Orthogonal rational functions and quadrature on the real half line

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http://www.elsevier.com/locate/jco

Orthogonal rational functions and quadrature on

the real half line

A. Bultheel,

a,1

P. Gonza´lez-Vera,

b,,2

E. Hendriksen,

c

and

Olav Nj

(astad

d

a_{Department of Computer Science, K.U.Leuven, Belgium} b_{Department of Mathematical Analysis, Univ. La Laguna, Tenerife, Spain}

c_{Department of Mathematics, University of Amsterdam, The Netherlands} d

Department of Mathematical Science, Norwegian University of Science and Technology, Trondheim, Norway

Received 28 March 2002; revised 11 September 2002; accepted 26 September 2002

Abstract

In this paper we generalize the notion of orthogonal Laurent polynomials to orthogonal rational functions. Orthogonality is considered with respect to a measure on the positive real line. From this, Gauss-type quadrature formulas are derived and multipoint Pade´ approximants for the Stieltjes transform of the measure. Convergence of both the quadrature formula and the multipoint Pade´ approximants is discussed.

1. Introduction

Laurent polynomials can be considered as rational functions with poles at zero and inﬁnity. In this paper we generalize this notion to rational functions that may have several negative real poles, which can be conveniently thought of as poles that are grouped around zero and inﬁnity.

_{Corresponding author.}

E-mail address:pglez@ull.es (P. Gonz!alez-Vera).

1_{A. Bultheel was partially supported by the Fund for Scientiﬁc Research (FWO), project ‘‘CORFU:}

Constructive study of orthogonal functions’’, grant #G.0184.02 and the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Ofﬁce for Science, Technology and Culture. The scientiﬁc responsibility rests with the author.

2_{P. Gonza´lez-Vera was partially supported by the research project B FM2001-3411 of the Spanish}

Ministry of Science and Technology.

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When for the integral Imðf Þ ¼

Z N

0

fðxÞ dmðxÞ;

we consider quadrature formulas of the form Inðf Þ ¼

Xn j¼1

AjfðxjÞ;

we may look for nodes xj and weights Ajsuch that the quadrature formula is exact

for a space of functions that is as large as possible. In the polynomial case, one obtains exactness in the space of polynomials of degree up to 2n 1 by choosing the nodes as the zeros of the nth orthogonal polynomial with respect to the measure m and taking the corresponding Gauss–Christoffel numbers as weights. These are the well-known Gauss formulas (see[13,16]). In[3](see also[10]) the polynomials were replaced by Laurent polynomials having a ﬁnite number of positive and negative powers of z: In that situation, exactness can be obtained for certain spaces of Laurent polynomials of dimension 2n; if we take as nodes the zeros of orthogonal Laurent polynomials. As mentioned above, we will now generalize this idea to rational functions whose poles need not all be located at zero and inﬁnity. Quadrature formulas for the integral over the positive half line exact in certain spaces of rational functions have been considered before. See [7,18]. For related results, see also[9,11,15].

It is also well known that these so-called Gauss-type quadrature formulas are closely related to rational approximants for the Stieltjes transform FmðzÞ ¼ Imð1=ðz

xÞÞ: More precisely, in the case of Laurent polynomials this quadrature formula FnðzÞ ¼ Inð1=ðz xÞÞ yields a two-point Pade´ approximant for Fmas was extensively

studied in[10]. In the present situation of more general rational functions we obtain multipoint Pade´ approximants.

The outline of the paper is as follows. In Section 2 we introduce our spaces of rational functions. The quadrature formulas are given in Section 3 where several expressions for the weights are proved. Error formulas and convergence for the Gauss-type formulas are discussed in Section 4 where we make use of results by Lopez-Lagomasino and Illa´n-Gonza´lez [14]. Section 5 introduces the multipoint Pade´ approximants and elaborates further on the connection between the quadrature formulas and the multipoint Pade´ approximants, which will give rise to new error estimates.

2. The spaces of rational functions

Let p and q be two given integers such that ppq: As usual, by a Laurent polynomial (L-polynomial in short) we mean a function of the form:

LðzÞ ¼X

q

j¼p

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We denote by Lp;q the space of Laurent polynomials as given by (2.1) and by L the

space of all Laurent polynomials. Also, Pk (k being a nonnegative integer) will

denote the space of polynomials of degree k at most, (observe that Pk¼ L0;k) and P

the space of all the polynomials. Laurent polynomials can be viewed as rational functions with prescribed poles at the origin and/or at inﬁnity. Here, we will deal with more general rational functions with prescribed poles at points a1; a2; y :As

will be explained later on, we will assume that the sequence of the poles a¼ fakgN1 is

in #C\ð0; NÞ: (Here and in the sequel #C denotes the Riemann sphere #C¼ C,fNg:) Furthermore, in order to reproduce the behavior of the Laurent polynomials (two-poles) we will assume that the sequence a is conveniently distributed around the origin and inﬁnity. More precisely, assume that a¼ a,b with a ¼ fajg

N

j¼1 and b¼

fbjg N

j¼1 so that both of them are contained in #C\ð0; NÞ with ajaN and bka0 for

any j; kX1: Although some of the results derived below can be obtained in this more general setting, we shall restrict ourselves in this paper to the situation where all the a’s are in ð1; 0 (these are the points around the origin) and all the b’s are in ½N; 1Þ (giving the points near inﬁnity). In fact, the introduction of two separate notations a’s and b’s is not strictly necessary for our results. Any non-speciﬁed basis that ‘‘respects the nesting of the Lnspaces’’ would do, but since some formulations

become simpler with the introduction of this notation, we shall use it in the rest of this text.

