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) =vγ,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-spaceL2ν, and whereu∈L2ν stands for the unknown solution. The Hilbert spaceL2ν 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=vα,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:L2ν−→L2νis continuous, i.e.S∈ L(L2ν)(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(xτjn)un(xτjn) + b(xτjn) µ(xτjn)
1 πi
Z 1
−1
µ(y)un(y) y−xτjn dy+
Z1
−1
k(xτjn, y)un(y)dy=f(xτjn), j = 1, . . . , n ,where the collocation pointsxτjnare chosen as the Chebyshev nodesxσjn = cos2j−12n πof first kind orxϕjn = cosn+1jπ 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=v14−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=Mnτwe denote the interpola- tion projection defined byMnf ∈imLnand(Mnf)(xτjn) =f(xτjn), 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−1n 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 theL2ν-norm. We call the method (1.3) convergent if, for any right-hand sidef ∈ L2ν 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 ofL2ν.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(L2ν) 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(L2ν), 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 pσn =Tnandpϕn=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=v14−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 inL2ν,the orthogonal projectionLn
takes the form
Lnu=
n−1X
k=0
hu,uekiνuek.
The interpolation operatorMn=Mnτcan be written asMnτ=ϑLτnϑ−1I ,whereLτndenotes the polynomial interpolation operator with respect to the nodesxjn=xτjn, 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
ξkηk.
Finally, we introduce the Christoffel numbers with respect to the weightsσandϕby λσkn:= π
n, λϕkn:= π[ϕ(xϕkn)]2
n+ 1 , k= 1, . . . , n , and the discrete weights
ωσkn:=
rπ
nv14+α2,14+β2(xσkn), ωknϕ :=
r π
n+ 1v14+α2,14+β2(xϕkn), 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
λµkn|q(xµkn)| |ν(xµkn)| ≤const Z 1
−1|q(x)|µ(x)ν(x)dx, where the constant does not depend onnandqand wherexµknand
λµkn= Z 1
−1
`µkn(x)µ(x)dx with `µkn(x) = Y
j=1,j6=k
x−xµjn xµkn−xµjn
are the nodes and the Christoffel numbers of the Gaussian rule with respect to the weightµ , respectively.
LetQµndenote the Gaussian quadrature rule with respect to the weightµ , Qµnf =
Xn k=1
λµknf(xµkn),
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→∞Qµnf = 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−Lµnfkµ = 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−Mnτfkν = 0forτ =σandτ =ϕ . Proof. Introduce the quadrature rule
Qnf = Z 1
−1
(Lσnf)(x)ϕ(x)dx= Xn k=1
σknf(xσkn), where
σkn= Z 1
−1
`σkn(x)ϕ(x)dx= Z 1
−1
`σkn(x)(1−x2)σ(x)dx= π
n[ϕ(xσkn)]2 forn >2.Consequently,
Qnf = π n
Xn k=1
[ϕ(xσkn)]2f(xσkn).
