation to study positive Jacobi polynomial sums, coming from positivequadratures, and for further information about these natural connections.The construction of positive quadrature rules is connected with the so-calledquasi-orthogonal polynomials. Let { P n } n > be a given sequence of monic orthogo-nal polynomials, generated by the three-term recurrence relation(1.1) x P n ( x ) = P n +1 ( x ) + β n P n ( x ) + γ n P n − ( x ) , n > , γ n = 0 , with P − ( x ) = 0 and P ( x ) = 1 . Then, given k ∈ N , the polynomials defined by(1.2) Q n ( x ) = P n ( x ) + k − X i =1 b i,n P n − i ( x ) , for n > k, are said to be a sequence of quasi-orthogonal polynomials of order k − k − b k − ,n = 0 . Here b i,n for n >
0, are realnumbers. By convention we set b ,n = 1, b − ,n = b − ,n = 0, b i,n = 0 when i > n , and also b i,n = 0 when n > k and i > k . Notice that for k = 1 we havethe standard orthogonality. This notion was introduced by Riesz while studyingthe moment problem and the reason for this nomenclature is rather simple: Q n is orthogonal to every polynomial of degree not exceeding n − k with respect tothe functional of orthogonality of { P n } n > . M. Riesz himself considered only thecase k = 2 while Fej´er [ ] concentrated his attention on the specific case when k = 3, P n are the Legendre polynomials and b ,n <
0. It seems that Shohat[ ] was the first who studied the general case. The renewed recent interest onthe quasi-orthogonal polynomials brought a large number of interesting results.Peherstorfer [
34, 35, 36 ] and Xu [ ] obtained results concerning the location ofthe zeros of the quasi-orthogonal polynomials and the positivity of the Christoffelnumbers when { P n } n > are orthogonal on [ − ,
1] with respect to a measure thatbelongs to Szeg˝o’s class. Xu [ ] established general properties of quasi-orthogonalpolynomials and, under the assumption that Q n is also orthogonal, studied therelation between the Jacobi matrices associated with both sequences. The zeros of RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 3 some quasi-orthogonal polynomials were studied recently by Beardon and Driver[ ] and Brezinski, Driver and Redivo-Zaglia [ ].Motivated by the relation between positive quadrature rules and quasi-ortho-gonal polynomials, we provide necessary and sufficient conditions in order that thesequence of polynomials { Q n } n > , obeying (1.2), is also orthogonal. The latterproblem is purely algebraic in nature. We solve it via a constructive approachby taking into account classical results on Sturm sequences. It becomes evidentthen that one may look at the solution in terms of a relation between the Jacobimatrices associated with the sequences of orthogonal polynomials. As a result thesolution is explicit in the sense that we establish the connection between the threeterm recurrence relations that generate the sequences { P n } n > and { Q n } n > aswell as between the linear functionals related to them. These results allow us tojudge about the nodes of two Gaussian type quadrature formulas whose locationcoincides with the zeros of the polynomials P n and Q n . Moreover, the Christoffelnumbers of the quadrature rules are obtained explicitly as a consequence of theclosed forms of the corresponding kernel polynomials which are also derived fromour general approach.The structure of the paper is as follows. In Section 2 we state the necessary andsufficient conditions of the orthogonality of a sequence of quasi-orthogonal polyno-mials of order k − h associated withthe Geronimus transformation of the initial linear functional. In Section 3, theproofs of those theorems are given as well as an algorithm to deduce the sequenceof connection coefficients. Section 4 is focussed on the relation between the corre-sponding Jacobi matrices. Thus, we have a computational approach to the zerosof Q n ( x ) since they are the eigenvalues of the n th principal leading submatricesof the corresponding Jacobi matrix. The Christoffel numbers are their normalizedeigenvectors. We also prove some results concerning the zeros of the polynomial Q n ( x ) as well as the expression of the kernel polynomials in terms of the initialones. In Section 5 we analyze some examples illustrating the problems consideredin the previous sections. First, the case when u is a symmetric linear functional isconsidered. The results are implemented for Chebyshev polynomials of the secondkind. Second, the non-symmetric case is studied and implemented for Laguerrepolynomials. Finally, we study the case of constant coefficients. In such a case, wesolve a problem posed in [ ] for k >
2. Orthogonality of quasi-orthogonal polynomials
The characterization of those quasi-orthogonal polynomials (1.2) which form asequence of orthogonal polynomials themselves can be approached from a generalpoint of view. Let P be the linear space of algebraic polynomials with complexcoefficients. Then h u, f i denotes the action of the linear functional u ∈ P ′ overthe polynomial f ∈ P , where P ′ denotes the algebraic dual of the linear space P .The sequence of monic orthogonal polynomials (SMOP) { P n } n > with respect tothe linear functional u obeys the conditions h u, P n P m i = K n δ nm , where K n = 0 BRACCIALI, MARCELL´AN, AND VARMA for all n >
0, and δ nm is the Kronecker delta. A linear functional u is said to beregular or quasi-definite (see [ ]) when the leading principal submatrices H n ofthe Hankel matrix H = ( u i + j ) i,j > composed by the moments u i = (cid:10) u, x i (cid:11) , i > n >
0. When the determinants of H n are positive forall nonnegative integers n the functional is called positive-definite. If the linearfunctional u is regular, then the SMOP { P n } n > satisfies the three-term recurrencerelation (1.1) with γ n = 0 and if u is positive-definite then γ n >
0. Conversely, ifa sequence of polynomials is generated by the recurrence relation (1.1) and γ n =0, then there is a linear functional u ∈ P ′ , such that { P n } n > is a sequence ofpolynomials orthogonal with respect to u and this is the statement of Favard’stheorem ([ ]). Moreover, if γ n > n ∈ N , then the linear functional u is positive-definite and it has an integral representation h u, f i = R R f dµ , f ∈ P ,where dµ is a positive Borel measure supported on an infinite subset of R (see [ ]).The linear functional v ∈ P ′ is called a rational perturbation of u ∈ P ′ , if thereexist polynomials p and q , such that q ( x ) v = p ( x ) u. Detailed information about the direct problems studied from several points of viewcan be found in [
2, 13, 23, 30, 46 ]. In particular, the connection formula be-tween the polynomials orthogonal with respect to v and u is called the generalisedChristoffel’s formula (see [ ]). The relation between the corresponding Jacobimatrices was studied in [ ].Let { P n } n > be a SMOP, m and k are positive integers. Let consider anothersequence of monic polynomials { Q n } n > related to { P n } n > by(2.1) Q n ( x ) + m − X j =1 a j,n Q n − j ( x ) = P n ( x ) + k − X i =1 b i,n P n − i ( x ) , n > , with a j,n , b i,n ∈ R , a m − ,n b k − ,n = 0. Then the problem to find necessary andsufficient conditions so that { Q n } n > is also a SMOP and to obtain the relationbetween the corresponding regular linear functionals is called an inverse problem.Observe that we adopt the convention that when either m or k is equal to one,then the corresponding sum does not appear, that is, we interpret it as an emptyone. A vast number of interesting results have been obtained on topics related tothe inverse problem (see [
1, 3, 4, 5, 10, 11, 25, 26, 32, 37 ]).In the present contribution we also focus our attention on the quasi-orthogonalpolynomials defined by (1.2) under the only natural restriction and b k − ,n = 0for n > k −
1. This corresponds to a very general situation when we set m = 1and k ∈ N in (2.1). Therefore, in what follows we consider this setting. Manyparticular results, when one looks for the relation between the functionals u and v , with respect to which the polynomial sequences { P n } n > and { Q n } n > are or-thogonal, are known [
13, 15, 18, 19, 29, 46 ] but the general case that we discussin the present contribution has not been approached in the literature yet. In thispaper we provide necessary and sufficient conditions so that the sequence of monicpolynomials { Q n } n > is also orthogonal. RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 5
Let { P n } n > be a SMOP corresponding to a regular linear functional u . Now wegive the necessary and sufficient conditions ensuring the orthogonality of the monicpolynomial sequence { Q n } n > that satisfies the three-term recurrence relation xQ n ( x ) = Q n +1 ( x ) + ˜ β n Q n ( x ) + ˜ γ n Q n − ( x ) , n > , with the initial conditions Q − ( x ) = 0 and Q ( x ) = 1, and the condition ˜ γ n = 0,for n > Theorem . Let { Q n } n > be a sequence of monic polynomials defined by (1 . . Then { Q n } n > is a SMOP with recurrence coefficients { ˜ β n } n > and { ˜ γ n } n > if and only if the coefficients b ,n = 1 , { b i,n } n > , i k − , satisfy the followingconditions (2.2) γ n + b ,n − b ,n +1 + b ,n ( β n − − β n − b ,n + b ,n +1 ) = 0 , for n > , (2.3) b ,n +1 = b ,n + β n − β n − k +1 + b k − ,n − b k − ,n − γ n − k +1 − b k − ,n b k − ,n γ n − k +2 , n > k, (2.4) b ,n +1 = b ,n + γ n − b k − ,n b k − ,n − γ n − k +1 + b ,n ( β n − − β n − b ,n + b ,n +1 ) , n > k, and b i +2 ,n +1 = b i +2 ,n + b i +1 ,n ( β n − − i − β n − b ,n + b ,n +1 ) + b i,n γ n − i − b i,n − [ γ n + b ,n − b ,n +1 + b ,n ( β n − − β n − b ,n + b ,n +1 )] , (2.5) for i k − and n > i + 1 .Moreover, the recurrence coefficients of { Q n } n > are given by ˜ β n = β n + b ,n − b ,n +1 , n > , (2.6) ˜ γ n = γ n + b ,n − b ,n +1 + b ,n ( β n − − β n − b ,n + b ,n +1 ) , n > , (2.7) and the coefficients ˜ γ n also satisfy (2.8) ˜ γ n = b k − ,n b k − ,n − γ n − k +1 , n > k. The above relations provide a complete characterization of the orthogonalityof the polynomial sequence { Q n } n > . When b j,n = b j , j = 1 , · · · , k − , you recoverTheorem 1 in [ ].On the other hand, a natural question arises about the relation between theregular linear functionals u and v such that { P n } n > and { Q n } n > are the corre-sponding SMOP. In this case, the functional v which describes the orthogonalityof the sequence { Q n } n > is a Geronimus spectral transformation of degree k − u . In other words, u = h ( x ) v , where h is a polynomial ofdegree k − ]). Our next result furnishes a method to determine h . Theorem . The coefficients of the polynomial (2.9) h ( x ) = h + h x + · · · + h k − x k − + h k − x k − , BRACCIALI, MARCELL´AN, AND VARMA such that u = h ( x ) v , are the unique solution of a system of k linear equations, wherethe entries of the corresponding matrix depend only on the sequences of connectioncoefficients { b i,n } n > k − , i = 1 , , . . . , k − . A detailed description of the linear system and about the explicit form of thecoefficients will be done in the sequel.It is worth pointing out that an alternative way to compute the coefficients of h is via a relation between the Jacobi matrices related to the sequences { P n } n > and { Q n } n > . We discuss this method in Section 4.Since the quasi-orthogonal polynomials arise naturally in the context of quad-rature formulae of Gaussian type, many properties that can be classified more thanas analytic rather than algebraic, such as the behaviour of their zeros and the pos-itivity of the Christoffel numbers have been analysed. Most of these results dealwith rather specific particular cases when either k is a small integer or the orthog-onal polynomials belong to classical families. In Section 4.2 we obtain some resultsabout the zeros of the polynomials P n and Q n .Many illustrative examples are analysed when the linear functional u is a sym-metric one, as well as when one deals with constant connection coefficients. Thelatter problem is motivated by a result in [ ] where { P n } n > is the sequence ofChebyshev polynomials.