Now, for any nonnegative integer n; we deﬁne U0¼ 1 and UnðxÞ ¼

x an

1 x=bn

; nX1: ð2:2Þ

In the rest of this paper, we set B0ðxÞ ¼ 1 while for nX1:

BnðxÞ ¼ UnðxÞBn1ðxÞ and BnðxÞ ¼

1 BnðxÞ

: ð2:3Þ

By introducing the polynomials

o0ðxÞ ¼ 1; onðxÞ ¼ ðx a1Þ?ðx anÞ; nX1;

p0ðxÞ ¼ 1; pnðxÞ ¼ ð1 x=b1Þ?ð1 x=bnÞ; nX1

we can write BnðxÞ ¼ onðxÞ=pnðxÞ for n ¼ 0; 1; 2; y :

On the other hand, for p and q nonnegative integers we deﬁne

Lp;q¼ spanfBk: ppkpqg: ð2:4Þ Thus, Lp;q¼ PðxÞ opðxÞpqðxÞ : PAPpþq

so that its dimension is pþ q þ 1:

Clearly, when all the an coincide at the origin and all the bn at inﬁnity, then

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In order to construct a sequence of nested spaces as deﬁned by (2.4) we will start from two nondecreasing sequencesfpðnÞg and fqðnÞg of nonnegative integers such that pðnÞ þ qðnÞ ¼ n (obviously pð0Þ ¼ qð0Þ ¼ 0). Thus, we deﬁne

Ln¼ LpðnÞ;qðnÞ¼ PðxÞ opðnÞðxÞpqðnÞðxÞ : PAPn :

So, we see that LnCL_nþ1 and dimðL_nÞ ¼ n þ 1: We denote L ¼SN_n¼0L_n: Setting DnðxÞ ¼ opðnÞðxÞpqðnÞðxÞ; then for any f ALn; one can write fðxÞ ¼

PðxÞ=DnðxÞ; PAPn:Now, in order to simplify our forthcoming statements it will be

convenient to introduce the factor lnðxÞ ¼

x apðnÞ if pðnÞ4pðn 1Þ;

1 x=bqðnÞ if qðnÞ4qðn 1Þ:

(

ð2:5Þ

Thus, DnðxÞ ¼ lnðxÞDn1ðxÞ for nX1 with D0ðxÞ ¼ 1:

3. Quadrature and orthogonality

Let m be a ﬁnite positive measure with an inﬁnite support contained in ½0; NÞ: Moreover, assume that the integrals Imðf Þ ¼R

N

0 fðxÞ dmðxÞ exist for all f AL L:

Then we can consider an inner product in the spaces Ln; deﬁned by /f ; gS¼

RN

0 fðxÞgðxÞ dmðxÞ; and construct an orthogonal basis. Our aim is to approximate

the integral Imðf Þ ¼

Z N

0

fðxÞ dmðxÞ ð3:1Þ

by a quadrature formula like Inðf Þ ¼

Xn j¼1

AjfðxjÞ ð3:2Þ

with distinct nodes fxjgnj¼1Cð0; NÞ and coefﬁcients or weights fAjgnj¼1 to be

determined so that Inðf Þ exactly matches Imðf Þ for f in LNwith N ¼ NðnÞ as large as

possible. We see we have at our disposal 2n unknown parameters so that it seems natural to ask for quadrature formulas exact in subspaces of dimension 2n: Recall that the poles a¼ fangCð1; 0 while b ¼ fbngC½N; 1Þ: Taking into account

that dimðLpþqÞ ¼ p þ q þ 1 we have the following general result (which actually

holds under the weaker assumption that all the poles are in½N; 0).

Theorem 3.1 (Gautschi [7]). Let p and q be nonnegative integers such that pþ q ¼ 2n 1: Then, Inðf Þ ¼Pnj¼1AjfðxjÞ ¼ Imðf Þ for any f ALp;q; if and only if,

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(i) fxjg are the zeros of the nth orthogonal polynomial with respect to the positive

measure

dm_p;qðxÞ ¼ dmðxÞ opðxÞpqðxÞ

: ð3:3Þ

(ii) Aj¼ CjopðxjÞpqðxjÞ40; fCjgnj¼1 being the nth Christoffel numbers for the

measure dm_p;qðxÞ:

Remark 3.1. The quadrature formula obtained in Theorem 3.1 is ‘‘optimal’’ in the sense that there does not exist a quadrature formula with n nodes inð0; NÞ that is exact in Lp;q with pþ q ¼ 2n:

Indeed, assume Inðf Þ ¼Pnj¼1AjfðxjÞ ¼ Inðf Þ for any f ALp;q with pþ q ¼ 2n:

Let QnðxÞ ¼ ðx x1Þ?ðx xnÞ and deﬁne RðxÞ ¼ Q2nðxÞ=½opðxÞpqðxÞ: Then,

clearly RALp;q and satisﬁes ImðRÞ40 and InðRÞ ¼ 0:

From Theorem 3.1, we see that when quadrature formulas of the form Inðf Þ ¼

Pn

j¼1AjfðxjÞ; exact in L2n1 are required, then the nodes are the zeros of the nth

orthogonal polynomial with respect to dmðxÞ=½opð2n1ÞðxÞpqð2n1ÞðxÞ: So, zeros of

orthogonal polynomials with respect to ‘‘varying’’ measures immediately arise. This is what some authors (see e.g.[18]) call the ‘‘Gaussian approach’’.