Since the nodesxσknof the quadrature ruleQnare the zeros of2Tn(x) =Un(x)−Un−2(x), the estimate
Z 1
−1|(Lσnf)(x)|2ϕ(x)dx≤2Qn|f|2 (2.1)
holds true (see [2, Hilfssatz 2.4,§III.2]). As an immediate consequence we obtain kMnσfk2ν=kLσnϑ−1fk2ϕ≤ 2π
n Xn k=1
|ϑ−1(xσkn)ϕ(xσkn)f(xσkn)|2= 2Qσn|ϑ−1ϕf|2. (2.2)
Now let >0be arbitrary andpbe a polynomial such thatkϑp−fkν < .Forn >degp we havekMnσf−fk2ν≤2
kMnσ(ϑp−f)k2ν+kϑp−fk2ν
.Since, in view of Lemma2.2,
n→∞lim Qσn|ϑ−1ϕ(ϑp−f)|2=kϑ−1ϕ(ϑp−f)k2σ =kϑp−fk2ν,we get in view of (2.2) that lim sup
n→∞ kMnσf−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→∞En(ω)An(En(ω))−1L(ω)n , ω∈T :={1,2,3,4},
in some Hilbert spacesXω.Here,L(ω)n :Xω−→Xωare 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, En(1):=Ln, En(2):=Wn, En(3)=En,τ(3) :=Vn =Vnτ, En(4)=En,τ(4) :=Ven=Venτ, and
Pn{ξ0, ξ1, ξ2, . . .}:={ξ0, . . . , ξn−1,0,0,0, . . .}, Wnu:=
n−1X
k=0
hu,eun−1−kiνeuk,
Vnτu:={ω1nτ u(xτ1n), . . . , ωτnnu(xτnn),0,0, . . .}, Venτu:={ωnnτ u(xτnn), . . . , ωτ1nu(xτ1n),0,0, . . .}. Immediately from the definitions, we conclude that
(En(1))−1=Ln, (En(2))−1=Wn,
(En,τ(3))−1ξ= Xn k=1
ξk−1
ωknτ e`τkn, (En,τ(4))−1ξ= Xn k=1
ξn−k
ωτkn e`τkn, where
e`τkn(x) := ϑ(x)
ϑ(xτkn)`τkn(x) = ϑ(x)pτn(x)
ϑ(xτkn)(x−xτkn)(pτn)0(xτkn).
Between the operatorsVnandVen,we have the relations VenVn−1Pn =VnVen−1Pn=WfnPn, (2.3)
whereWfn ∈ L(imPn)is defined by
Wfn{ξ0, ξ1, . . . , ξn−1}=Wfn−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.
(En,τ(ω))∗= (E(ω)n,τ)−1. (2.4)
ForEn,σ(ω), ω∈ {3,4},we have the following result.
LEMMA2.4. LetVn =VnσandVen=Venσ.Then (Vn−1)∗= 1
2Vn(Ln+Ln−1), (Ven−1)∗= 1
2Ven(Ln+Ln−1), and, consequently,
Vn∗= ((Vn−1)∗)−1= (2Ln−Ln−1)Vn−1, Ven∗= ((Ven−1)∗)−1= (2Ln−Ln−1)Ven−1.
Proof. For symmetry reasons, we may restrict our considerations to the operator(Vn−1)∗. Letj= 0,1, . . . , n−1.Then
Vn−1ξ, u
ν =
* n X
k=1
ξk−1
ωσknϑ(xσkn)ϑ`σkn, ϑUj
+
ν
=
* n X
k=1
ξk−1
ωknσ ϑ(xσkn)`σkn, ϕ2Uj
+
σ
, and, forj= 0, . . . , n−2,we obtain
Vn−1ξ,euj
ν= π n
Xn k=1
ξk−1[ϕ(xσkn)]2
ωknσ ϑ(xσkn) Uj(xσkn)
= Xn k=1
ξk−1ωσknϑ(xσkn)Uj(xσkn) =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(xσkn) =−Tn−1(xσkn), n >1, (2.6)
we get, forn >1,
Vn−1ξ,uen−1
ν = 1 2
* n X
k=1
ξk−1
ωknσ ϑ(xσkn)`σkn, Tn−1−Tn+1
+
σ
= π 2n
Xn k=1
ξk−1
ωσknϑ(xσkn)Tn−1(xσkn)
= π 2n
Xn k=1
ξk−1
ωσknϑ(xσkn)[ϕ(xσkn)]2Un−1(xσkn)
=1 2
Xn k=1
ξk−1ωσknϑ(xσkn)Un−1(xσkn)
=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 {Vnσ},{(Vnσ)−1},and{Venσ},{(Venσ)−1}.But, this follows, by using Lemma2.1, relation (2.2), and the notationu=ϑpn∈imLn,from
kVnσuk2l2 =π n
Xn k=1
ϕ2(xσkn)|pn(xσkn)|2
≤const Z 1
−1
ϑ(x)pn(x) ϑ(x)
2
[ϕ(x)]2σ(x)dx= constkuk2ν
and
(Vnσ)−1ξ2
ν =
Xn k=1
ξk−1
rn π
ϑ(xσkn) ϕ(xσkn)`eσkn
2
ν
≤2Qσn
Xn k=1
rn
πξk−1`eσkn(x)
2
= 2 Xn k=1
|ξk−1|2= 2kξk2`2.