3. Proofs of Theorems 2.1 and 2.2 and the direct problem3.1. Proof of Theorem 2.1.
The core of the overall approach is a classicalresult of Sturm [ ] on counting the number of real zeros of an algebraic polynomial.We refer to [ , Section 10.5] and [ , Sections 2.4, 2.5] for detailed informationabout various versions of Sturm’s result as well as about the historical background.We state the general version of Sturm’s theorem in the setting we need. Let R n +1 and R n be polynomials of exact degree n +1 and n , respectively, with monic leadingcoefficients. Execute the Euclidean algorithm(3.1) R k +1 ( x ) = ( x − c k ) R k ( x ) − d k R k − ( x ) , k = n, n − , . . . , . A careful inspection of the general version of Sturm’s theorem shows that thefollowing holds:
Theorem A . (Sturm) Under the above assumptions, the polynomials R n +1 and R n have real and strictly interlacing zeros if and only if d k , k = n, n − , . . . , , are positive real numbers. Furthermore, the zeros of the polynomial R k , k = n, n − , . . . , , are all real and the zeros of two consecutive polynomials are strictlyinterlacing.It follows immediately from Theorem A and Favard’s theorem that, given twopolynomials R n +1 and R n with positive leading coefficients and with real andstrictly interlacing zeros, the Euclidean algorithm (3.1) generates the sequence R k , k = 0 , . . . , n + 1, such that these are the first n + 1 terms of a sequence of or-thogonal polynomials, which can be constructed by using the standard three termrecurrence relation. In other words, any two polynomials of consecutive degrees RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 7 and interlacing zeros may be “embedded” in a sequence of orthogonal polynomi-als. This straightforward but beautiful observation was pointed out by Wendroff[ ] and the statement is nowadays called Wendroff’s theorem. Observe that R n +1 and R n generate R k , k = n − , . . . , R k , k = 0 , . . . , n + 1 of all the polynomials can be extended “forward” invarious ways. The complete characterization of the sequences of orthogonal poly-nomials P n and Q n that are related by the relation (1.2) is obtained via Theorem A. Proof of Theorem 2.1 .Applying the Euclidean algorithm (3.1) with “initial” polynomials R n +1 ( x ) = Q n +1 ( x ) and R n ( x ) = Q n ( x ) and setting c n = ˜ β n , we obtain Q n +1 ( x ) = ( x − ˜ β n ) Q n ( x ) − R n − ( x ) , where R n − ( x ) is a polynomial of degree at most n −
1. Using (1.2) together withthe recurrence relation (1.1) we conclude that(3.2) R n − ( x ) = k X i =0 h b i,n ( β n − i − ˜ β n ) − b i +1 ,n +1 + b i +1 ,n + b i − ,n γ n − ( i − i P n − i ( x ) , where b − ,n = 0 and b ,n = 1. Moreover, when n > k , we have b i,n = 0 for all i > k .Now we can determine necessary and sufficient conditions in order to the poly-nomial R n − ( x ) coincides with the polynomial ˜ γ n Q n − ( x ), i.e.,(3.3) R n − ( x ) = ˜ γ n P n − ( x ) + k − X i =1 b i,n − P n − − i ( x ) ! . Comparing the coefficients that multiply P n ( x ) and P n − ( x ) in (3.2) and (3.3)we derive the conditions β n − ˜ β n − b ,n +1 + b ,n = 0 , n > ,b ,n ( β n − − ˜ β n ) − b ,n +1 + b ,n + γ n = ˜ γ n , n > , and the latter obviously correspond to (2.6) and (2.7). This means that(3.4) ˜ γ n = γ n + b ,n − b ,n +1 + b ,n ( β n − − β n − b ,n + b ,n +1 ) , n > . Since ˜ γ n = 0, we obtain the constraint γ n + b ,n − b ,n +1 + b ,n ( β n − − β n − b ,n + b ,n +1 ) = 0 , for n > , which is exactly (2.2).Similarly, comparing the coefficients of P n − ( x ) , ..., P n − k ( x ) in (3.2) and (3.3),we obtain the following conditions: b i,n − ˜ γ n = b i,n γ n − i + b i +2 ,n − b i +2 ,n +1 + b i +1 ,n ( β n − − i − β n − b ,n + b ,n +1 ) ,1 i k − , n > i + 1 , (3.5) b k − ,n − ˜ γ n = b k − ,n γ n − k +2 + b k − ,n ( β n − k +1 − β n − b ,n + b ,n +1 ) ,n > k − BRACCIALI, MARCELL´AN, AND VARMA and b k − ,n − ˜ γ n = b k − ,n γ n − k +1 , n > k. (3.7)Now (2.3) follows from (3.6) and (3.7) while (2.4) is a consequence of (3.4) and(3.7). Finally, (3.4) and (3.5) imply (2.5).It is important to check that at the last step the coefficient b k − ,n +1 must bedifferent from zero in order to be consistent with the quasi-orthogonality condition.This completes the proof.Theorem 2.1 provides also a forward algorithm to compute the coefficients b i,n for n > k + 1. Starting with coefficients b i,k − , i = 1 , , . . . , k − , from the linearcombination Q k − ( x ) = P k − ( x ) + b ,k − P k − ( x ) + · · · + b k − ,k − P ( x ) , we choose the coefficients b i,k for i = 1 , , . . . , k − , and write Q k ( x ) = P k ( x ) + b ,k P k − ( x ) + · · · + b k − ,k P ( x ) . Then we compute b ,n +1 , for n > k , using equation (2.3) and b ,n , b k − ,n − , b k − ,n − , b k − ,n and b k − ,n (see the first scheme in Fig. 1). We compute b ,n +1 ,for n > k , using equation (2.4) and b ,n , b ,n , b ,n +1 , b k − ,n and b k − ,n − (see thesecond scheme in Fig. 1).We compute b i +2 ,n +1 , for n > k and 1 i k −
3, using equation (2.5) and b i +2 ,n , b i +1 ,n , b i,n , b i,n − , and also b ,n , b ,n +1 , b ,n , and b ,n +1 . This is illustratedas the first scheme in Fig. 2. Alternatively, b i +2 ,n +1 , for n > k and 1 i k − b i +2 ,n +1 = b i +2 ,n + b i +1 ,n ( β n − − i − β n − b ,n + b ,n +1 ) + b i,n γ n − i − b i,n − b k − ,n b k − ,n − γ n − k +1 , using b i +2 ,n , b i +1 ,n , b i,n , b i,n − , and also b ,n , b ,n +1 , b k − ,n − , b k − ,n , (see thesecond scheme in Fig. 2). n−1 n n+112k−2k−1 n−1 n n+112k−2k−1 Figure 1.
Scheme for the calculation of b ,n +1 and b ,n +1 , n > k .As we have pointed out above, after the computations at level n + 1, it isnecessary to verify if b k − ,n +1 = 0, for n > k . RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 9 n−1 n n+112ii+1i+2 n−1 n n+11ii+1i+2k−1
Figure 2.
