Moreover, we discuss stability of operator sequences belonging to algebras generated by the sequences of the collocation methods for the above described operators

COLLOCATION METHODS FOR CAUCHY SINGULAR INTEGRAL EQUATIONS ON THE INTERVAL^∗

P. JUNGHANNS^†ANDA. ROGOZHIN^‡

Abstract. In this paper we consider polynomial collocation methods for the numerical solution of a singular integral equation over the interval, where the operator of the equation is supposed to be of the formaI+bµ⁻¹SµI+ KwithSthe Cauchy singular integral operator, with piecewise continuous coefficientsaandb ,and with a Jacobi weightµ . Kdenotes an integral operator with a continuous kernel function. To the integral equation we apply two collocation methods, where the collocation points are the Chebyshev nodes of the first and second kind and where the trial space is the space of polynomials multiplied by another Jacobi weight. For the stability and convergence of this collocation scheme in weightedL²-spaces, we derive necessary and sufficient conditions. Moreover, we discuss stability of operator sequences belonging to algebras generated by the sequences of the collocation methods for the above described operators. Finally, the so-called splitting property of the singular values of the sequences of the matrices of the discretized equations is proved.

Key words. Cauchy singular integral equation, polynomial collocation method, stability, singular values, split- ting property.

AMS subject classifications. 45L10, 65R20, 65N38.

1. Introduction and preliminaries. The present paper can be considered as an im- mediate continuation of [7], where the stability of the collocation method with respect to Chebyshev nodes of second kind for Cauchy singular integral equations (CSIEs) is investi- gated. Here we purpose, firstly, to establish analogous results for collocation with respect to Chebyshev nodes of first kind (and to compare them with the results of [7]) and, secondly, to study the stability of operator sequences belonging to an algebra generated by the sequences of the collocation methods applied to Cauchy singular integral operators (CSIOs). Moreover, we will be able to prove results on the singular value distribution of the respective matrix sequences related to the collocation methods.

A functiona : [−1,1] −→ Cis called piecewise continuous if it has one-sided limits a(x ±0) for all x ∈ (−1,1) and is continuous at±1.For definiteness, we assume that the function values coincide with the limits from the left. The set of piecewise continuous functions on[−1,1]is denoted by PC.

We analyze polynomial collocation methods for CSIEs on the interval(−1,1)of the type

a(x)u(x) + b(x) µ(x)

1 πi

Z 1

−1

µ(y)u(y) y−x dy+

Z1

−1

k(x, y)u(y)dy=f(x), (1.1)

wherea, b: [−1,1]−→Cstand for given piecewise continuous functions, where the weight functionµis of the formµ(x) =v^γ,δ(x) := (1−x)^γ(1+x)^δwith real numbers−1< γ, δ <

1,where the kernelk: (−1,1)×(−1,1)−→Cis supposed to be continuous (comp. Lemma 2.10), where the right-hand side functionf is assumed to belong to a weightedL²-spaceL²_ν, and whereu∈L²_ν stands for the unknown solution. The Hilbert spaceL²_ν is defined as the space of all (classes of) functionsu: (−1,1)−→Cwhich are square integrable with respect

∗Received April 11, 2003. Accepted for publication September 23, 2003. Recommended by Sven Ehrich.

†Fakult¨at f ¨ur Mathematik, Technische Universit¨at Chemnitz, D-09107 Chemnitz, Germany. E-mail:

peter.junghanns@mathematik.tu-chemnitz.de

‡Fakult¨at f ¨ur Mathematik, Technische Universit¨at Chemnitz, D-09107 Chemnitz, Germany. E-mail:

rogozhin@mathematik.tu-chemnitz.de 11

to the weightν=v^α,β,−1< α, β <1.The inner product in this space is defined by hu, viν:=

Z 1

−1

u(x)v(x)ν(x)dx and the norm bykukν:=p

hu, uiν.In short operator notation (1.1) takes the form Au:= (aI+bµ⁻¹SµI+K)u=f.

(1.2) HereaI :L²

ν −→L²

ν denotes the multiplication operator defined by(au)(x) :=a(x)u(x), the operatorS:L²

ν −→L²

ν is the CSIO given by (Su)(x) := 1

πi Z 1

−1

u(y) y−x dy, and K : L²

ν −→ L²

ν stands for the integral operator with kernel k(x, y). Note that the condition−1< α, β < 1for the exponents of the classical Jacobi weightν(x)guarantees that the CSIOS:L²_ν−→L²_νis continuous, i.e.S∈ L(L²_ν)(see [3]).