On the other hand, one could think of using ‘‘orthogonal rational functions’’. In this respect, as usual, the measure m induces an inner product on L; deﬁned by

/f ; gS ¼Z N

0

fðxÞgðxÞ dmðxÞ; f ; gAL: Letfjkg

n

0be the orthonormal basis for Lnobtained by applying the Gram–Schmidt

orthogonalization process to the basisfBk: pðnÞpkpqðnÞg: We suppose thereby

that the ordering of the Ln is respected, i.e. jnALn\Ln1: When the process is

repeated for n¼ 1; 2; 3; y an essentially unique orthonormal sequence fjng N 0 is

achieved. Each j_n can be expressed as j_nðxÞ ¼ PnðxÞ

opðnÞðxÞpqðnÞðxÞ

¼PnðxÞ DnðxÞ

; PnAP_n: ð3:4Þ

Since jn>Ln1 it follows that Pn represents (up to a multiplicative factor) the nth

orthonormal polynomial with respect to the positive measure dmðxÞ

DnðxÞDn1ðxÞ

: ð3:5Þ

Thus, the polynomial PnðxÞ and hence jnðxÞ has n distinct zeros in ð0; NÞ so that the

sequencefjng N

0 is regular (see[4]).

Let x1; y; xnbe the zeros of jnðxÞ: Since Ln1is a Chebyshev space (i.e. spanned

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weights A1; y; An so that

Inðf Þ ¼

Xn j¼1

AjfðxjÞ ¼ Imðf Þ 8f ALn1: ð3:6Þ

Furthermore, it also holds that

Inðf Þ ¼ ImðRn1ðf ; ÞÞ; ð3:7Þ

Rn1ðf ; xÞ being the unique element in Ln1 interpolating f at nodes x1; y; xn:

Formula (3.6) corresponds to the so-called ‘‘orthogonal approach’’ (see[18]) and it is known [2,12] that in general both approaches, ‘‘the Gaussian’’ and ‘‘the orthogonal’’ do not coincide, i.e. that Inðf Þ given by (3.6) is not, in general exact

in L2n1(observe that this fact does hold in the polynomial situation i.e. pðnÞ ¼ 0 for

any n ðqðnÞ ¼ nÞ and bk¼ N so that now Ln¼ Pn and the well-known Gauss–

Christoffel formulas, exact in L2n1¼ P2n1;arise).

However, taking into account that P2n1 can be written as Pn Pn1;paralleling

the polynomial situation (see[13]) we can prove the following.

Theorem 3.2. An n-point quadrature formula of the form Inðf Þ ¼Pnj¼1AjfðxjÞ with

distinct nodes inð0; NÞ is exact in Ln Ln1; if and only if

(i) Inðf Þ is exact in Ln1;

(ii) fxjgnj¼1 are the zeros of the nth orthogonal function jnðxÞ with respect to dm:

Proof. ‘‘)’’

(i) Since Ln1CL_n L_n1 this part trivially follows. (ii) Set QnðxÞ ¼ ðx x1Þ?ðx xnÞ and deﬁne

RnðxÞ ¼ QnðxÞ opðnÞðxÞpqðnÞðxÞ ¼QnðxÞ DnðxÞ AL_n:

Since for any rALn1; r RnAL_n L_n1; it follows that /r; RnS ¼ Imðr RnÞ ¼

Xn j¼1

AjrðxjÞRnðxjÞ ¼ 0:

Thus, RnðxÞ coincides with jnðxÞ up to a multiplicative factor.

‘‘(’’.

Letfxjgnj¼1be the zeros of jnðxÞ: As already said, weights or coefﬁcients A1; y; An

can be uniquely determined so that Inðf Þ ¼

Xn j¼1

AjfðxjÞ ¼ Imðf Þ 8f ALn1: ð3:8Þ

Take now RALn Ln1;then one can write

RðxÞ ¼ PðxÞ

DnðxÞDn1ðxÞ

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and ImðRÞ ¼ Z N 0 RðxÞ dmðxÞ ¼ Z N 0 PðxÞ dmðxÞ DnðxÞDn1ðxÞ¼ Xn j¼1 ˜ BjPðxjÞ ˜

Bj being the nth Christoffel numbers for the measure dmðxÞ=½DnðxÞDn1ðxÞ:

Hence, ˜InðRÞ ¼ ImðRÞ if we deﬁne ˜ InðRÞ ¼ Xn j¼1 ˜ AjRðxjÞ; A˜j¼ ˜BjDnðxjÞDn1ðxjÞ since ImðRÞ ¼ Xn j¼1 ˜ BjDnðxjÞDn1ðxjÞ DnðxjÞDn1ðxjÞ PðxjÞ ¼ Xn j¼1 ˜ AjRðxjÞ ¼ ˜InðRÞ:

Thus, a quadrature formula with nodes the zeros fxjgnj¼1 of jnðxÞ being exact in

Ln Ln1 has been constructed. Thus, by (3.8) it follows that

˜

Inðf Þ ¼ Imðf Þ ¼ Inðf Þ 8f ALn1:

Since both quadrature formulas Inðf Þ and ˜Inðf Þ have the same nodes it holds that

Aj ¼ ˜Aj; j¼ 1; y; n: &

Remark 3.2. The quadrature formula deduced from Theorem 3.2 is also ‘‘optimal’’ in the sense that there cannot exist quadrature formulas Inðf Þ ¼Pnj¼1AjfðxjÞ with

nodes inð0; NÞ being exact in Ln Ln:

In the sequel, we are going to restrict ourselves to these quadratures and they will be called of ‘‘Gauss-type’’ for Ln Ln1 or more simply, when no confusion is

possible of ‘‘Gauss-type’’.