Alternative schemes for calculation of b i +2 ,n +1 , n > k .The initial coefficients b ,n = 1, b ,n , b ,n , . . . , b n,n , for 1 n k −
2, startingfrom Q k and Q k − , are uniquely determined by the “backward” process describedby the Euclidean algorithm and by Theorem A.Let us notice the key role played by the connection coefficients for the polyno-mials Q k − and Q k as initial data to run the above algorithm.As a summary, you can generate the coefficients of quasi-orthogonal polynomi-als in a recursive way, assuming some initial conditions. The dual basis { ω n } n > ∈ P ′ of { P n } n > isdefined, as usual, by the conditions (see [ ]) h ω n , P m i = δ nm . It is easy to see that the elements of the basis, dual to SMOP { P n } n > withrespect to the regular linear functional u , are ω n = P n u h u,P n i . Let us define theleft-multiplication of a linear functional u ∈ P ′ by any polynomial f ∈ P via h f u, p i = h u, f p i , p ∈ P . Let { Q n } n > , given by relation (1 . v . According to [ ], if we use the expansion of the linear functional u in terms of the dual basis { Q j v h v,Q j i } j > of the SMOP { Q n } n > , in view of orthog-onality properties and relation (1 . Lemma . u = k − X j =0 h u, Q j i (cid:10) v, Q j (cid:11) Q j v, i.e. , u = h ( x ) v, (3.8) where h ( x ) = h k − x k − + h k − x k − + · · · + h x + h is a polynomial of degree ( k − because its leading coefficient is h k − = b k − ,k − h u, i h v,Q k − i 6 = 0 . Proof of Theorem 2.2
For n > k we have h u, P m Q n i = h h ( x ) v, P m Q n i = h h v, P m Q n i + h h v, xP m Q n i + · · · + h k − (cid:10) v, x k − P m Q n (cid:11) . For m = n, n − , . . . , n − ( k − h u, P n Q n i = h h v, P n Q n i + h h v, xP n Q n i + · · · + h k − (cid:10) v, x k − P n Q n (cid:11) h u, P n − Q n i = h h v, P n − Q n i + h h v, xP n − Q n i + · · · + h k − (cid:10) v, x k − P n − Q n (cid:11) ... ...(3.9) (cid:10) u, P n − ( k − Q n (cid:11) = h (cid:10) v, P n − ( k − Q n (cid:11) + h (cid:10) v, xP n − ( k − Q n (cid:11) + · · · ++ h k − (cid:10) v, x k − P n − ( k − Q n (cid:11) . Since, for j = 0 , , . . . , k − , (cid:10) v, x l P n − j Q n (cid:11) = (cid:26) , if l < j, (cid:10) v, Q n (cid:11) , if l = j, assuming b ,n = 1 and using (1.2), we derive h u, P n − j Q n i = * u, P n − j k − X i =0 b i,n P n − i + = b j,n (cid:10) u, P n − j (cid:11) , for j = 0 , . . . , k − . Now we write the equations (3.9) as a system of k linear equations T ¯ h = b, where T = (cid:10) v, Q n (cid:11) h v, xP n Q n i · · · (cid:10) v, x k − P n Q n (cid:11) (cid:10) v, x k − P n Q n (cid:11) (cid:10) v, Q n (cid:11) · · · (cid:10) v, x k − P n − Q n (cid:11) (cid:10) v, x k − P n − Q n (cid:11) · · · (cid:10) v, x k − P n − Q n (cid:11) (cid:10) v, x k − P n − Q n (cid:11) ... ... · · · ... ...0 0 · · · (cid:10) v, Q n (cid:11) (cid:10) v, x k − P n − ( k − Q n (cid:11) · · · (cid:10) v, Q n (cid:11) , ¯ h = h h h ... h k − h k − and b = b ,n (cid:10) u, P n (cid:11) b ,n (cid:10) u, P n − (cid:11) b ,n (cid:10) u, P n − (cid:11) ... b k − ,n D u, P n − ( k − E b k − ,n D u, P n − ( k − E . The latter can be rewritten in the form h j (cid:10) v, Q n (cid:11) + k − X l = j +1 h l (cid:10) v, x l P n − j Q n (cid:11) = b j,n (cid:10) u, P n − j (cid:11) , for j = 0 , , . . . , k − . RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 11
Using the backward technique for solution of systems of linear equations, we obtain,for j = k − , k − , ..., , h j = b j,n (cid:10) u, P n − j (cid:11) − k − X l = j +1 h l (cid:10) v, x l P n − j Q n (cid:11), (cid:10) v, Q n (cid:11) , for n > k. In order to simplify (3.10), let J P be the tridiagonal matrix corresponding tothe SMOP { P n } n > , that is, x P = J P P , where P = ( P , P , ... ) T and J P = β . . . . . .γ β . . . . . . γ β . . . . . . ... ... ... ... . . . ... ...0 0 0 0 . . . β n −
10 0 0 0 . . . γ n − β n − . . .... ... ... ... . . . . . . . Notice that, for j = 0 , , . . . , k − , and l > j we have x l P n − j ( x ) = n + l − j X i =0 ( J lP ) n − j,i P i ( x ) , where ( J lP ) n − j,i denotes the ( n − j, i ) entry of the matrix J lP . Then the equalities (cid:10) v, x l P n − j Q n (cid:11) = * v, n + l − j X i = n ( J lP ) n − j,i P i Q n + (3.11) = n + l − j X i = n ( J lP ) n − j,i h v, P i Q n i hold for l > j .Now it is clear that the inner products h v, P n + r Q n i , r = 0 , , , . . . , l − j , can beexpressed in terms of the coefficients b i,n + i , i = 1 , , . . . , l − j , and from the valueof (cid:10) v, Q n (cid:11) . Indeed, we rewrite (1.2) in the form P n + r ( x ) = Q n + r ( x ) − k − X i =1 b i,n + r P n + r − i ( x ) , which implies h v, P n + r Q n i = * v, Q n + r − k − X i =1 b i,n + r P n + r − i ! Q n + = − r X i =1 b i,n + r h v, P n + r − i Q n i , for r = 1 , , . . . , l − j , so that(3.12) h v, P n + r Q n i + r − X i =1 b i,n + r h v, P n + r − i Q n i = − b r,n + r (cid:10) v, Q n (cid:11) . Using equations (3.12), for r = 1 , , . . . , l − j, and including the equation h v, P n Q n i = (cid:10) v, Q n (cid:11) , we obtain the following system of ( l − j + 1) equations: · · · · · · b ,n +2 · · · b ,n +3 b ,n +3 · · · b l − j − ,n + l − j b l − j − ,n + l − j b l − j − ,n + l − j · · · b ,n + l − j × h v, P n Q n ih v, P n +1 Q n ih v, P n +2 Q n i ... h v, P n + l − j Q n i = − b ,n +1 − b ,n +2 ... − b l − j,n + l − j (cid:10) v, Q n (cid:11) . (3.13)Let us denote by A l − j +1 the matrix of the latter system. Then the solution h v, P n + r Q n i , r = 0 , , , . . . , l − j , is obtained in terms of the coefficients b i,n , i = 1 , , . . . , l − j , and (cid:10) v, Q n (cid:11) .Replacing the solution of (3.13) into (3.11) we conclude that (cid:10) v, x l P n − j Q n (cid:11) = (cid:0) ( J lP ) n − j,n , ( J lP ) n − j,n +1 , . . . , ( J lP ) n − j,n + l − j (cid:1) × A − l − j +1 − b ,n +1 − b ,n +2 ... − b l − j,n + l − j (cid:10) v, Q n (cid:11) , where A − l − j +1 is the inverse of the matrix A l − j +1 . Finally we solve the system (3.10)and find all coefficients h j , j = 0 , , . . . , k − , of the polynomial h as functions of β n , γ n and b i,n . Thus, Theorem 2.2 is proved.The above result shows that the sequences { b j,n } n > k , j = 0 , , . . . , k − h ( x ) given in Theorem 2. In other words, the sequences { b j,n } n > k ,j = 0 , , . . . , k −
1, together with the coefficients of the three term recurrence re-lation, determine uniquely the polynomial h ( x ). Moreover, since the matrix T isnonsingular, any polynomial h ( x ) of the form (2.9) determines uniquely the coeffi-cients b ,n , b ,n , . . . , b k − ,n , for n > k . We discuss this question thoroughly in thenext section. RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 13
Notice that the latter observations provide not only an algorithm to calcu-late h ( x ), but also an alternative proof about the relation between the Geronimustransformation and the quasi-orthogonal polynomials.It is easy to see from (3.10) that the leading coefficient of h ( x ) is given, in analternatively way, by(3.14) h k − = b k − ,n D u, P n − ( k − E h v, Q n i , for n > k − . Considering n = k − h u, i = 1, we obtain h k − = b k − ,k − (cid:10) v, Q k − (cid:11) = b k − ,k − ˜ γ ˜ γ · · · ˜ γ k − h v, i 6 = 0 , where ˜ γ , ˜ γ , ..., ˜ γ k − are given by (2 . h ( x ) can also be obtained in an explicit form. Indeed,it follows from (3.10) that(3.15) (cid:10) v, Q n (cid:11) h k − = b k − ,n D u, P n − ( k − E − (cid:10) v, x k − P n − ( k − Q n (cid:11) h k − . Now (3.11), with l = k − j = k −
2, yields (cid:10) v, x k − P n − ( k − Q n (cid:11) = n +1 X i = n ( J k − P ) n − ( k − ,i h v, P i Q n i = ( J k − P ) n − ( k − ,n h v, P n Q n i + ( J k − P ) n − ( k − ,n +1 h v, P n +1 Q n i = k − X i =0 β n − i (cid:10) v, Q n (cid:11) + h v, P n +1 Q n i . Since Q n +1 ( x ) = P n +1 ( x ) + b ,n +1 P n ( x ) + b ,n +1 P n − ( x ) + · · · + b k − ,n +1 P n − ( k − ( x ) , then h v, P n +1 Q n i = − b ,n +1 (cid:10) v, Q n (cid:11) . Therefore (3.15) becomes (cid:10) v, Q n (cid:11) h k − = b k − ,n D u, P n − ( k − E − k − X i =0 β n − i − b ,n +1 ! (cid:10) v, Q n (cid:11) h k − h k − h k − = b ,n +1 − k − X i =0 β n − i + b k − ,n h k − D u, P n − ( k − E h v, Q n i . Then (3.14) implies h k − h k − = b ,n +1 − k − X i =1 β n +1 − i + b k − ,n D u, P n − ( k − E b k − ,n D u, P n − ( k − E = b ,n +1 − k − X i =1 β n +1 − i + b k − ,n b k − ,n γ n − k +2 . The computations of the remaining coefficients of h ( x ) are rather involved andyield extremely complex explicit expressions so that we omit them. Remark . Notice that the above result shows that you can find a direct re-lation between the coefficients of the polynomial h, the connection coefficients of thesequences { P n } n > and { Q n } n > and the coefficients of the three term recurrencerelation of the sequence { P n } n > .