Letσ(x) = (1−x²)⁻¹² andϕ(x) = (1−x²)¹² denote the Chebyshev weights of first and second kind, respectively. For the numerical solution of the CSIE (1.2), we consider the polynomial collocation method

a(x^τ_jn)un(x^τ_jn) + b(x^τ_jn) µ(x^τ_jn)

1 πi

Z 1

−1

µ(y)un(y) y−x^τ_jn dy+

Z1

−1

k(x^τ_jn, y)un(y)dy=f(x^τ_jn), j = 1, . . . , n ,where the collocation pointsx^τ_jnare chosen as the Chebyshev nodesx^σ_jn = cos^2j−1_2n πof first kind orx^ϕ_jn = cos_n+1^jπ of second kind and where the trial functionun is sought in the space of all functions un =ϑpn with a polynomialpn of degree less thann and with the Jacobi weightϑ=v^{14−α2,14−β2} .We write the above method in operator form as

Anun=Mnf , un∈imLn. (1.3)

HereLndenotes the orthogonal projection ofL²

νonto thendimensional trial spaceimLnof all polynomials of degree less thannmultiplied byϑ .ByMn=M_n^τwe denote the interpola- tion projection defined byMnf ∈imLnand(Mnf)(x^τ_jn) =f(x^τ_jn), j= 1, . . . , n .Finally, the discretized integral operatorAn : imLn −→ imLn is given byAn := MnA|imLn. In accordance with e.g. [11], we call the collocation method stable if the operatorsAn are invertible at least for all sufficiently largenand if the norms of the inverse operatorsA⁻¹_n are bounded uniformly with respect ton .Of course, the norm is the operator norm in the space imLn if the last is equipped with the restriction of theL²_ν-norm. We call the method (1.3) convergent if, for any right-hand sidef ∈ L²_ν and for any approximating sequence{fn}, fn ∈ imLn, withkf−fnkν −→ 0, the approximate solutionsun obtained by solving Anun=fnconverge to the exact solutionuof (1.2) in the norm ofL²_ν.Note that the stabil- ity implies bounded condition numbers for the matrix representation ofAn in a convenient basis, and, together with the consistency relationAnLn−→A ,it implies the convergence.

In all what follows, for the exponents in the weight functionsµandν ,we suppose

−1< α−2γ <1, −1< β−2δ <1, (1.4)

and

α0:=γ+1 4−α

2 6= 0, β0:=δ+1 4−β

2 6= 0. (1.5)

Note that condition (1.4) ensures the boundedness of the integral operator A ∈ L(L²_ν) whereas (1.5) is needed to derive strong limits for the discrete operators (see Lemma3.4).

In the subsequent analysis, we will show that there exist four limit operatorsWω{An}, ω = 1,2,3,4, introduced in the Lemmata 3.2–3.4. Moreover, we show that the map- pings{An} 7→ Wω{An}can be extended to *-homomorphismsWω : A⁰ −→ L(L²_ν), whereA⁰ denotes a C^∗-algebra of operator sequences including all sequences{Mn(aI + bµ⁻¹SµI)Ln}, a, b∈PC.The invertibility ofWω{An}, ω= 1,2,3,4,will turn out to be necessary and sufficient for the stability of{An} ∈ A⁰.

2. AC^∗-algebra of operator sequences and stability. In this section we will introduce one of theC^∗-algebras of operator sequences under consideration here. For n ≥ 0, let p^σ_n =Tnandp^ϕ_n=Unstand for the orthonormal polynomials of degreenwith respect to the weight functionsσandϕ ,respectively. That means that

T0(x) = 1

√π, Tn(coss) = r2

πcosns, n≥1, s∈(0, π), and

Un(coss) = r2

π

sin(n+ 1)s

sins , n≥0, s∈(0, π). We set

e

un(x) :=ϑ(x)Un(x), n= 0,1,2, . . . , withϑ=p

ν⁻¹ϕ=v^{14−α2,14−β2} .Then the solution of (1.3) can be represented by un(x) =

n−1X

k=0

ξkneuk(x),

and, with respect to the orthonormal system{eun}^∞n=0 inL²_ν,the orthogonal projectionLn

takes the form

Lnu=

n−1X

k=0

hu,uekiνuek.

The interpolation operatorMn=M_n^τcan be written asM_n^τ=ϑL^τ_nϑ⁻¹I ,whereL^τ_ndenotes the polynomial interpolation operator with respect to the nodesxjn=x^τ_jn, j = 1, . . . , n .By

`²we denote the Hilbert space of all square summable sequencesξ ={ξk}^∞k=0of complex numbers equipped with the inner product

hξ, ηi^`2 :=

X∞ k=0

ξkη_k.

Finally, we introduce the Christoffel numbers with respect to the weightsσandϕby λ^σ_kn:= π

n, λ^ϕ_kn:= π[ϕ(x^ϕ_kn)]²

n+ 1 , k= 1, . . . , n , and the discrete weights

ω^σ_kn:=

rπ

nv^{14+α2,14+β2}(x^σ_kn), ω_kn^ϕ :=

r π

n+ 1v^{14+α2,14+β2}(x^ϕ_kn), k= 1, . . . , n .

The proof of the approximation properties of the interpolation operatorsMnis based on the following auxiliary results.

LEMMA 2.1 ([10],Theorem 9.25). Let µ, ν be classical Jacobi weights with µν ∈ L¹(−1,1)and letj ∈Nbe fixed. Then for each polynomialqwithdegq≤jn ,

Xn k=1

λ^µ_kn|q(x^µ_kn)| |ν(x^µ_kn)| ≤const Z 1

−1|q(x)|µ(x)ν(x)dx, where the constant does not depend onnandqand wherex^µ_knand

λ^µ_kn= Z 1

−1

`<sup>µ</sup><sub>kn</sub>(x)µ(x)dx with `^µ_kn(x) = Y

j=1,j6=k

x−x^µ_jn x^µ_kn−x^µ_jn

are the nodes and the Christoffel numbers of the Gaussian rule with respect to the weightµ , respectively.