Next, from Theorem 3.2, we can deduce the following proposition where positivity of the weights is clearly displayed.

Proposition 3.3. Let Inðf Þ ¼Pnj¼1AjfðxjÞ be the n-point Gauss-type formula for Imðf Þ

and let j_nðxÞ be the nth orthonormal rational functions as defined above, then Aj¼ Z N 0 lnðxÞjnðxÞ lnðxjÞðx xjÞj0nðxjÞ 2 dmðxÞ; j¼ 1; y; n ð3:9Þ with lnðxÞ as given by (2.5).

Proof. For 1pjpn; let LjAL_n1 be such that LjðxkÞ ¼ dj;k¼

1 if j¼ k; 0 if jak; (

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where fxkgnk¼1 are the zeros of jnðxÞ: Clearly, L2jALn1 Ln1: Therefore, by

Theorem 3.2

ImðLjÞ ¼ InðLjÞ and ImðL2jÞ ¼ InðL2jÞ; j¼ 1; 2; y; n

and from this, we obtain Aj¼ Z N 0 LjðxÞ dmðxÞ ¼ Z N 0 L2 jðxÞ dmðxÞ; j¼ 1; 2; y; n: ð3:11Þ

On the other hand, LjAL_n1 is uniquely determined by condition (3.10) and it can be checked easily that

LjðxÞ ¼ lnðxÞ lnðxjÞ j_nðxÞ ðx xjÞj0nðxjÞ ; j¼ 1; 2; y; n: ð3:12Þ

Thus, from (3.11) and (3.12) the proof follows. &

To end this section, let us introduce the associated rational functions: snðzÞ ¼ Z N 0 j_nðzÞ jnðxÞ z x dmðxÞ; n¼ 1; 2; y ð3:13Þ Then, we have

Proposition 3.4. The weights of the n-point Gauss-type quadrature formula are given by Aj¼ snðxjÞ j0 nðxjÞ ; j¼ 1; y; n; ð3:14Þ

where j_n is the nth orthonormal rational function and sn is the associated rational

function.

Proof. Let LjAL_n1 be as in the proof of Proposition 3.3, i.e. L_jðx_kÞ ¼ d_j;k: Hence Aj¼ Z N 0 LjðxÞ dmðxÞ ¼ Z N 0 lnðxÞ lnðxjÞ j_nðxÞ ðx xjÞ dmðxÞ j0 nðxjÞ dmðxÞ; j¼ 1; y; n with lnðxÞ given by (2.5). Thus

Aj¼ 1 lnðxjÞj0nðxjÞ Z N 0 1 ðxj xÞ ½jnðxjÞ jnðxÞlnðxÞ dmðxÞ: ð3:15Þ

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Assume ﬁrst that lnðxÞ ¼ x apðnÞ;then Z N 0 jnðxÞ xj x lnðxÞ dmðxÞ ¼ Z N 0 jnðxÞ xj x ðx apðnÞ xjþ xjÞ dmðxÞ ¼ lnðxjÞ Z N 0 jnðxÞ xj x dmðxÞ: ð3:16Þ

So, (3.15) and (3.16) imply (3.14).

Suppose next that lnðxÞ ¼ 1 x=bqðnÞ: If bqðnÞ¼ N; then lnðxÞ ¼ 1 and (3.13)

trivially follows. If bqðnÞ is a ﬁnite number, then lnðxÞ=lnðxjÞ ¼ ðx bqðnÞÞ=ðxj

bqðnÞÞ: Hence 1 lnðxjÞ Z N 0 j_nðxÞlnðxÞ xj x dmðxÞ ¼ 1 xj bqðnÞ Z N 0 j_nðxÞ xj x ðx bqðnÞÞ dmðxÞ ¼ 1 xj bqðnÞ Z N 0 j_nðxÞ xj x ðx xjþ xj bqðnÞÞ dmðxÞ ¼ Z N 0 jnðxÞ xj x dmðxÞ: Again, from (3.15) we obtain

Aj¼ 1 j0 nðxjÞ Z N 0 j_nðxjÞ jnðxÞ xj x dmðxÞ ¼snðxjÞ j0 nðxjÞ : &

4. Errors and convergence for Gauss-type formulas

In this section we will ﬁrst deal with the error expression for an n-point Gauss-type formula, that is, we shall be concerned with

Enðf Þ ¼ Z N 0 fðxÞ dmðxÞ X n j¼1 AjfðxjÞ ¼ Imðf Þ Inðf Þ: ð4:1Þ

Following the polynomial case (see e.g. [13]) we will start with a certain rational-Hermite interpolant in Ln Ln1:Indeed, let t1; y; tnbe n distinct nodes inð0; NÞ

and f a function admitting a derivative on ð0; NÞ: Since Ln Ln1 represents a

Chebyshev space of dimension 2n; there exists a unique R2n1ðf ; xÞALn Ln1

satisfying the following interpolation conditions: R2n1ðf ; tjÞ ¼ f ðtjÞ

R0

2n1ðf ; tjÞ ¼ f0ðtjÞ

)

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Recall now that for any RALn Ln1;

RðxÞ ¼ PðxÞ

DnðxÞDn1ðxÞ

¼ PðxÞ

C2n1ðxÞ

with C2n1ðxÞ ¼ DnðxÞDn1ðxÞAP2n1 and PAP2n1:

Let Lj;nðxÞ and ˜Lj;nðxÞ be polynomials of degree 2n 1 at most, satisfying

Lj;nðtkÞ ¼ dj;k; L0j;nðtkÞ ¼ 0; j; k¼ 1; y; n;

˜

Lj;nðtkÞ ¼ 0; L˜0j;nðtkÞ ¼ dj;k; j; k¼ 1; y; n:

Then, by virtue of uniqueness for the interpolant R2n1ðf ; xÞ we have

R2n1ðf ; xÞ ¼ Xn j¼1 Hj;nðxÞf ðtjÞ þ Xn j¼1 ˜ Hj;nðxÞf0ðtjÞ; ð4:3Þ where Hj;nðxÞ ¼ Lj;nðxÞC2n1ðtjÞ þ ˜Lj;nðxÞC02n1ðtjÞ C2n1ðxÞ AL_n L_n1 ð4:4Þ and ˜ Hj;nðxÞ ¼ ˜ Lj;nðxÞC2n1ðtjÞ C2n1ðxÞ AL_n L_n1: ð4:5Þ

Therefore, by integrating with respect to dm one obtains ImðR2n1ðf ; ÞÞ ¼ Xn j¼1 ImðHj;nÞf ðtjÞ þ Xn j¼1 Imð ˜Hj;nÞf0ðtjÞ ¼X n j¼1 ˜ AjfðtjÞ þ Xn j¼1 ˜ Bjf0ðtjÞ ð4:6Þ with ˜Aj ¼ ImðHj;nÞ and ˜Bj¼ Imð ˜Hj;nÞ; j ¼ 1; y; n:

In short, (4.6) represents a quadrature formula for Imðf Þ involving values of f ðxÞ

and its derivative and exactly integrating any functions in Ln Ln1:

Now, it should be taken into account that we have started from n distinct points t1; y; tnonð0; NÞ: Let us see what happens when the zeros fx1; y; xng of jnðxÞ are

taken as nodes in the above approach.

Setting as usual j_nðxÞ ¼ PnðxÞ=DnðxÞ and using the fact that ˜Lj;nðxkÞ ¼ 0 for

k¼ 1; 2; y; n; it follows that ˜ Lj;nðxÞ ¼ PnðxÞqðxÞ; qAPn1: Thus, ˜ Bj¼ C2n1ðxjÞ Z N 0 PnðxÞ DnðxÞ dmðxÞ ¼ C2n1ðxjÞ Z N 0 j_nðxÞrðxÞ dm ¼ 0; j¼ 1; y; n;

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since rðxÞ ¼ qðxÞ=Dn1ðxÞALn1:Therefore, formula (4.6) becomes ImðR2n1ðf ; ÞÞ ¼ Xn j¼1 ˜ AjfðxjÞ: ð4:7Þ

Thus, ˜Inðf Þ ¼Pnj¼1A˜jfðxjÞ represents an n-point quadrature formula exact in Ln

Ln1: By virtue of Theorem 3.2, ˜Inðf Þ must coincide with the n-point Gauss-type

formula. Therefore,

Enðf Þ ¼ Imðf Þ Inðf Þ ¼ Imðf R2n1ðf ; ÞÞ: ð4:8Þ

Now, from (4.8) and using the error term in the Hermite interpolation (see e.g.[5]), the following can be easily proved.

Theorem 4.1. Let fðxÞ be a real function for which fð2nÞ_{ðxÞ exists on ½0; NÞ: Then}

Enðf Þ ¼ ðf ðxÞC2n1ðxÞÞð2nÞðynÞ ð2nÞ! Z N 0 *j2 nðxÞlnðxÞ dmðxÞ; 0pynpN; ð4:9Þ

where C2n1ðxÞ ¼ DnðxÞDn1ðxÞ; lnðxÞ is given by (2.5) and *jnðxÞ ¼ ˜PnðxÞ=DnðxÞ

represents the nth orthogonal function in Ln; normalized so that its numerator ˜PnðxÞ is

a monic polynomial of degree n:

Remark 4.1. When ak¼ 0 and bj¼ N for all k; j ¼ 1; 2; y; then C2n1ðxÞ ¼

xpðnÞþpðn1Þ; lnðxÞ ¼ xpðnÞpðn1Þ and *jnðxÞ ¼ xpðnÞP˜nðxÞALpðnÞ;qðnÞ: Thus, from

expression (4.9) we can recover the error formula in the n-point Gauss-type quadrature formula for Laurent polynomials as given in[3](see e.g. also[10]). On the other hand, when taking pðnÞ ¼ 0 for each n ðqðnÞ ¼ nÞ and bk¼ N for any

kX1; then Ln¼ L0;n¼ Pn;*jnðxÞ ¼ ˜PnðxÞ (monic) and (4.9) now yields (recall that

lnðxÞ ¼ 1) the well-known formula

Enðf Þ ¼ fð2nÞ_ðy nÞ ð2nÞ! Z N 0 ˜ P2_nðxÞ dmðxÞ;

which gives the error in the n-point Gauss–Christoffel formula.