4. Gaussian type quadrature formulas4.1. An interpretation in terms of Jacobi matrices.
In this section weprovide an alternative approach to the above problems based on the matrix form ofthe three-term recurrence relations as well as of the connection coefficients betweenthe two sequences of polynomials. Let J P and J Q be the tridiagonal matrices cor-responding to the SMOP { P n } n > and { Q n } n > , respectively. Then the three-termrecurrence relations satisfied by the SMOP { P n } n > and { Q n } n > are equivalentto(4.1) x P = J P P , x Q = J Q Q , where P = ( P , P , ... ) T and Q = ( Q , Q , ... ) T .On the other hand, (1 .
2) reads as(4.2) Q = ˜AP , where ˜A = (˜ a s,l ) s,l > is a banded lower triangular matrix with entries ˜ a s,s = 1 and˜ a s,l = 0, s − l > k −
1. Combining (4 .
1) and (4 .
2) we obtain x ˜AP = J Q ˜AP and then(4.3) ˜AJ P = J Q ˜A , i.e., J Q = ˜AJ P ˜A − . These represent a succinct matrix form of the relations obtained in Theorem 2.1.On the other hand, Christoffel formula [ ] is equivalent to(4.4) ˜ h ( x ) P = ˜BQ where ˜B = (˜ b s,l ) s,l > is a banded upper triangular matrix with entries ˜ b s,s + k − = 1,˜ b s,l = 0, l − s > k −
1, and ˜ h ( x ) = h ( x ) /h k − , where h ( x ) is the polynomial definedin (3.8).Substituting (4.1) and (4.2) into (4.4), we obtain(4.5) ˜ h ( J P ) = ˜B ˜A , where ˜ h ( J P ) is a diagonal matrix of size (2 k − ˜B isuniquely determined from (4 . h ( J Q ) = ˜A ˜ h ( J P ) ˜A − = ˜A ˜B , the matrix J Q can be determined from (4.6). Notice that (4.6) is the LU factor-ization of the matrix ˜ h ( J Q ) while (4 .
5) is a UL factorization of the matrix ˜ h ( J P ). RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 15
We also describe relations between the corresponding finite dimensional tridi-agonal matrices which appear in the three-term recurrence relations (4.1) as wellas on (4.2). If ( P ) n = { P , P , ..., P n } T and ( Q ) n = { Q , Q , ..., Q n } T , then (4.1)and (4.2) reduce to x ( P ) n = ( J P ) n +1 ( P ) n + P n +1 e n +1 , (4.7) x ( Q ) n = ( J Q ) n +1 ( Q ) n + Q n +1 e n +1 , (4.8) ( Q ) n = ( ˜A ) n +1 ( P ) n , (4.9)where ( . ) n denotes the leading principal submatrix of size n × n of the correspondinginfinite one, while here and in what follows, e j is the j -th vector of the canonicalbasis in R n +1 with all entries zeros except for the j -th one, which is one. Replacing(4.9) and (1.2) in (4.8) yields x ( ˜A ) n +1 ( P ) n = " ( J Q ) n +1 ( ˜A ) n +1 + e n +1 k − X i =1 b i,n +1 e Tn +2 − i ! ( P ) n + P n +1 e n +1 . Having in mind (4.7), the latter simplifies to( ˜A ) n +1 ( J P ) n +1 = ( J Q ) n +1 ( ˜A ) n +1 + ( ˜A ) n +1 e n +1 k − X i =1 b i,n +1 e Tn +2 − i ! . Thus, we obtain( J Q ) n +1 = ( ˜A ) n +1 " ( J P ) n +1 − e n +1 k − X i =1 b i,n +1 e Tn +2 − i ! ( ˜A ) − n +1 . This result means that ( J Q ) n +1 is a rank-one perturbation of the matrix ( J P ) n +1 . Remark . The particular cases k = 2, k = 3 , and k = 4 of the above matrixmethod are considered in [ ], and [ ], respectively. Remark . Having in mind that the zeros of the polynomial Q n +1 are theeigenvalues of the matrix ( J Q ) n +1 , the above expression means that they are theeigenvalues of a rank one perturbation of the matrix ( J P ) n +1 . Therefore, one mayestimate them using the classical theory of eigenvalue perturbations (see [ ]). Onthe other hand, the corresponding Christoffel numbers are the first component ofthe normalized eigenvector associated with each eigenvalue. In this sectionwe discuss some properties of these zeros and of their location with respect tothose of P n provided that both { P n } n > and { Q n } n > are sequences of orthogonalpolynomials and they are related by (1.2).In order to obtain inequalities for the number of zeros of Q n which are greaterthan the largest zero of P n we need a theorem on Descartes rule of signs for or-thogonal polynomials due to Obrechkoff. Given a finite sequence α , . . . , α n ofreal numbers, let S ( α , . . . , α n ) be the number of its sign changes. Recall that S ( α , . . . , α n ) is counted in the following natural way. First we discard the zeroentries from the sequence and then count a sign change if two consecutive terms in the remaining sequence have opposite signs. By Z ( f ; ( a, b )) we denote the numberof the zeros, counting their multiplicities, of the function f ( x ) in ( a, b ). Definition . The sequence of functions f , . . . , f n obeys the general Descartes’rule of signs in the interval ( a, b ) if the number of zeros in ( a, b ), where the multiplezeros are counted with their multiplicities, of any real nonzero linear combination α f ( x ) + . . . + α n f n ( x )does not exceed the number of sign changes in the sequence α , . . . , α n .More precisely, this property states that Z ( α f ( x ) + . . . + α n f n ( x ); ( a, b )) S ( α , . . . , α n )for any ( α , . . . , α n ) = (0 , . . . , Theorem
B (Obrechkoff [ ]) . If the sequence of polynomials { p n } n > is de-fined by the recurrence relation xp n ( x ) = a n p n +1 ( x ) + b n p n ( x ) + c n p n − ( x ) , n > , with p − ( x ) = 0 and p ( x ) = 1, where a n , b n , c n ∈ R , a n , c n > z n denotes thelargest zero of p n ( x ), then the sequence of polynomials p , . . . , p n obeys Descartes’rule of signs in ( z n , ∞ ).Since, by Favard’s theorem [ ], the requirements on p k ( x ) in Theorem B areequivalent to the fact that { p n } n > is a sequence of orthogonal polynomials, weobtain Corollary . Suppose that the orthogonal polynomials p k ( x ) , k = 0 , , . . . , n be normalized in such a way that their leading coefficients are all of the same signand let z n be the largest zero of p n ( x ) . Then, for any set of real numbers α , . . . , α n ,which are not identically zero, we get Z ( α p ( x ) + · · · + α n p n ( x ); ( z n , ∞ )) S ( α , . . . , α n ) . Some applications of Theorem B and Corollary 4.1 to zeros of orthogonal poly-nomials were discussed in [ ]Now we are ready to formulate a result concerning inequalities for largest zerosof the polynomials Q n . Theorem . Let { P n } n > be a sequence of monic orthogonal polynomialsand let { Q n } n > k be defined by (1.2). If the zeros of P n ( x ) are x n, < · · · < x n,n ,then Z ( Q n ( x ) , ( x n,n , ∞ )) S (1 , b ,n , . . . , b k − ,n ) . Despite that in this paper we are interested in the situation when { Q n } n > is another sequence of orthogonal polynomials, the above result about the largestzeros of Q n does not depend on the fact that the sequence of polynomials obeys anorthogonality property or not. RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 17
Corollary . If the zeros of Q n are also real and simple, denoted by y n, < · · · < y n,n and S (1 , b ,n , . . . , b k − ,n ) = ℓ , then y n,n − ℓ < x n,n . In particular y n,n − k +1 < x n,n independently of the signs of b i,n > for i =1 , . . . , n − k + 1 . Moreover, if b i,n > for i = 1 , . . . , n − k + 1 , then y n,n < x n,n which means that all zeros of Q n precede x n,n . Finally, we obtain a relation between the Stieltjes functions of u and v. Indeed,let define S u ( z ) = ∞ X n =0 u n z n +1 and S v ( z ) = ∞ X n =0 v n z n +1 , where u n = h u, x n i and v n = h v, x n i .Since h u, x n i = h v, h ( x ) x n i then u n = k − X j =0 h j v j + n , and S u ( z ) = ∞ X n =0 z n +1 k − X j =0 h j v j + n = k − X j =0 h j z j ∞ X n =0 v j + n z j + n +1 ! = k − X j =0 h j z j S v ( z ) − j − X s =0 v s z s +1 ! = k − X j =0 h j z j S v ( z ) − k − X j =0 h j z j j − X s =0 v s z s +1 ! . Therefore, S u ( z ) = h ( z ) S v ( z ) − T ( z ) , where T ( z ) = k − X j =0 h j z j j − X s =0 v s z s +1 ! is apolynomial of degree at most k −