LetQ^µ_ndenote the Gaussian quadrature rule with respect to the weightµ , Q^µnf =

Xn k=1

λ^µ_knf(x^µ_kn),

and writeR=R(−1,1)for the set of all functionsf : (−1,1)−→C,which are bounded and Riemann integrable on each closed subinterval of(−1,1).

LEMMA 2.2 ([2], Satz III.1.6b and Satz III.2.1). Letµ(x) = (1−x)^γ(1 +x)^δ with γ, δ >−1.Iff ∈Rsatisfies

|f(x)| ≤const (1−x)^ε−1−γ(1 +x)^ε−1−δ, −1< x <1, for someε >0,then lim

n→∞Q^µ_nf = Z 1

−1

f(x)µ(x)dx .If even

|f(x)| ≤const (1−x)^ε−1+γ2 (1 +x)^ε−1+δ2 , −1< x <1, then lim

n→∞kf−L^µnfkµ = 0.

COROLLARY2.3. Letf ∈Rand, for someε >0,

|f(x)| ≤const (1−x)^ε−1+α2 (1 +x)^ε−1+β2 , −1< x <1. Then lim

n→∞kf−M_n^τfk^ν = 0forτ =σandτ =ϕ . Proof. Introduce the quadrature rule

Qnf = Z 1

−1

(L^σ_nf)(x)ϕ(x)dx= Xn k=1

σknf(x^σ_kn), where

σkn= Z 1

−1

`^σ_kn(x)ϕ(x)dx= Z 1

−1

`^σ_kn(x)(1−x²)σ(x)dx= π

n[ϕ(x^σ_kn)]² forn >2.Consequently,

Qnf = π n

Xn k=1

[ϕ(x^σ_kn)]²f(x^σ_kn).

Since the nodesx^σ_knof the quadrature ruleQnare the zeros of2Tn(x) =Un(x)−Un−2(x), the estimate

Z 1

−1|(L^σ_nf)(x)|²ϕ(x)dx≤2Qn|f|² (2.1)

holds true (see [2, Hilfssatz 2.4,§III.2]). As an immediate consequence we obtain kM_n^σfk²ν=kL^σ_nϑ⁻¹fk²ϕ≤ 2π

n Xn k=1

|ϑ⁻¹(x^σ_kn)ϕ(x^σ_kn)f(x^σ_kn)|²= 2Q^σ_n|ϑ⁻¹ϕf|². (2.2)

Now let >0be arbitrary andpbe a polynomial such thatkϑp−fk^ν < .Forn >degp we havekM_n^σf−fk²ν≤2

kM_n^σ(ϑp−f)k²ν+kϑp−fk²ν

.Since, in view of Lemma2.2,

n→∞lim Q^σ_n|ϑ⁻¹ϕ(ϑp−f)|²=kϑ⁻¹ϕ(ϑp−f)k²σ =kϑp−fk²ν,we get in view of (2.2) that lim sup

n→∞ kM_n^σf−fk²ν <6².

The proof for the caseτ =ϕis analogous (see also [2, Satz III.2.1]).

Now we start to prepare the definition of a certainC^∗-algebra of operator sequences, which is closely related to the above mentioned four limit operators defined as strong limits

Wω{An}:= lim

n→∞E_n^(ω)An(E_n^(ω))⁻¹L^(ω)_n , ω∈T :={1,2,3,4},

in some Hilbert spacesX_ω.Here,L^(ω)n :X_ω−→X_ωare projections andEn^(ω): imLn−→

imL^(ω)n are certain operators defined by X₁:=X₂:=L²

ν, X₃:=X₄:=`², L⁽¹⁾_n :=L⁽²⁾_n :=Ln, L⁽³⁾_n :=L⁽⁴⁾_n :=Pn, E_n⁽¹⁾:=Ln, E_n⁽²⁾:=Wn, E_n⁽³⁾=E_n,τ⁽³⁾ :=Vn =V_n^τ, E_n⁽⁴⁾=E_n,τ⁽⁴⁾ :=Ven=Ve_n^τ, and

Pn{ξ0, ξ1, ξ2, . . .}:={ξ0, . . . , ξn−1,0,0,0, . . .}, Wnu:=

n−1X

k=0

hu,eun−1−ki^νeuk,

V_n^τu:={ω_1n^τ u(x^τ_1n), . . . , ω^τ_nnu(x^τ_nn),0,0, . . .}, Ven^τu:={ωnn^τ u(x^τnn), . . . , ω^τ1nu(x^τ1n),0,0, . . .}. Immediately from the definitions, we conclude that

(E_n⁽¹⁾)⁻¹=Ln, (E_n⁽²⁾)⁻¹=Wn,

(E_n,τ⁽³⁾)⁻¹ξ= Xn k=1

ξk−1

ω_kn^τ e`^τ_kn, (E_n,τ⁽⁴⁾)⁻¹ξ= Xn k=1

ξn−k

ω^τ_kn e`^τ_kn, where

e`^τ_kn(x) := ϑ(x)

ϑ(x^τ_kn)`^τ_kn(x) = ϑ(x)p^τ_n(x)

ϑ(x^τ_kn)(x−x^τ_kn)(p^τ_n)⁰(x^τ_kn).