Concerning convergence of the sequencefInðf Þg of Gauss-type formulas we ﬁrst

need the following lemma (which is proved in[14]as a direct consequence of[1]). Lemma 4.2. Let fpðnÞg and fqðnÞg be two nondecreasing sequences of nonnegative integers such that pðnÞ þ qðnÞ ¼ n; n ¼ 0; 1; y : Let a ¼ fangCð1; 0 and b ¼

fb_ngC½N; 1Þ be two sequences. Assume that at least one of the following two conditions hold lim n-NpðnÞ ¼ N and XN n¼1 ffiffiffiffiffiffiffi janj p 1þ janj ¼ þN ð4:10Þ

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or lim n-NqðnÞ ¼ N and XN n¼1 ffiffiffiffiffiffiffiffi jbnj p 1þ jb_nj¼ þN: ð4:11Þ Then, L ¼SN

0 Ln is dense (with respect to the uniform norm) in the class

CþN¼ ff : ½0; NÞ-R : f a continuous function and limx-þNfðxÞ exists and is

finiteg:

Note that these conditions mean that the sequences an and bn may tend to 0 or

inﬁnity but not too fast. If they tend to zero faster than 1=n2_{or to inﬁnity faster than}

n2_; _{then the inﬁnite sums above would converge and the conditions would not be}

satisﬁed.

Then, from the density of L in CþN and the positivity of the weights for the

Gauss-type formulas, one can prove the following.

Theorem 4.3. Under the same conditions as in Lemma 4.2, it holds that lim

n-NInðf Þ ¼ Imðf Þ ¼

Z N

0

fðxÞ dmðxÞ

for any function f in CþN where fInðf Þg is the sequence of Gauss-type quadrature

formulas as defined in Section 3.

To conclude this section, we prove the following theorem.

Theorem 4.4. Let g be an increasing function on½0; NÞ such that dm ¼ dg and suppose that f is a bounded function on½0; NÞ for which limt-NfðtÞ exists and such that its

improper Riemann–Stieltjes integralR₀NfðxÞ dgðxÞ converges. Then, under the same hypothesis as in Lemma 4.2, it holds that

lim

n-NInðf Þ ¼

Z N

0

fðxÞ dgðxÞ;

wherefInðf Þg is the sequence of Gauss-type formulas as defined in Section 3.

Proof. The proof follows closely the proof of Theorem 2 in [14]. The main ingredients are Theorem 1.5.4 in[17], the density of the space L of rational functions in the space of continuous functions on½0; NÞ; and the positivity of the weights of our quadrature formulas. Because[14] is not easily available, we include the proof for the sake of completeness.

Take a40 and consider the change of variable t¼ jðxÞ ¼ 1

xþa:Now deﬁne

hðtÞ ¼ ðf 3j

1_ÞðtÞ; ₀_otp1=a;

limt-0ðf 3j1ÞðtÞ; t¼ 0:

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Thus, hðtÞ represents a bounded Riemann–Stieltjes integrable function on ½0; 1=aÞ with respect toðg3j1_{Þ satisfying}

Z 1=a 0 hðtÞ dðg3j1ÞðtÞ ¼ Z N 0 fðxÞ dgðxÞ ¼ Z N 0 fðxÞ dmðxÞ:

Now, making use of[17, Theorem 1.5.4], for a given e40; there exist polynomials P and p such that

pðtÞphðtÞpPðtÞ in ½0; 1=a and Z 1=a 0 ðPðtÞ pðtÞÞ dðg3j1ÞðtÞ ! oe: Hence, one can write

p 1 xþ a pf ðxÞpP 1 xþ a for xX0 ð4:12Þ and Z N 0 ðP pÞ 1 xþ a dgðxÞ oe: ð4:13Þ

Now, because of the positivity of the weights in Inðf Þ from (4.12) it follows

In p 1 xþ a pInðf ÞpIn P 1 xþ a : ð4:14Þ

On the other hand, sinceðP3jÞðxÞ and ðp3jÞðxÞ belong to CþNfrom Theorem 4.3

and (4.14) one obtains

Z N 0 p 1 xþ a dgðxÞp lim inf n-N Inðf Þp lim sup_n-N Inðf Þ pZ N 0 P 1 xþ a dgðxÞ: ð4:15Þ Thus, 0p lim sup n-N Inðf Þ lim infn-n Inðf Þp Z N 0 P 1 xþ a p 1 xþ a dgðxÞoe by (4.13). Since e is arbitrary, we conclude that limn-NInðf Þ exists. Finally by (4.12)

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5. Multipoint Pade´ Approximants

Throughout this section we shall be concerned with the function in the variable z: FmðzÞ ¼ Im 1 z x ¼ Z N 0 dmðxÞ z x; ze½0; NÞ: ð5:1Þ

Here, FmðzÞ represents the well-known Stieltjes transform of the measure dm:

For this purpose, suppose that in order to approximate FmðzÞ we make use

of the n-point Gauss-type formula deﬁned in Section 3. Thus, we have (z is a parameter) In 1 z x ¼X n j¼1 Aj z xj ¼ FnðzÞ: ð5:2Þ

Clearly, FnðzÞ represents a rational function of type ðn 1; nÞ i.e. the numerator is a

polynomial of degree n 1 at most and the denominator a polynomial of degree n at most.