Analogously we get the uniform boundedness of the sequencesn Venσo
andn
(Venσ)−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(Xω) such that, for all ω∈T ,
En(ω)An(En(ω))−1L(ω)n −→Wω{An},
(2.7)
En(ω)An(En(ω))−1L(ω)n ∗
−→Wω{An}∗
holds inXωin 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(Xω),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
En(ω1)A(ωn2)(En(ω1))−1L(ωn1)−→0,
En(ω1)A(ωn2)(En(ω1))−1L(ωn 1)∗
−→0.
COROLLARY 2.8. The algebraF is a C∗-algebra and the mappings Wω : F −→
L(Xω), ω∈T ,are *-homomorphisms.
Proof. Of course, the mappingsWω : F −→ L(Xω), 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∗nVn−1Pn
= 1 2
Vn(2Ln−WnL1Wn)An(Ln+WnL1Wn)Vn−1Pn
∗
= Pn+Vn−1WnL1WnVn−1Pn∗
VnAnVn−1Pn∗1
2 2Pn−Vn−1WnL1WnVnPn∗
−→(W3{An})∗.
The proof for the respective sequence{(VnA∗nVn−1Pn)∗}is analogous.
Using Corollary2.7, we define the subsetJ ⊂ F,of all sequences of the form X4
ω=1
n(En(ω))−1L(ω)n TωEn(ω)o
+{Cn}
whereTω∈ K(Xω)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}:Xω−→Xω, 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
(En(ω))−1L(ω)n TωEn(ω)o
+{Cn}
whereTω ∈ K(Xω)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= (En(3))−1 e
ωnτk(xτj+1,n, xτk+1,n)ρ(xτj+1,n)ϑ(xτk+1,n)n−1
j,k=0En(3)Ln,
whereωenσ =π/nandeωnϕ=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
`σkn(y)ϕ(y)dy= Z 1
−1
`σkn(y)ϕ2(y)σ(y)dy= π
n[ϕ(xσkn)]2, the operatorsKncan be written asMnσKn,where
(Knun)(x) = Z 1
−1
ϕ(y)Lσn[k(x,·)ϕ−1un](y)dy .
Obviously, due to the Arzela-Ascoli theorem the operatorK : L2ν →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→∞kKnLn −KLnkL(L2ν,C[−1,1]) = 0. To this end, we introduce operators e
Kn : imLn−→C[−1,1]by (Kenun)(x) =
Z 1
−1
ϕ(y)Lσn[k(x,·)ρ−1](y)(ϑ−1un)(y)dy . Due to the exactness of the Gaussian rule we have, forj = 0, . . . , n−2,
e
Knuej=
Lσn[k(x,·)ρ−1], ϕ2Uj
σ=
Lσn[k(x,·)ρ−1Uj], ϕ2
σ =Knuej, and, in view of relations (2.5), (2.6),
2Kenuen−1=
Lσn[k(x,·)ρ−1],2ϕ2Un−1
σ
=
Lσn[k(x,·)ρ−1], Tn−1−Tn+1
σ
=
Lσn[k(x,·)ρ−1Un−1], ϕ2
σ
=Knuen−1. Consequently,KnLn=Ken(2Ln−Ln−1).