Between the operatorsVnandVen,we have the relations VenV_n⁻¹Pn =VnVe_n⁻¹Pn=WfnPn, (2.3)

whereWfn ∈ L(imPn)is defined by

Wfn{ξ0, ξ1, . . . , ξn−1}=Wf_n⁻¹{ξ0, ξ1, . . . , ξn−1}={ξn−1, ξn−2, . . . , ξ0}.

Furthermore, the operatorsEn,σ^(ω), ω∈ {1,2},andEn,ϕ^(ω), ω∈ {1,2,3,4},are unitary opera- tors, i.e.

(E_n,τ^(ω))^∗= (E^(ω)_n,τ)⁻¹. (2.4)

ForEn,σ^(ω), ω∈ {3,4},we have the following result.

LEMMA2.4. LetVn =V_n^σandVen=Ve_n^σ.Then (V_n⁻¹)^∗= 1

2Vn(Ln+Ln−1), (Ve_n⁻¹)^∗= 1

2Ven(Ln+Ln−1), and, consequently,

V_n^∗= ((V_n⁻¹)^∗)⁻¹= (2Ln−Ln−1)V_n⁻¹, Ve_n^∗= ((Ve_n⁻¹)^∗)⁻¹= (2Ln−Ln−1)Ve_n⁻¹.

Proof. For symmetry reasons, we may restrict our considerations to the operator(V_n⁻¹)^∗. Letj= 0,1, . . . , n−1.Then

V_n⁻¹ξ, u

ν =

* _n X

k=1

ξk−1

ω^σ_knϑ(x^σ_kn)ϑ`^σ_kn, ϑUj

+

ν

=

* _n X

k=1

ξk−1

ω_kn^σ ϑ(x^σ_kn)`^σ_kn, ϕ²Uj

+

σ

, and, forj= 0, . . . , n−2,we obtain

V_n⁻¹ξ,euj

ν= π n

Xn k=1

ξk−1[ϕ(x^σ_kn)]²

ω_kn^σ ϑ(x^σ_kn) Uj(x^σ_kn)

= Xn k=1

ξk−1ω^σ_knϑ(x^σ_kn)Uj(x^σ_kn) =hξ, Vnueji`2.

Forj=n−1,using the relation (1−x²)Un−1(x) = 1

2[γn−1Tn−1(x)−γn+1Tn+1(x)], (2.5)

whereγ0=√

2andγn = 1forn≥1,and the fact that

Tn+1(x^σkn) =−Tn−1(x^σkn), n >1, (2.6)

we get, forn >1,

V_n⁻¹ξ,uen−1

ν = 1 2

* _n X

k=1

ξk−1

ω_kn^σ ϑ(x^σ_kn)`^σ_kn, Tn−1−Tn+1

+

σ

= π 2n

Xn k=1

ξk−1

ω^σ_knϑ(x^σ_kn)Tn−1(x^σ_kn)

= π 2n

Xn k=1

ξk−1

ω^σ_knϑ(x^σ_kn)[ϕ(x^σ_kn)]²Un−1(x^σ_kn)

=1 2

Xn k=1

ξk−1ω^σ_knϑ(x^σ_kn)Un−1(x^σ_kn)

=1

2hξ, Vneun−1i`2 . LEMMA 2.5. The sequencesn

E^(ωn ¹⁾ En^(ω2)−1

L^(ωn²⁾

o

converge weakly to zero for all indicesω1, ω2∈Twithω16=ω2.

Proof. The proof for the caseτ =ϕone can find in [7, Lemma 2.1]. The caseτ =σcan be dealt with completely analogous after checking the uniform boundedness of the sequences {V_n^σ},{(V_n^σ)⁻¹},and{Ve_n^σ},{(Ve_n^σ)⁻¹}.But, this follows, by using Lemma2.1, relation (2.2), and the notationu=ϑpn∈imLn,from

kV_n^σuk²l² =π n

Xn k=1

ϕ²(x^σ_kn)|pn(x^σ_kn)|²

≤const Z 1

−1

ϑ(x)pn(x) ϑ(x)

2

[ϕ(x)]²σ(x)dx= constkuk²ν

and

(V_n^σ)⁻¹ξ²

ν =

Xn k=1

ξk−1

rn π

ϑ(x^σ_kn) ϕ(x^σ_kn)`e^σ_kn

2

ν

≤2Q^σ_n

Xn k=1

rn

πξk−1`e^σ_kn(x)

2

= 2 Xn k=1

|ξk−1|²= 2kξk²`².

Analogously we get the uniform boundedness of the sequencesn Ve_n^σo

andn

(Ve_n^σ)⁻¹o . COROLLARY2.6. The sequences

n

E^(ωn ¹⁾−∗

En^(ω2)∗

L^(ωn²⁾

o

converge weakly to zero for all indicesω1, ω2∈T withω16=ω2.