2, and S v ( z ) = S u ( z ) h ( z ) + T ( z ) h ( z ) . Since the Stieltjes function S v is a linear spectral modification of S u ([ ]),assuming that u is a positive definite linear functional and h is a positive polynomialon the support of a positive Borel measure dµ associated with u , it is well known(see [ ] and [ ]) that for n large enough each zero ζ of h with multiplicity j attracts j zeros of Q n . On the other hand, for every fixed n , at most k − Q n can lie outside supp ( µ ). These facts allow us to judge about the location ofthe zeros of h that lie outside the support of dµ . In [ ] quadratureformulas on the real line with the highest degree of accuracy, with positive weights,and with one or two prescribed nodes anywhere on the interval of integration arecharacterized. Next we will consider a more general problem when we deal with more prescribed nodes. We are interested in the study of Christoffel numbers as-suming they are positive numbers, i.e. by choosing those nodes outside the intervalof orthogonality of the initial measure.Let K n ( x, y ; u ) and K n ( x, y ; v ) be the kernel polynomials associated with thepositive definite linear functionals u and v , respectively, i.e. K n ( x, y ; u ) = n X j =0 P j ( x ) P j ( y ) || P j || and K n ( x, y ; v ) = n X j =0 Q j ( x ) Q j ( y ) || Q j || , where || P m || = h u, P m ( x ) P m ( x ) i and || Q m || = h v, Q m ( x ) Q m ( x ) i . First of all, we will find an algebraic relation between K n ( x, y ; u ) and K n ( x, y ; v ).Writing K n ( x, y ; v ) as K n ( x, y ; v ) = n X m =0 α n,m ( y ) P m ( x ) , we get α n,m ( y ) = h u, K n ( x, y ; v ) P m ( x ) i|| P m || . If m n − k + 1 , then h u, K n ( x, y ; v ) P m ( x ) i = h v, K n ( x, y ; v ) h ( x ) P m ( x ) i . Fromthe reproducing property of the kernel polynomial we get α n,m ( y ) = h ( y ) P m ( y ) || P m || , f or m n − k + 1 . On the other hand, α n,n − k +2 ( y ) = h u, K n ( x, y ; v ) P n − k +2 ( x ) i|| P n − k +2 || = D v, h K n +1 ( x, y ; v ) − Q n +1 ( x ) Q n +1 ( y ) || Q n +1 || i h ( x ) P n − k +2 ( y ) E || P n − k +2 || = h ( y ) P n − k +2 ( y ) || P n − k +2 || − Q n +1 ( y ) || Q n +1 || b k − ,n +1 .α n,n − k +3 ( y ) = h u, K n ( x, y ; v ) P n − k +3 ( x ) i|| P n − k +3 || = D v, h K n +2 ( x, y ; v ) − Q n +2 ( x ) Q n +2 ( y ) || Q n +2 || − Q n +1 ( x ) Q n +1 ( y ) || Q n +1 || i h ( x ) P n − k +3 ( x ) E || P n − k +3 || = h ( y ) P n − k +3 ( y ) || P n − k +3 || − Q n +2 ( y ) || Q n +2 || b k − ,n +2 − Q n +1 ( y ) || Q n +1 || b k − ,n +1 . RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 19
Finally, α n,n ( y ) = h u, K n ( x, y ; v ) P n ( x ) i|| P n || = * v, K n + k − ( x, y ; v ) − n + k − X j = n +1 Q j ( x ) Q j ( y ) || Q j || h ( x ) P n ( x ) + || P n || = h ( y ) P n ( y ) || P n || − n + k − X j = n +1 Q j ( y ) || Q j || b j − n,j . In other words, K n ( x, y ; v ) = h ( y ) K n ( x, y ; u ) − [ P ( k − n − k +2 ( x )] T T n,k − D k − Q ( k − n +1 ( y ) , where P ( k − n − k +2 ( x ) = ( P n − k +2 ( x ) , P n − k +3 ( x ) , . . . , P n ( x )) T , Q ( k − n +1 ( y ) = ( Q n +1 ( y ) , Q n +2 ( y ) , . . . , Q n + k − ( y )) T , T n,k − = b k − ,n +1 · · · b k − ,n +1 b k − ,n +2 · · · b k − ,n +1 b k − ,n +2 b k − ,n +3 · · · b ,n +1 b ,n +2 b ,n +3 · · · b k − ,n + k − and D k − = diag (cid:18) || Q n +1 || , || Q n +2 || , . . . , || Q n + k − || (cid:19) . By setting L n,k − = T n,k − D k − we get(4.10) K n ( x, y ; v ) = h ( y ) K n ( x, y ; u ) − [ P ( k − n − k +2 ( x )] T L n,k − Q ( k − n +1 ( y ) . If we commute the variables in (4.10),(4.11) K n ( y, x ; v ) = h ( x ) K n ( y, x ; u ) − [ P ( k − n − k +2 ( y )] T L n,k − Q ( k − n +1 ( x ) , since the kernel polynomials are symmetric with respect to the variables, thensubtracting (4.11) from (4.10), we get(4.12) K n ( x, y ; u ) = [ P ( k − n − k +2 ( y )] T L n,k − Q ( k − n +1 ( x ) − [ P ( k − n − k +2 ( x )] T L n,k − Q ( k − n +1 ( y ) h ( x ) − h ( y ) . Substituting (4.12) in (4.10) we obtain(4.13) K n ( x, y ; v ) = h ( y )[ P ( k − n − k +2 ( y )] T L n,k − Q ( k − n +1 ( x ) − h ( x )[ P ( k − n − k +2 ( x )] T L n,k − Q ( k − n +1 ( y ) h ( x ) − h ( y ) . In particular, the confluent formula holds K n ( x, x ; v ) = [ h ( x )[ P ( k − n − k +2 ( x )] T ] ′ L n,k − Q ( k − n +1 ( x ) − h ( x )[ P ( k − n − k +2 ( x )] T L n,k − [ Q ( k − n +1 ( x )] ′ − h ′ ( x ) . or, alternatively from (4.10) K n ( x, x ; v ) = h ( x ) K n ( x, x ; u ) − [ P ( k − n − k +2 ( x )] T L n,k − Q ( k − n +1 ( x ) . On the other hand, from (4.10) and taking into account that[ P ( k − n − k +2 ( x )] T T n,k − = [ Q ( k − n +1 ( x )] T − [ P ( k − n +1 ( x )] T Z n,k − where Z n,k − = b ,n +2 b ,n +3 · · · b k − ,n + k − b ,n +3 · · · b k − ,n + k − · · · b k − ,n + k − ... ... ... ...0 0 0 · · · b ,n + k − · · · , we get K n + k − ( x, y ; v ) = h ( y ) K n ( x, y ; u ) + [ P ( k − n − k +2 ( x )] T Z n,k − D k − Q ( k − n +1 ( y ) , and using the same arguments as above to obtain formula (4.13), we get the fol-lowing compact expression for the kernel polynomial. Proposition . K n + k − ( x, y ; v ) = h ( x )[ P ( k − n +1 ( x )] T M n,k − Q ( k − n +1 ( y ) − h ( y )[ P ( k − n +1 ( y )] T M n,k − Q ( k − n +1 ( x ) h ( x ) − h ( y ) , where M n,k − = Z n,k − D k − . Remark . Proceeding as above one has the expression for the confluentformula K n + k − ( x, x ; v ). Remark . If h ( x ) = x − a, then Z n, = 1. Thus K n +1 ( x, y ; v ) = ( x − a ) P n +1 ( x ) Q n +1 ( y ) − ( y − a ) P n +1 ( y ) Q n +1 ( x )( x − y ) || Q n +1 || . If we denote by y n +1 ,j , j = 1 , , . . . , n + 1 , the zeros of the polynomial Q n +1 , we de-duce in a straightforward way the value of the Christoffel numbers in the quadratureformula by using the above zeros as nodes. Indeed,1 K n +1 ( y n +1 ,j , y n +1 ,j ; v ) = 1 b ,n +1 ( y n +1 ,j − a ) P n ( y n +1 ,j ) Q ′ n +1 ( y n +1 ,j ) .
5. Examples
In this section we analyze some examples which illustrate the problems con-sidered in the previous sections. First we focus our attention on the symmetriccase which is less complex than the general one. The case when the connectioncoefficients are constant real numbers is also studied.
RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 21
Let us consider the symmetric SMOP { P n } n > , thatis the case when β n = 0 for n >
0. According to Theorem 2.1, equations (2.3),(2.4) and (2.5) become(5.1) b ,n +1 = b ,n + b k − ,n − b k − ,n − γ n − k +1 − b k − ,n b k − ,n γ n − k +2 , n > k, (5.2) b ,n +1 = b ,n + γ n − b k − ,n b k − ,n − γ n − k +1 + b ,n ( b ,n +1 − b ,n ) , n > k, and for 1 i k − b i +2 ,n +1 = b i +2 ,n + b i +1 ,n ( b ,n +1 − b ,n ) + b i,n γ n − i − b i,n − [ γ n + b ,n − b ,n +1 + b ,n ( b ,n +1 − b ,n )] . (5.3)Equations (2.6) and (2.7) become˜ β n = b ,n − b ,n +1 , n > , ˜ γ n = γ n + b ,n − b ,n +1 + b ,n ( b ,n +1 − b ,n ) , n > , or alternatively˜ β n = b k − ,n b k − ,n γ n − k +2 − b k − ,n − b k − ,n − γ n − k +1 , n > k, ˜ γ n = b k − ,n b k − ,n − γ n − k +1 n > k. (5.4) Step 1.