On the other hand, let Rn1ðx; zÞ be the interpolant to 1=ðz xÞ in Ln1;

satisfying

Rn1ðxj; zÞ ¼

1 z xj

; j¼ 1; y; n: ð5:3Þ

Now, by recalling that the nodes x1; y; xn are the zeros of jnðxÞ; it can be easily

checked that Rn1ðx; zÞ ¼ 1 z x 1 j_nðxÞ j_nðzÞ lnðxÞ lnðzÞ : ð5:4Þ It follows that ImðRn1ðx; zÞÞ ¼ In 1 z x ¼ FnðzÞ: ð5:5Þ

Therefore we may write

FnðzÞ ¼ ImðRn1ðx; zÞÞ ¼ Im 1 z x Im j_nðxÞlnðxÞ ðz xÞj_nðzÞlnðzÞ ¼ FmðzÞ Im j_nðxÞlnðxÞ ðz xÞjnðzÞlnðzÞ : ð5:6Þ We also get FnðzÞ ¼ Im j_nðzÞlnðzÞ jnðxÞlnðxÞ ðz xÞj_nðzÞlnðzÞ ¼ 1 j_nðzÞlnðzÞ Im ½jnðzÞ jnðxÞlnðzÞ jnðxÞ½lnðxÞ lnðzÞ z x ¼ 1 j_nðzÞIm jnðzÞ jnðxÞ z x þ 1 j_nðzÞlnðzÞ Im lnðzÞ lnðxÞ z x jnðxÞ :

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Notice thatðlnðzÞ lnðxÞÞ=ðz xÞ is constant. So by the orthogonality of the jnwe have Im lnðzÞ lnðxÞ z x jnðxÞ ¼ 0 for n¼ 1; 2; 3; y : Therefore FnðzÞ ¼ 1 jnðzÞ Im j_nðzÞ jnðxÞ z x ¼snðzÞ jnðzÞ ; n¼ 1; 2; 3; y ð5:7Þ

snðzÞ being the ‘‘associated’’ function as deﬁned by (3.13).

Remark 5.1. Eq. (5.7) can also be obtained from (3.14) and the partial fraction decomposition of FnðzÞ:

Conversely, (3.14) follows directly from (5.2) and (5.7).

Considering the error FmðzÞ FnðzÞ; for ze½0; NÞ; it follows from (5.6) that

FmðzÞ FnðzÞ ¼ Z N 0 j_nðxÞ j_nðzÞ lnðxÞ lnðzÞ dmðxÞ ðz xÞ: ð5:8Þ

On the other hand, the function (in the variable x) ðjnðzÞ jnðxÞÞ

z x

lnðxÞ

lnðzÞ

is in Ln1:Therefore, because of orthogonality,

Z N 0 j_nðzÞ jnðxÞ z x _l nðxÞ lnðzÞ jnðxÞ dmðxÞ ¼ 0:

From here and (5.8) one arrives at the following expression for ze½0; NÞ: FmðzÞ FnðzÞ ¼ 1 lnðzÞj2nðzÞ Z N 0 j2_nðxÞ ðz xÞlnðxÞ dmðxÞ: ð5:9Þ

On the other hand, since 1ALn1;it holds that Pnj¼1Aj¼R N 0 dmðxÞ ¼ c0:Hence, FnðzÞ ¼ A1 z x1 þ A2 z x2 þ ? þ An z xn ¼A1þ ? þ An z þ O 1 z2 ¼c0 z þ O 1 z2 :

The Oðz2Þ terms hold of course for z-N: Thus, since FmðzÞ ¼ c0z1þ Oðz2Þ;

which is an asymptotic expansion for z-N in the cut plane S ¼ C\½0; NÞ; we see that FmðzÞ FnðzÞ ¼ Oðz2Þ asymptotically for z-N in S: In this

sense FnðzÞ interpolates FmðzÞ twice at inﬁnity. Interpolating the value zero at

inﬁnity is implied by the fact that the rational interpolant Fn is of type ðn 1; nÞ:

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pointsfa1;a1; y;apðn1Þ;apðn1Þ;b1;b1; y;bqðn1Þ;bqðn1Þ;gng where

g_n¼ apðnÞ if pðnÞ4pðn 1Þ; bqðnÞ if qðnÞ4qðn 1Þ:

(

This gives an additional 2ðqðn 1Þ þ pðn 1ÞÞ þ 1 ¼ 2n 1 interpolation condi-tions. Since Fnis of typeðn 1; nÞ; we have these 2n 1 conditions plus 1 additional

interpolation condition at inﬁnity, giving exactly 2n interpolation conditions for the 2n parameters. In other words, FnðzÞ represents a multipoint Pade´ approximant

(MPA) to FmðzÞ at points of the tables a ¼ fang N

1 and *b¼ b,fNg ¼ fbkg N

1 ,fNg:

We have pðnÞ þ pðn 1Þ interpolation conditions for a and 2n ðpðnÞ þ pðn 1ÞÞ ¼ qðnÞ þ qðn 1Þ þ 1 for *b: For this reason and according to the notation used in the two-point situation (see e.g.[6]) i.e. a¼ f0g and *b ¼ fNg; we will sometimes write,

FnðzÞ ¼

snðzÞ

j_nðzÞ¼ ½pðnÞ þ pðn 1Þ=nFmðzÞ: ð5:10Þ

Remark 5.2. When taking b_n¼ N and pðnÞ ¼ 0 for each nX0; then we have seen that Ln¼ Pnso that jnðzÞ ¼ PnðzÞ coincides with the nth orthonormal polynomial

with respect to the measure dm: Then, (5.9) becomesðlnðzÞ ¼ 1Þ

FmðzÞ FnðzÞ ¼ 1 P2 nðzÞ Z N 0 P2_nðxÞ z xdmðxÞ ð5:11Þ

and the well-known formula for the errors in the one-point Pade´ approximant (at N) to F_mðzÞ is recovered.