Of course, all constructions in what follows depend on the choice ofτ =σorτ =ϕ . Nevertheless, we will omit the subscriptτ if there is no possibility of misunderstandings.

ByF we denote the set of all sequences{An} = {An}^∞n=1 of linear operatorsAn : imLn −→ imLn,for which there exist operatorsWω{An} ∈ L(X_ω) such that, for all ω∈T ,

E_n^(ω)An(E_n^(ω))⁻¹L^(ω)_n −→Wω{An},

(2.7)

E_n^(ω)An(E_n^(ω))⁻¹L^(ω)_n ∗

−→Wω{An}^∗

holds inX_ωin the sense of strong convergence forn−→ ∞.If we define, forλ1, λ2∈C, λ1{An}+λ2{Bn}:={λ1An+λ2Bn},

{An}{Bn}:={AnBn}, {An}^∗:={A^∗_n}, and

k{An}kF:= supn

kAnLnkL(L2

ν):n= 1,2, . . .o ,

then it is not hard to see thatF becomes a Banach algebra with unit element{Ln}.From Lemma2.5and Corollary2.6we conclude

COROLLARY 2.7. For allω ∈ T and all compact operatorsTω ∈ K(X_ω),the se- quences{A^(ω)n } = n

(En^(ω))⁻¹L^(ω)n TωEn^(ω)

o

belong toF,and for ω1 6= ω2, we get the strong limits

E_n^(ω1)A^(ω_n²⁾(E_n^(ω1))⁻¹L^(ω_n¹⁾−→0,

E_n^(ω1)A^(ω_n²⁾(E_n^(ω1))⁻¹L^(ω_n ¹⁾∗

−→0.

COROLLARY 2.8. The algebraF is a C^∗-algebra and the mappings Wω : F −→

L(X_ω), ω∈T ,are *-homomorphisms.

Proof. Of course, the mappingsWω : F −→ L(X_ω), ω ∈ T ,are homomorphisms.

Hence, it suffices to show that the operator sequences{En^(ω)A^∗_n(En^(ω))⁻¹L^(ω)n }and the re- spective sequences of adjoint operators are strongly convergent for all sequences{An} ∈ F and thatWω{A^∗_n}= (Wω{An})^∗ , ω∈T .In case(En^(ω))⁻¹ = (En^(ω))^∗ this can be easily verified. Consequently, due to (2.4), it remains to consider the caseτ =σ , ω= 3,4.

For symmetry reasons we may restrict the proof to the caseτ =σ , ω= 3.Let{An} ∈ F.Using Lemma2.4, the relationLn−Ln−1=WnL1Wn,the compactness ofL1:L²

ν−→

L²

ν,and Corollary2.7, we get VnA^∗_nV_n⁻¹Pn

= 1 2

Vn(2Ln−WnL1Wn)An(Ln+WnL1Wn)V_n⁻¹Pn

∗

= Pn+V_n⁻¹WnL1WnV_n⁻¹Pn∗

VnAnV_n⁻¹Pn∗1

2 2Pn−V_n⁻¹WnL1WnVnPn∗

−→(W3{An})^∗.

The proof for the respective sequence{(VnA^∗_nV_n⁻¹Pn)^∗}is analogous.

Using Corollary2.7, we define the subsetJ ⊂ F,of all sequences of the form X4

ω=1

n(E_n^(ω))⁻¹L^(ω)_n TωE_n^(ω)o

+{Cn}

whereTω∈ K(X_ω)and where{Cn}is in the idealN ⊂ F of all sequences{Cn}tending to zero in norm, i.e. of all sequences withkCnLnkL(L²

ν)−→0.Now, the following theorem is crucial for our stability and convergence analysis.

THEOREM2.9 ([11], Theorem 10.33). The setJ forms a two-sided closed ideal ofF. A sequence{An} ∈ Fis stable if and only if the operatorsWω{An}:X_ω−→X_ω, ω∈T , are invertible and if the coset{An}+J is invertible inF/J .

Furthermore, we will need the auxiliary algebraF²of sequences{An}of linear opera- torsAn : imLn −→imLn,for which (2.7) holds true forω = 1,2.Moreover, we define

the subsetJ²⊂ F²of all sequences of the form X2

ω=1

n

(E_n^(ω))⁻¹L^(ω)_n TωE_n^(ω)o

+{Cn}

whereTω ∈ K(X_ω)and where{Cn}is in the idealN ⊂ F.Obviously, the setJ²forms a two-sided closed ideal ofF²,andF ⊂ F²,J²⊂ J.

In addition to the operator sequences corresponding to the collocation method applied to compact operators, the sequences of quadrature discretizations of integral operators with continuous kernels are contained inJ ,too. Indeed, we can formulate the following lemma.

LEMMA 2.10. Suppose the functionk(x, y)/ρ(y),whereρ= √νϕ =ϑ⁻¹ϕ ,is con- tinuous on[−1,1]×[−1,1]and thatK is the integral operator with kernelk(x, y).Then {MnKLn} ∈ J²⊂ J.Moreover, if the approximationsKn∈ L(imLn)are defined by

Kn= (E_n⁽³⁾)⁻¹ e

ω_n^τk(x^τ_j+1,n, x^τ_k+1,n)ρ(x^τ_j+1,n)ϑ(x^τ_k+1,n)n−1

j,k=0E_n⁽³⁾Ln,

whereωe_n^σ =π/nandeω_n^ϕ=π/(n+ 1),then the operator norm ofKn−LnK|^imLntends to zero and{Kn}is inJ².