If we fix b ,n = b for n > k , then from (5.1) b k − ,n b k − ,n γ n − k +2 = b k − ,n − b k − ,n − γ n − k +1 = · · · = b k − ,k − b k − ,k − γ , and it is easy to conclude that ˜ β n = 0 , for n > k. Relation (5.2) yields(5.5) b ,n +1 = b ,n + γ n − b k − ,n b k − ,n − γ n − k +1 . Step 2.
If we impose the restrictions b ,n = b and b ,n = b , for n > k , thenfrom (5.5) and (5.4), we obtain γ n = b k − ,n b k − ,n − γ n − k +1 = ˜ γ n , for n > k. Proposition . If b ,n = b and b ,n = b , for n > k , then ˜ β n = 0 , n > k, ˜ γ n = γ n , n > k. This means that Q [ k +1] n ( x ) = P [ k +1] n ( x ) . Here, for a fixed positive integer number s , we denote by { P [ s ] n } n > the sequenceof polynomials satisfying the three-term recurrence relation xP [ s ] n ( x ) = P [ s ] n +1 ( x ) + β n + s P [ s ] n ( x ) + γ n + s P [ s ] n − ( x ) , n > , with initial conditions P [ s ] − ( x ) = 0, P [ s ]0 ( x ) = 1 . It is said to be the sequence ofassociated monic polynomials of order s for the linear functional u (see [ ]). Step 3.
We keep b ,n = b and b ,n = b , for n > k , and we add the constrain b ,n = b , for n > k . Since from (5.3), with i = 1, b ,n +1 = b ,n + b ( γ n − − γ n ) , n > k + 1 , then b ( γ n − − γ n ) = 0 . Thus, either b = 0 or γ n remains constant for n > k , thatis, γ n = γ k for n > k. If the coefficients γ n are constants for n > k , then { P n } n > is the sequence ofanti-associated polynomials of order k for the Chebyshev polynomials of the secondkind (see [ ]). Step 4.
The other possibility is that b = 0, b ,n = b and b ,n = b , for n > k + 1.Now we add the restriction b ,n = b , for n > k + 1. Since, from (5.3) with i = 2, b ,n +1 = b ,n + b ( γ n − − γ n ) , n > k + 1 , we obtain b ( γ n − − γ n ) = 0 , n > k + 1 , and, again, either b = 0 or the sequence { γ n } n > k − is a periodic sequence withperiod 2. Thus { P n } n > is the sequence of anti-associated polynomials of order k − ]). We refer to [ , p.91] for the explicitexpression of symmetric orthogonal polynomials defined by recurrence relationswhose coefficients are 2-periodic sequences. Let S n ( x ) = xS n − ( x ) − γ n S n − ( x ),where γ n = a > γ n +1 = b >
0. Then S n ( x ) = ( ab ) n/ h U n ( z ) + p b/a U n − ( z ) i ,S n +1 ( x ) = ( ab ) n/ xU n ( z ) , where z = ( x − ( a + b )) / (4 ab ) / . Step 5.
Yet another possibility is b = 0, b = 0, b ,n = b and b ,n = b , for n > k + 1. Following the previous reasoning let to add the restriction b ,n = b , for n > k + 1. Then (5.3), for i = 3, reads b ,n +1 = b ,n + b ( γ n − − γ n ) , n > k + 1 . Hence, b ( γ n − − γ n ) = 0 , n > k + 1 . Then either b = 0 or the sequence { γ n } n > k − is a 3-periodic one. RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 23
We can proceed in this way up to i = k − Example . Let { P n } n > be the sequence of monic Chebyshev polynomialsof second kind { ˜ U n } n > orthogonal with respect to dµ ( x ) = (1 − x ) / dx on ( − , β n = 0, γ n = 1 / n > . .
2) and (5 .
3) become b ,n +1 = b ,n + 14 (cid:18) b k − ,n − b k − ,n − − b k − ,n b k − ,n (cid:19) , n > kb ,n +1 = b ,n + 14 (cid:18) − b k − ,n b k − ,n − (cid:19) + b ,n ( b ,n +1 − b ,n ) , n > k,b i +2 ,n +1 = b i +2 ,n + 14 b i,n + b i +1 ,n ( b ,n +1 − b ,n ) − b i,n − (cid:20)
14 + b ,n − b ,n +1 + b ,n ( b ,n +1 − b ,n ) (cid:21) , for 1 i k − b ,n = b for n > k , and b ,n = b for n > k . Then we have b i +2 ,n +1 = b i +2 ,n + 14 ( b i,n − b i,n − ) , i k − , n > k. In particular, according to the fact that b ,n = b and b ,n = b for n > k , then b ,n +1 = b ,n , n > k + 1 ,b ,n +1 = b ,n , n > k + 1 , and, as a consequence, for every 1 i k − b i +2 ,n +1 = b i +2 ,n , n > k + 1 . On the other hand, if you assume, instead of b ,n = b and b ,n = b for n > k ,that b k − ,n = b k − and b k − ,n = b k − for n > k , a reverse situation in terms of theconnection coefficients, then b ,n +1 = b ,n , n > k + 1 ,b ,n +1 = b ,n , n > k + 1 , and b i +2 ,n +1 = b i +2 ,n + 14 ( b i,n − b i,n − ) , i k − , n > k + 1 , In particular, this means that b i +2 ,n +1 = b i +2 , i k − , n > k + 1 . Notice that in this case b ,k +1 = b ,k + 14 (cid:18) b k − ,k − b k − ,k − − b k − ,k b k − ,k (cid:19) ,b ,k +1 = b ,k + 14 (cid:18) − b k − ,k b k − ,k − (cid:19) ,b i +2 ,k +1 = b i +2 ,k + 14 ( b i,k − b i,k − ) , i k − . In other words, we have constant connection coefficients, but they appear for n > k + 1. Proposition . Let assume that { Q n } n > is a sequence of quasi-orthogonalpolynomials of order k − with respect to the sequence { ˜ U n } n > . If either b ,n = b and b ,n = b for n > k , or b k − ,n = b k − and b k − ,n = b k − for n > k , then allthe remaining connection coefficients are constant for n > k + 1 . Notice that if theinitial conditions are b k − ,k = b k − ,k − and b k − ,k = b k − ,k − then all coefficientsare constant for n > k . In this case, ˜ β n = 0 , n > k, ˜ γ n = 14 , n > k + 1 . This means that the SMOP { Q n } n > has the same sequence of ( k +1)-associatedpolynomials that the SMOP { ˜ U n } n > . In other words it is an anti-associated SMOPof order k + 1 of the Chebyshev polynomials of second kind. Notice that key information for the sequence { Q n } n > is given by the sequences { b ,n } n > and { b ,n } n > or, alternatively, bythe sequences { b k − ,n } n > k − and { b k − ,n } n > k − because˜ β n = β n + b ,n − b ,n +1 , n > , ˜ γ n = γ n + b ,n − b ,n +1 + b ,n ( β n − − β n − b ,n + b ,n +1 ) , n > , ˜ γ n = γ n − k +1 b k − ,n b k − ,n − , n > k. If for n > k the coefficients b ,n and b ,n do not depend on n , i.e. b ,n = b and b ,n = b , we have ˜ β n = β n , n > k, ˜ γ n = γ n + b ( β n − − β n ) , n > k. On the other hand, if b k − ,n and b k − ,n are constant coefficients, for n > k , i.e. b k − ,n = b k − and b k − ,n = b k − , it follows from (2.6), (3.6) and (3.7) that˜ β n = β n − k +1 + b k − b k − ( γ n − k +2 − γ n − k +1 ) , n > k, ˜ γ n = γ n − k +1 , n > k. RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 25
Example . Let { P n } n > be the sequence of either monic Chebyshev poly-nomials of third kind { ˜ V n } n > , orthogonal with respect to dµ ( x ) = (1 + x ) / (1 − x ) − / dx on ( − , { ˜ W n } n > ,orthogonal with respect to dµ ( x ) = (1 − x ) / (1 + x ) − / dx on ( − , P n ( x ) = ˜ U n ( x ) + a ˜ U n − ( x ) , n > , where the coefficient a depends on the choice of P . For the Chebyshev polyno-mials of the third kind, with ˜ V ( x ) = x − / a = − /
2, and for the Chebyshevpolynomials of the fourth kind, with ˜ W ( x ) = x + 1 / a = 1 /
2, (see [ , p.89]).Then Q n ( x ) = P n ( x ) + b ,n P n − ( x ) + · · · + b k − ,n P n − k +1 ( x )= ˜ U n ( x ) + ( a + b ,n ) ˜ U n − ( x ) + ( ab ,n + b ,n ) ˜ U n − ( x ) + · · · +( ab k − ,n + b k − ,n ) ˜ U n − k +1 ( x ) + ab k − ,n ˜ U n − k ( x ) . Thus, this problem is reduced to the one concerning Chebyshev polynomials ofthe second kind.Notice that if b ,n = b and b ,n = b , for n > k , according to Example 5.1,this yields ab i,n + b i +1 ,n = ˜ b i +1 i k − ,ab k − ,n = ˜ b k − . The same analysis applies when we assume b k − ,n = b k − and b k − ,n = b k − , for n > k . Example . Let { P n } n > be the sequence of monic Laguerre polynomials { ˜ L ( α ) n } n > , orthogonal with respect to dµ ( x ) = x α e − x dx on (0 , ∞ ), α > −