On the other hand, if we take b_k¼ N and ak¼ 0 for any kX1; then

j_nðzÞ ¼PnðzÞ zpðnÞALpðnÞ;qðnÞ and (5.9) reduces to FmðzÞ FnðzÞ ¼ zpðnÞþpðn1Þ P2 nðzÞ Z N 0 P2 nðxÞ xpðnÞþpðn1Þ dmðxÞ ðz xÞ: ð5:12Þ

Thus, the error formula (see e.g. [3]) for the ½pðnÞ þ pðn 1Þ=n two-point Pade´ approximant to Fmis also recovered.

Connections between multipoint Pade´ approximants and Gauss-type quadrature formulas will be given in the following results. We ﬁrst have,

Theorem 5.1. Under the same assumptions as in Lemma 4.2, the following two statements are equivalent:

(a) lim_n-NInðf Þ ¼ Imðf Þ; 8f ACþN; where fInðf Þg is the sequence of Gauss-type

quadrature formulas for dm:

(b) limn-N½pðnÞ þ pðn 1Þ=nFmðzÞ ¼ FmðzÞ ¼

RN 0

dmðxÞ

zx uniformly on any compact

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Proof. (a)) (b). We know that ½pðnÞ þ pðn 1Þ=n_F_mðzÞ ¼ Inð1=ðz xÞÞ: Since for

ze½0; NÞ; 1=ðz xÞACþN;the pointwise convergence of the MPAis assured for any

zAC\½0; NÞ:

Let K be compact in C\½0; NÞ and take zAK: Then, j½pðnÞ þ pðn 1Þ=n_F_mðzÞj ¼ In 1 z x ¼ Xn j¼1 Aj z xj pX n j¼1 Aj jz xjjp c0 distðK; ½0; NÞÞ: Recall that c0is the zeroth moment c0¼R

N

0 dmðxÞ: Thus, we see that the sequence of

MPAis normal and the conclusion follows by applying the Stieltjes–Vitali Theorem [8, Theorem 1.5.3.1].

(b)) (a). Let fzkg be a sequence of complex numbers in C\½0; NÞ satisfying zjazk

if jak and X Im ffiffiffizn p 40 Im ffiffiffiffiffizn p 1þ jznj ¼ þN:

Then, it is known (see[1]) that spanf1; 1 z1x;

1

z2x; yg is dense in CþNwith respect to

the uniform norm. Thus, for f ACþNand for given e40; there exists an NAN such

that

jf ðxÞ SNðxÞjoe 8xA½0; NÞ;

where for certain values of a0; y; aN:

SNðxÞ ¼ a0þ a1 z1 x þ a2 z2 x þ ? þ aN zN x : Clearly, limn-NInðSNÞ ¼ ImðSNÞ: Therefore, it holds that

jImðSNÞ InðSNÞjoe 8n4n0:

Thus, by taking n4n0;one can write

jImðf Þ Inðf Þj ¼ jImðf Þ ImðSNÞ þ ImðSnÞ Inðf Þj

p jImðf SNÞj þ jImðSNÞ Inðf Þj

p c0eþ jImðSNÞ InðSNÞ þ InðSNÞ Inðf Þj

p c0eþ jImðSNÞ InðSNÞj þ jInðSN f Þj

o ð2c0þ 1Þe: &

Theorem 5.2. Let f be an analytic function in a simply connected domain G which contains the half line½0; N and whose boundary G is a rectifiable Jordan curve. Let

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Inðf Þ be the n-point Gauss-type formula for Ln Ln1: Then, Imðf Þ Inðf Þ ¼ 1 2pi Z G ½FmðzÞ ½pðnÞ þ pðn 1Þ=nFmðzÞf ðzÞ dz: ð5:13Þ

Proof. By Cauchy’s formula for the exterior of G in #C; we have that for xA½0; N: 1 2pi Z G fðzÞ z xdz¼ f ðxÞ f ðNÞ:

Now we can prove our theorem following standard arguments. Let m_n be the measure on½0; NÞ ¼ Rþ_{corresponding to the quadrature formula I}

n:By the Fubini theorem we have 1 2pi Z G fðzÞFmðzÞ dz ¼ 1 2pi Z G Z RþfðzÞ 1 z xdmðxÞ dz ¼ Z Rþ 1 2pi Z G fðzÞ z xdz dmðxÞ ¼ Z Rþðf ðxÞ f ðNÞÞ dmðxÞ ¼ Imðf Þ c0fðNÞ: Similarly, 1 2pi Z G fðzÞFnðzÞ dz ¼ Z Rþ ðf ðxÞ f ðNÞÞ dm_nðxÞ ¼ Inðf Þ c0fðNÞ: Hence, Enðf Þ ¼ Imðf Þ Inðf Þ ¼ 1 2pi Z G fðzÞ½FmðzÞ FnðzÞ dz: ð5:14Þ

Finally, observe that Fn¼ ½pðnÞ þ pðn 1Þ=nFm;and that this result also holds when

G is replaced by any rectiﬁable curve which is homotopic to G in G\½0; N: & Thus when considering analytic integrands, the errors for the Gauss-type formulas are ‘‘controlled’’ by the errors of the MPA. Indeed, by (5.9) and (5.14) it readily follows that the following is true.

Corollary 5.3. Under the same assumptions as in Theorem 5.1, it holds that jImðf Þ Inðf Þjp Cðf ; GÞ infzAGjlnðzÞj2nðzÞj Z N 0 j2_nðxÞlnðxÞ dmðxÞ; ð5:15Þ

Cðf ; GÞ being a positive constant depending on f ; G; and lnðxÞ as given by (2.5).

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