Proof. Consider the caseτ =σ .Since Z 1

−1

`^σ_kn(y)ϕ(y)dy= Z 1

−1

`^σ_kn(y)ϕ²(y)σ(y)dy= π

n[ϕ(x^σ_kn)]², the operatorsKncan be written asM_n^σK_n,where

(K_nun)(x) = Z 1

−1

ϕ(y)L^σ_n[k(x,·)ϕ⁻¹un](y)dy .

Obviously, due to the Arzela-Ascoli theorem the operatorK : L²_ν →C[−1,1]is compact.

Hence, lim

n→∞kMnKLn −LnKLnkL(L2

ν) = 0 (see Corollary 2.3), and it is sufficient to show that lim

n→∞kK_nLn −KLnk^L(L2ν,C[−1,1]) = 0. To this end, we introduce operators e

K_n : imLn−→C[−1,1]by (Ke_nun)(x) =

Z 1

−1

ϕ(y)L^σ_n[k(x,·)ρ⁻¹](y)(ϑ⁻¹un)(y)dy . Due to the exactness of the Gaussian rule we have, forj = 0, . . . , n−2,

e

K_nuej=

L^σ_n[k(x,·)ρ⁻¹], ϕ²Uj

σ=

L^σ_n[k(x,·)ρ⁻¹Uj], ϕ²

σ =K_nuej, and, in view of relations (2.5), (2.6),

2Ke_nuen−1=

L^σ_n[k(x,·)ρ⁻¹],2ϕ²Un−1

σ

=

L^σ_n[k(x,·)ρ⁻¹], Tn−1−Tn+1

σ

=

L^σ_n[k(x,·)ρ⁻¹Un−1], ϕ²

σ

=K_nuen−1. Consequently,K_nLn=Ke_n(2Ln−Ln−1).

Now, we deal with lim

n→∞kKe_nLn−KLnk^L(L2ν,C[−1,1]).We take an arbitraryu ∈ L²_ν and getLnu= ϑpn,wherepn is a certain polynomial of degree less thann .By kn(x, y) we refer to the best uniform approximation tok(x, y)/ρ(y)in the space of polynomials with degree less thennin both variables. Using (2.1) we get, forx∈[−1,1],

|(Ke_nLnu−KLnu)(x)|

=

Z 1

−1

ϕ(y) L^σ_n[k(x,·)ρ⁻¹](y)−k(x, y)/ρ(y)

pn(y)dy

≤

Z 1

−1

ϕ(y)L^σn[k(x,·)ρ⁻¹−kn(x, .)](y)pn(y)dy +

Z 1

−1

ϕ(y)[k(x, y)/ρ(y)−kn(x, y)](y)pn(y)dy

≤ Z 1

−1

L^σ_n[k(x,·)ρ⁻¹−kn(x, y)](y)

2

ϕ(y)dy ^1/2

kpnk^ϕ +

Z 1

−1

k(x, y)/ρ(y)−kn(x, y)

2

ϕ(y)dy ^1/2

kpnk^ϕ

≤ 2π n

Xn k=1

|k(x, x^σ_kn)/ρ(x^σ_kn)−kn(x, x^σ_kn)|²[ϕ(x^σ_kn)]²

!1/2

kLnuk^ν

+kk(·,·)ρ⁻¹−kn(·,·)k^∞k1k^ϕkLnuk^ν

≤3kk(·,·)ρ⁻¹−kn(·,·)k^∞k1k^ϕkLnuk^ν. Thus, since lim

n→∞kk(·,·)ρ⁻¹−kn(·,·)k^∞= 0,we obtain

n→∞lim kK_nLn−KLnk^L(L2ν,C[−1,1])

≤ lim

n→∞kKe_nLn−KLnkL(L2

ν,C[−1,1])k2Ln−Ln−1kL(L2 ν)

+ lim

n→∞kK(Ln−Ln−1)k^L(L2ν,C[−1,1])= 0.

The proof in case ofτ =ϕis similar and can be found in the proof of [7, Lemma 2.4].

3. The operator sequence of the collocation method. We will show that the sequence {MnAPn}corresponding to the singular integral operatorA ∈ L(L²_ν)(cf. (1.2) belongs to the algebraF,and we will computeWω{An}, ω∈ T .We do this separately for multipli- cation operators, for the singular integral operatorµ⁻¹Sµwith a special weightµ=ρ(see Lemma2.10), and forµ⁻¹Sµwith a generalµ .