1. Inthis situation, β n = 2 n + α + 1 for n >
0, and γ n = n ( n + α ) for n >
1. Considerthe case when b ,n = b and b ,n = b , for n > k . It follows from (2 .
4) that b k − ,n b k − ,n − γ n − k +1 = γ n + b ((2( n −
1) + α + 1) − (2 n + α + 1)) , n > k,b k − ,n b k − ,n − = γ n − b γ n − k +1 . n > k, Step 1. If b = 0, then b k − ,n = (cid:18) nk − (cid:19)(cid:18) n + αk − (cid:19) A ( k, α ) , n > k. where A ( k, α ) does not depend on n . Therefore b k − ,n is a polynomial of degree2 k − n . Step 2. If b = 0, then b k − ,n b k − ,n − = ( n − α )( n − α )( n − k + 1)( n − k + 1 + α ) , where α , α are, in general, complex numbers such that ( n − α )( n − α ) = n ( n + α ) − b . Thus, b k − ,n b k − ,k − = (cid:0) n − α n − k +1 (cid:1)(cid:0) n − α n − k +1 (cid:1)(cid:0) n − k +1+ αn − k +1 (cid:1) , n > k. Then b k − ,n is a rational function.From (2 .
3) we have b k − ,n b k − ,n γ n − k +2 = b k − ,n − b k − ,n − γ n − k +1 + 2 k − . Then b k − ,n b k − ,n γ n − k +2 = (2 k − n + c , where c does not depend on n , and b k − ,n = ((2 k − n + c ) (cid:0) n − α n − k +2 (cid:1)(cid:0) n − α n − k +2 (cid:1)(cid:0) n − k +2+ αn − k +2 (cid:1) b k − ,k − ( k − − α )( k − − α ) . Then b k − ,n is also a rational function.Now we look at the behaviour of the coefficients b i,n for 3 i k −
3, and n > k .From (2 .
8) and (2 .
5) with i = 1, we have b ,n +1 = b ,n + b ( β n − − β n ) − b ( β n − − β n ) + b ( γ n − − γ n )= b ,n − b + b (2 b + ( n − n − α ) − n ( n + α )) , we see that b ,n = c , n + c , n + c , is a polynomial of degree two in n .Also, from (2 .
8) and (2 .
5) with i = 2, we have b ,n +1 = b ,n + b ,n ( β n − − β n ) + b γ n − − b ( γ n − b )= b ,n − b ,n + b ( γ n − − γ n ) + 2 b b = b ,n − (cid:0) c , n + c , n + c , (cid:1) + b [( n − n − α ) − n ( n + α )] + 2 b b , and b ,n = c , n + c , n + c , n + c , is a polynomial of degree three in n .Suppose that k = 5. If b = 0, then the above relations yield that b k − ,n = b ,n is a polynomial of degree eight. On the other hand, b ,n is a polynomial of degreethree, which is a contradiction. Otherwise, if b = 0, according to the abovecalculations, b k − ,n = b ,n and b k − ,n = b ,n are rational functions of the variable n . However, b ,n and b ,n are polynomials of degrees two and three, respectively.This is a contradiction again.We conclude that it is not possible that b ,n = 0 and b ,n = 0, n > k , areconstant real numbers when you deal with Laguerre orthogonal polynomials. RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 27
Now we consider the special case when allthe coefficients in (1.2) do not depend on n . Let us apply Theorem 2.1 to obtain thenecessary and sufficient conditions for the orthogonality of the monic polynomialsequence { Q n } n > . Let { P n } n > be a SMOP with respect to a linear functional u and(5.6) Q n ( x ) = P n ( x ) + b P n − ( x ) + · · · + b k − P n − k +1 ( x ) , n > k, where { b i } k − i =1 are real numbers, and b k − = 0. The above necessary and sufficientconditions become γ n − k +1 − γ n = b ( β n − − β n ) , n > k + 1 , (5.7) b i − ( γ n − k +1 − γ n − i +1 ) = b i ( β n − i − β n ) , n > k + 1 , i k − , (5.8) ˜ β n = β n , n > k + 1 , ˜ γ n = γ n − k +1 , n > k + 1 , ˜ γ n = γ n + b ( β n − − β n ) = 0 , n > k, where { ˜ β n } n > and { ˜ γ n } n > are the coefficients of the three term recurrence relationsatisfied by the SMOP { Q n } n > , for n > k .These results were obtained [ ]. In that paper the authors provide also adetailed study of the case k = 3 with constant coefficients. The case k = 4 withconstant coefficients was analysed thoroughly in [ ].Now we focus our attention on the case when the sequence { P n } n > is symmet-ric, i.e., β n = 0, for all n >
0. The conditions (5 .
7) and (5 .
8) yield the necessaryand sufficient conditions, for n > k + 1, γ n − ( k − − γ n = 0 , (5.9) b i − (cid:0) γ n − ( k − − γ n − ( i − (cid:1) = 0 , i k − . (5.10)Then, as a consequence of Theorem 2.1, we obtain Corollary . Let { P n } n > be a symmetric monic polynomial sequence andlet { Q n } n > be a monic polynomial sequence defined by relation (5 . , for n > k .Then { Q n } n > is a SMOP with recurrence coefficients { ˜ β n } n > and { ˜ γ n } n > if andonly if the sequence { γ n } n > satisfies (5 . and (5 . . Furthermore, the recurrencecoefficients of SMOP { Q n } n > satisfy ˜ β n = 0 and ˜ γ n = γ n , for n > k + 1 . Inother words, Q [ k +1] n ( x ) = P [ k +1] n ( x ) , where Q [ k +1] n and P [ k +1] n are the associatedpolynomials of order k + 1 for the SMOP { Q n } n > and { P n } n > , respectively (see [ ] ). Our next result characterizes { γ n } n > as a periodic sequence and we also discussits possible periods. Theorem . Under the hypothesis of Corollary 5.1, the sequence of the co-efficients of the three-term recurrence relation { γ n } n > must be a periodic sequencewith period j , where j is a divisor of k − . Furthermore, if | b j | + | b k − − j | = 0 ,for j ⌊ ( k − / ⌋ , then the period of the sequence { γ n } n > is k − . If ( | b r | + | b k − − r | )( | b s | + | b k − − s | ) . . . ( | b t | + | b k − − t | ) = 0 , for any r, s, ..., t , such that r, s, ..., t ⌊ ( k − / ⌋ , then the period of the sequence { γ n } n > is the greatestcommon divisor of r, s, ..., t , and k − . Proof.
Conditions (5 .
9) and (5 . n > k +1, tell us that if any coefficient b j = 0, for 1 j k −
2, then γ n − j = γ n − ( k − = γ n . Hence, we conclude that • if any coefficient b j = 0, for 1 j ⌊ ( k − / ⌋ , then γ n − j = γ n implies that { γ n } n > is a periodic sequence with period j ; • if any coefficient b k − − j = 0, for 1 j ⌊ ( k − / ⌋ , then γ n − ( k − − j ) = γ n − ( k − implies that { γ n } n > is a periodic sequence with period j .As a summary, if | b j | + | b k − − j | 6 = 0, for any 1 j ⌊ ( k − / ⌋ , then { γ n } n > is a j -periodic sequence.The condition (5 . γ n − ( k − = γ n , for n > k + 1, tell us that the sequence { γ n } n > is also a k − k − j ⌊ ( k − / ⌋ has, in fact, period equals to the greatest common divisor of k − j . Since all divisors of k − j ⌊ ( k − / ⌋ , all choicesof b j such that | b j | + | b k − − j | 6 = 0 yield the divisors in 1 j ⌊ ( k − / ⌋ . Alsothe choice | b j | + | b k − − j | = 0 for 1 j ⌊ ( k − / ⌋ yields k − (cid:3) Remark . i) If | b j | + | b k − − j | 6 = 0 for only one j such that 1 j ⌊ ( k − / ⌋ , then if k − j , the period of the sequence { γ n } n > isexactly j .ii) Observe that to choose values for | b r | and | b k − − r | one needs k > r + 1.iii) Notice that the coefficient γ > Remark . If we consider a SMOP { P n } n > , such that β n = β , for n > .
7) and (5 .