We will use the well-known relations between the Chebyshev polynomials of first and second kind

SϕUn=iTn+1, Sϕ⁻¹Tn=−iUn−1, n= 0,1,2, . . . , U−1≡0, (3.1)

and

Tn+1= 1

2(Un+1−Un−1), n= 0,1,2, . . . , U−1≡0. (3.2)

Furthermore, for the description of the occuring strong limits we need the operators Jν∈ L(L²_ν,L²_σ), u7→

X∞ n=0

γnhu,euniνTn, (3.3)

J_ν⁻¹∈ L(L²_σ,L²_ν), u7→

X∞ n=0

1

γnhu, Tniσeun, (3.4)

V ∈ L(L²_ν), u7→

X∞ n=0

hu,euniνeun+1, (3.5)

withγnas in (2.5), and their adjoint operators J_ν^∗∈ L(L²_σ,L²_ν), u7→

X∞ n=0

γnhu, Tniσeun,

J_ν^−∗ ∈ L(L²_ν,L²_σ), u7→

X∞ n=0

1

γnhu,euniνTn, V^∗∈ L(L²_ν), u7→

X∞ n=0

hu,uen+1iνuen.

Finally, we will use the following special case of Lebesgue’s dominated convergence theorem.

REMARK3.1. Ifξ, η∈`², ξⁿ ={ξ_kⁿ},|ξ_kⁿ| ≤ |ηk|for alln > n0,and if lim

n→∞ξⁿ_k =ξk

for allk= 0,1,2, ...,then lim

n→∞kξⁿ−ξk`² = 0.

LEMMA 3.2. Leta ∈PC, A=aI , An =MnaLn.Then{An} ∈ F.In particular, W1{An}=A , W3{An}=a(1)I , W4{An}=a(−1)I ,and

W2{An}=





J_ν⁻¹aJν , τ =σ , aI =A , τ=ϕ .

Proof. The proof in case ofτ =ϕis given in [7, Lemma 3.8], and the proof in case of τ =σ is very similar. Thus, here we only pay attention to the proof of the convergence of (M_n^σaLn)^∗and ofWnM_n^σaWn.

We writeM_n^σf =

n−1X

j=0

α^σ_jn(f)uejand get, forj= 0,1, . . . , n−2,

α^σ_jn(f) =hM_n^σf,eujiν=hL^σ_nϑ⁻¹f, ϕ²Ujiσ

= π n

Xn k=1

f(x^σ_kn)

ϑ(x^σ_kn)[ϕ(x^σ_kn)]²Uj(x^σ_kn)

= π n

Xn k=1

f(x^σ_kn)ν(x^σ_kn)ϕ(x^σ_kn)uej(x^σ_kn). Forj=n−1, n≥2,we use relations (2.5) and (2.6) to obtain

α^σ_n−1,n(f) =hM_n^σf,eun−1iν =hL^σ_nϑ⁻¹f, ϕ²Un−1iσ

= 1

2hL^σ_nϑ⁻¹f, Tn−1iσ

= π 2n

Xn k=1

f(x^σ_kn)

ϑ(x^σ_kn)Tn−1(x^σ_kn)

= π 2n

Xn k=1

f(x^σ_kn)

ϑ(x^σ_kn)[ϕ(x^σ_kn)]²Un−1(x^σ_kn)

= π 2n

Xn k=1

f(x^σ_kn)ν(x^σ_kn)ϕ(x^σ_kn)uen−1(x^σ_kn).

Hence,

α^σ_jn(f) =εjn

π n

Xn k=1

f(x^σ_kn)ν(x^σ_kn)ϕ(x^σ_kn)euj(x^σ_kn), (3.6)

whereεjn = 1forj = 0,1, . . . , n−2andεn−1,n= 1/2.As an immediate consequence of (3.6) we obtain, foru, v ∈L²_ν,

hM_n^σaLnu, viν=

n−1X

j=0

hv,uejiν n−1X

l=0

hu,ueliνhM_n^σauel,uejiν

=

n−1X

j=0

εjn

π n

Xn k=1

a(x^σ_kn)

n−1X

l=0

hu,ueliνeul(x^σ_kn)ν(x^σ_kn)ϕ(x^σ_kn)euj(x^σ_kn)hv,eujiν

=

n−1X

l=0

π n

Xn k=1

a(x^σ_kn)

n−1X

j=0

εjnhv,uejiνeuj(x^σ_kn)ν(x^σ_kn)ϕ(x^σ_kn)uel(x^σ_kn)hu,ueliν

=1

2hu,(2Ln−Ln−1)M_n^σa(Ln+Ln−1)viν. Thus,

(M_n^σaLn)^∗=1

2(2Ln−Ln−1)M_n^σa(Ln+Ln−1), (3.7)

whence we have the strong convergence of(Mn^σaLn)^∗toaI inL²_ν.

We verify the convergence ofWnM_n^σaWneumfor each fixedm≥0.Letn > m .With the help of (3.6), the identity

e

un−1−m(x^σ_kn) = ϑ(x^σ_kn)

ϕ(x^σ_kn)ϕ(x^σ_kn)Un−1−m(x^σ_kn)

= 1

ρ(x^σ_kn) r2

πsin(n−m)(2k−1)π (3.8) 2n

=(−1)^k+1

ρ(x^σ_kn) γmTm(x^σ_kn),

and the formula for the Fourier coefficients of the interpolating polynomialL^σ_nf , L^σ_nf =

n−1X

j=0

e

α^σ_jn(f)Tj with αe^σ_jn(f) =π n

Xn k=1

f(x^σ_kn)Tj(x^σ_kn), (3.9)

we get, using Lemma2.2, WnM_n^σaWnuem=

n−1X

j=0

α^σ_n−1−j,n(aeun−1−m)euj

=

n−1X

j=0

εn−1−j,n

π n

Xn k=1

a(x^σ_kn)eun−1−m(x^σ_kn)ν(x^σ_kn)ϕ(x^σ_kn)eun−1−j(x^σ_kn)uej

=

n−1X

j=0

εn−1−j,n

π n

Xn k=1

a(x^σ_kn)γmTm(x^σ_kn)γjTj(x^σ_kn)euj

=

n−1X

j=0

π n

Xn k=1

a(x^σ_kn)(Jνeum)(x^σ_knTj(x^σ_kn)J_ν⁻¹Tj

=J_ν⁻¹L^σ_naJνeum−→J_ν⁻¹aJνuem in L²_ν. Thus,

WnM_n^σaWn =J_ν⁻¹L^σ_naJνLn −→J_ν⁻¹aJν in L²

ν. (3.10)

LEMMA3.3. SupposeA =ρ⁻¹SρI ,whereρ=ϑ⁻¹ϕ=√νϕ ,andAn =MnALn. Then{An} ∈ Fand

W1{An}=A , W2{An}=





iJ_ν⁻¹ρV^∗ , τ =σ ,

−A , τ=ϕ , andW3/4{An}=±Swith

S=

















1−(−1)^j−k

πi(j−k) −1−(−1)^j+k+1 πi(j+k+ 1)

∞ j,k=0

, τ =σ ,

2(k+ 1)

1−(−1)^j−k πi[(j+ 1)²−(k+ 1)²]

!∞ j,k=0

, τ=ϕ .

Proof. The caseτ =ϕis considered in [7, Lemma 3.9]. Thus, let us consider the case τ =σ .

From (3.1) it follows thatSρun is a polynomial of degree not greater thannif un ∈ imLn.Hence, applying (2.2), Lemma2.1, and the boundedness of the operatorS:L²

σ−→

L²

σ,we obtain, forun ∈imLn,

kM_n^σρ⁻¹Sρunk²ν ≤2Q^σ_n|Sρun|²

≤const Z 1

−1|(Sρun)(x)|²σ(x)dx

≤constkρunk²σ= constkunk²ν,

which shows the uniform boundedness of{An}.Again with the help of (3.1) as well as with the help of Corollary2.3we see that, forn > m ,

M_n^σρ⁻¹Sρuem=iM_n^σρ⁻¹Tm+1→iρ⁻¹Tm+1=ρ⁻¹Sρuem in L²_ν. Whence, the strong convergence of{An}toAis proved.

The well-known Poincar`e-Bertrand commutation formula implies that, foru∈L²_ν and v∈L²

ν⁻¹,

hSu, vi=hu, Svi,

whereh., .idenotes theL²(−1,1)inner product without weight. Consequently, the adjoint operator ofS :L²

ν −→L²

νis equal toν⁻¹Sν :L²

ν −→L²

ν.Again, taking into account that SρLnuis a polynomial with a degree≤n(cf. (3.1), we get, forj= 0, ..., n−2andu∈L²_ν,

hM_n^σρ⁻¹SρLnu,eujiν =hL^σ_nϕ⁻¹SρLnu, ϕ²Ujiσ

=π n

Xn k=1

(SρLnu)(x^σ_kn)ϕ(x^σ_kn)Uj(x^σ_kn) =hSρLnu, L^σ_nϕUjiσ

=hSρLnu, σν⁻¹L^σ_nϕUjiν =hρLnu, ν⁻¹SσL^σ_nϕUjiν

=hu, LnϑSσL^σ_nρeujiν

and, using relations (2.5) and (2.6),

hM_n^σρ⁻¹SρLnu,uen−1iν=hL^σ_nϕ⁻¹SρLnu, ϕ²Un−1iσ

= π 2n

Xn k=1

(SρLnu)(x^σ_kn)

ϕ(x^σ_kn) Tn−1(x^σ_kn)

= 1

2hSρLnu, L^σ_nϕUn−1iσ

= 1

2hu, LnϑSσL^σ_nρeun−1iν. Hence, in view of (3.1)

(M_n^σρ⁻¹SρLn)^∗=1

2LnϑSσL^σ_nρ(Ln+Ln−1) =1

2ϑSσL^σ_nρ(Ln+Ln−1). Using Lemma2.2, we obtain the strong convergence of(M_n^σρ⁻¹SρLn)^∗toϑSϑ⁻¹I .

In view of (3.1), (3.2), (3.10), (2.5), and Lemma2.2, we have, forn > m+ 1, WnM_n^σρ⁻¹SρWneum=WnM_n^σρ⁻¹Sρeun−1−m

=iWnM_n^σρ⁻¹Tn−m

= i

2WnM_n^σρ⁻¹ϑ⁻¹(eun−m−eun−m−2)

=−i

2WnM_n^σϕ⁻¹Wn(eum+1−uem−1)

=−i

2J_ν⁻¹L^σ_nϕ⁻¹Jν(eum+1−uem−1)