8) yield the same behaviour for { γ n } n > as in Theorem5 . k = 4, when the sequence { P n } n > is not symmetric, in [ ] theauthors also consider the choice b = b = 0 and they prove that both sequences { γ n } ∞ n =2 and { β n } ∞ n =2 must be 3-periodic. When one considers either only b = 0or only b = 0, the behaviour of { γ n } ∞ n =2 and { β n } ∞ n =2 is one-periodic. Finally,with both b = 0 and b = 0 the behaviour of { γ n } ∞ n =2 and { β n } ∞ n =2 depends onthe values of b , b and b . Remark . Grinshpun [ ] showed that Bernstein-Szeg˝o’s orthonormal poly-nomials of i -th kind, i = 1 , , , , and only them, can be represented as a linearcombination of Chebyshev orthonormal polynomials of i -th kind, respectively, withconstant coefficients, namelyˆ Q n ( x ) = k − X j =0 t j ˆ P n − j ( x ) , n > k, where { ˆ Q n } n > denote the Bernstein-Szeg˝o orthonormal polynomials of i -th kindand { ˆ P n } n > are the Chebyshev orthonormal polynomials of i th kind. RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 29
Sequences of Bernstein-Szeg˝o polynomials are orthogonal with respect to theweight functions ω i ( x ) = µ i ( x ) σ k − ( x ) , i = 1 , , , , where µ i ( x ) is the Chebyshev weight function of the i th kind, i = 1 , , , , and σ k − ( x ) is a positive polynomial of degree k − − , t j are given as the real coefficients of a polynomial t ( z ) of degree k −
1, that ap-pears as the Fej´er-normalized representation of the positive polynomials σ k − ( x ) . Moreover, Grinshpun proves that if { P n } n > are the classical Chebyshev orthonor-mal polynomials of one of the four kinds, ˆ Q n ( x ) = P k − j =0 b j ˆ P n − j ( x ) , n > k, with b b k − = 0 , and the polynomial g ( z ) = P k − j =0 b j z j either does not have any zerosin the unit disc or all its zeros are located on the unit circle, then either ˆ Q n ( x )or ˆ Q ∗ n ( x ) = P k − j =0 b j ˆ P n − k +1+ j ( x ) , n > k , are Bernstein-Szeg˝o polynomials of thecorresponding kind. Acknowledgements.
We thank Dr. D. K. Dimitrov by his continued support.His comments and criticism have contributed to improve the presentation of themanuscript.
References
1. M. Alfaro, F. Marcell´an, A. Pe˜na, M. L. Rezola,
On linearly related orthogonal polynomialsand their functionals , J. Math. Anal. Appl. (2003), 307–319.2. M. Alfaro, F. Marcell´an, A. Pe˜na, M. L. Rezola,
On rational transformations of linear func-tionals: direct problem , J. Math. Anal. Appl. (2004), 171–183.3. M. Alfaro, F. Marcell´an, A. Pe˜na, M. L. Rezola,
When do linear combinations of orthogonalpolynomials yield new sequences of orthogonal polynomials? , J. Comput. Appl. Math. (2010), 1446–1452.4. M. Alfaro, A. Pe˜na, M. L. Rezola, F. Marcell´an,
Orthogonal polynomials associated with aninverse quadratic spectral transform , Comput. Math. Appl. (2011), 888–900.5. M. Alfaro, A. Pe˜na, J. Petronilho, M. L. Rezola, Orthogonal polynomials generated by a linearstructure relation: inverse problem , J. Math. Anal. Appl. (2013), 182–197.6. R. Askey,
Positive quadrature methods and positive polynomial sums , in: C. K. Chui et al.(eds.),
Approximation Theory, V (College Station, Tex., 1986), Academic Press, Boston, MA,1986, 1–29.7. R. Askey, G. Gasper,
Positive Jacobi polynomial sums. II , Amer. J. Math. (1976), 709–737.8. R. Askey, G. Gasper, Inequalities for polynomials , In: A. Baernstein et al. (eds.),
The Bieber-bach Conjecture (West Lafayette, Ind., 1985), Math. Surveys Monogr., 21, Amer. Math. Soc.,Providence, 1986, 7–32.9. A. F. Beardon, K. A. Driver,
The zeros of linear combinations of orthogonal polynomials , J.Approx. Theory (2005), 179–186.10. D. Beghdadi, P. Maroni,
On the inverse problem of the product of a semiclassical form by apolynomial , J. Comput. Appl. Math. (1998), 377–399.11. A. Branquinho, F. Marcell´an, Generating new classes of orthogonal polynomials , Int. J. Math.Math. Sci. (1996), 643–656.12. C. Brezinski, K. A. Driver, M. Redivo-Zaglia, Quasi-orthogonality with applications to somefamilies of classical orthogonal polynomials , Appl. Numer. Math. (2004), 157–168.13. M. I. Bueno, F. Marcell´an, Darboux transformations and perturbations of linear functionals ,Linear Algebra Appl. (2004), 215–242.
14. A. Bultheel, R. Cruz–Barroso, M. Van Barel,
On Gauss-type quadrature formulas with pre-scribed nodes anywhere on the real line , Calcolo (2010), 21–48.15. T. S. Chihara, On quasi-orthogonal polynomials , Proc. Amer. Math. Soc. (1957), 765–767.16. T. S. Chihara, An Introduction to Orthogonal Polynomials , Gordon and Breach, New York,1978.17. D. K. Dimitrov,
Connection coefficients and zeros of orthogonal polynomials , J. Comput.Appl. Math. (2001), 331–340.18. A. Draux,
On quasi-orthogonal polynomials , J. Approx. Theory (1990), 1–14.19. D. Dickinson, On quasi-orthogonal polynomials , Proc. Amer. Math. Soc. (1961), 185–194.20. S. Elhay, J. Kautsky, Jacobi matrices for measures modified by a rational factor , Numer.Algorithms (1994), 205–227.21. J. Favard, Sur les polynˆomes de Tchebycheff , C. R. Acad. Sci. Paris (1935), 2052–2053.22. L. Fej´er,
Mechanische quadraturen mit positiven Cotesschen zahlen , Math. Z. (1933),287–309.23. W. Gautschi, Orthogonal Polynomials: Computation and Approximation , Oxford UniversityPress, New York, 2004.24. Z. Grinshpun,
Special linear combinations of orthogonal polynomials , J. Math. Anal. Appl. (2004), 1–18.25. C. Hounga, M. N. Hounkonnou, A. Ronveaux,
New families of orthogonal polynomials , J.Comput. Appl. Math. (2006), 474–483.26. K. H. Kwon, D. W. Lee, F. Marcell´an, S. B. Park,
On kernel polynomials and self-perturbationof orthogonal polynomials , Ann. Mat. Pura Appl. (2001), 127–146.27. G. L´opez Lagomasino,
Convergence of Pad´e approximants of Stieltjes type meromorphic func-tions and comparative asymptotics for orthogonal polynomial , Math. USSR Sb. (1989),207–227.28. G. L´opez Lagomasino, Relative asymptotics for orthogonal polynomials on the real axis , Math.USSR Sb. (1990), 505–529.29. F. Marcell´an, S. Varma, On an inverse problem for a linear combination of orthogonal poly-nomials , J. Difference Equ. Appl. (2014), 570–585.30. P. Maroni, Sur la suite de polynˆomes orthogonaux associ´ee `a la forme u = δ c + λ ( x − c ) − L ,Period. Math. Hungar. (1990), 223–248.31. P. Maroni, Une th´eorie alg´ebrique des polynˆomes orthogonaux. Application aux polynˆomesorthogonaux semi-classiques . In: C. Brezinski et al. (eds.),
Orthogonal polynomials and theirapplications (Erice, 1990), IMACS Ann. Comput. Appl. Math., 9, Baltzer, Basel, 1991, 95–130.32. P. Maroni,
Semi-classical character and finite type relations between polynomial sequences ,Appl. Numer. Math. (1999), 295–330.33. N. Obrechkoff, Zeros of Polynomials , 1963 (in Bulgarian); English translation by I. Dimovskyand P. Rusev, Marin Drinov Acad. Publ., Sofia, 2003.34. F. Peherstorfer,
Linear combinations of orthogonal polynomials generating positive quadratureformulas , Math. Comp. (1990), 231–241.35. F. Peherstorfer, On orthogonal polynomials with perturbed recurrence relations , J. Comput.Appl. Math. (1990), 203–212.36. F. Peherstorfer, Zeros of linear combinations of orthogonal polynomials , Math. Proc. Cam-bridge Philos. Soc. (1995), 533–544.37. J. Petronilho,
On the linear functionals associated to linearly related sequences of orthogonalpolynomials , J. Math. Anal. Appl. (2006), 379–393.38. G. P´olya, ¨Uber die konvergenz von quadraturverfahren , Math. Z. (1933), 264–286.39. Q. I. Rahman, G. Schmeisser, Analytic Theory of Polynomials , Oxford University Press, Ox-ford, 2002.40. A. Ronveaux, W. Van Assche,
Upward extension of the Jacobi matrix for orthogonal polyno-mials , J. Approx. Theory (1996), 335–357. RTHOGONALITY OF QUASI-ORTHOGONAL POLYNOMIALS 31
41. J. A. Shohat,
On mechanical quadratures, in particular, with positive coefficients , Trans.Amer. Math. Soc. (1937), 491–496,42. C. Sturm, M´emoire sur la r´esolution des ´equations num´eriques , M´emoires divers pr´esent´espar des savants ´etrangers `a l’Acad´emie Royale des Sciences de l’Institut de France (1835),273–318.43. B. Wendroff, On orthogonal polynomials , Proc. Amer. Math. Soc. (1961), 554–555.44. Y. Xu, A characterization of positive quadrature formulae , Math. Comp. (1994), 703–718.45. Y. Xu, Quasi-orthogonal polynomials, quadrature and interpolation , J. Math. Anal. Appl. (1994), 779–799.46. A. Zhedanov,
Rational spectral transformations and orthogonal polynomials , J. Comput. Appl.Math. (1997), 67–83. Departamento de Matem´atica Aplicada, UNESP-Univ Estadual Paulista, S˜ao Jos´edo Rio Preto, SP, Brazil
E-mail address : [email protected] Departamento de Matem´aticas, Universidad Carlos III de Madrid, Legan´es, Spain,and Instituto de Ciencias Matem´aticas (ICMAT), Cantoblanco, Spain
E-mail address : [email protected] Department of Mathematics, Faculty of Science, Ankara University, Tando˘gan,Ankara, Turkey
E-mail address ::