Now, we deal with lim
n→∞kKenLn−KLnkL(L2ν,C[−1,1]).We take an arbitraryu ∈ L2ν 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],
|(KenLnu−KLnu)(x)|
=
Z 1
−1
ϕ(y) Lσn[k(x,·)ρ−1](y)−k(x, y)/ρ(y)
pn(y)dy
≤
Z 1
−1
ϕ(y)Lσn[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
Lσn[k(x,·)ρ−1−kn(x, y)](y)
2
ϕ(y)dy 1/2
kpnkϕ +
Z 1
−1
k(x, y)/ρ(y)−kn(x, y)
2
ϕ(y)dy 1/2
kpnkϕ
≤ 2π n
Xn k=1
|k(x, xσkn)/ρ(xσkn)−kn(x, xσkn)|2[ϕ(xσkn)]2
!1/2
kLnukν
+kk(·,·)ρ−1−kn(·,·)k∞k1kϕkLnukν
≤3kk(·,·)ρ−1−kn(·,·)k∞k1kϕkLnukν. Thus, since lim
n→∞kk(·,·)ρ−1−kn(·,·)k∞= 0,we obtain
n→∞lim kKnLn−KLnkL(L2ν,C[−1,1])
≤ lim
n→∞kKenLn−KLnkL(L2
ν,C[−1,1])k2Ln−Ln−1kL(L2 ν)
+ lim
n→∞kK(Ln−Ln−1)kL(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(L2ν)(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(L2ν,L2σ), u7→
X∞ n=0
γnhu,euniνTn, (3.3)
Jν−1∈ L(L2σ,L2ν), u7→
X∞ n=0
1
γnhu, Tniσeun, (3.4)
V ∈ L(L2ν), u7→
X∞ n=0
hu,euniνeun+1, (3.5)
withγnas in (2.5), and their adjoint operators Jν∗∈ L(L2σ,L2ν), u7→
X∞ n=0
γnhu, Tniσeun,
Jν−∗ ∈ L(L2ν,L2σ), u7→
X∞ n=0
1
γnhu,euniνTn, V∗∈ L(L2ν), 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 ={ξkn},|ξkn| ≤ |ηk|for alln > n0,and if lim
n→∞ξnk =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}=
Jν−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 (MnσaLn)∗and ofWnMnσaWn.
We writeMnσf =
n−1X
j=0
ασjn(f)uejand get, forj= 0,1, . . . , n−2,
ασjn(f) =hMnσf,eujiν=hLσnϑ−1f, ϕ2Ujiσ
= π n
Xn k=1
f(xσkn)
ϑ(xσkn)[ϕ(xσkn)]2Uj(xσkn)
= π n
Xn k=1
f(xσkn)ν(xσkn)ϕ(xσkn)uej(xσkn). Forj=n−1, n≥2,we use relations (2.5) and (2.6) to obtain
ασn−1,n(f) =hMnσf,eun−1iν =hLσnϑ−1f, ϕ2Un−1iσ
= 1
2hLσnϑ−1f, Tn−1iσ
= π 2n
Xn k=1
f(xσkn)
ϑ(xσkn)Tn−1(xσkn)
= π 2n
Xn k=1
f(xσkn)
ϑ(xσkn)[ϕ(xσkn)]2Un−1(xσkn)
= π 2n
Xn k=1
f(xσkn)ν(xσkn)ϕ(xσkn)uen−1(xσkn).
Hence,
ασjn(f) =εjn
π n
Xn k=1
f(xσkn)ν(xσkn)ϕ(xσkn)euj(xσkn), (3.6)
whereεjn = 1forj = 0,1, . . . , n−2andεn−1,n= 1/2.As an immediate consequence of (3.6) we obtain, foru, v ∈L2ν,
hMnσaLnu, viν=
n−1X
j=0
hv,uejiν n−1X
l=0
hu,ueliνhMnσauel,uejiν
=
n−1X
j=0
εjn
π n
Xn k=1
a(xσkn)
n−1X
l=0
hu,ueliνeul(xσkn)ν(xσkn)ϕ(xσkn)euj(xσkn)hv,eujiν
=
n−1X
l=0
π n
Xn k=1
a(xσkn)
n−1X
j=0
εjnhv,uejiνeuj(xσkn)ν(xσkn)ϕ(xσkn)uel(xσkn)hu,ueliν
=1
2hu,(2Ln−Ln−1)Mnσa(Ln+Ln−1)viν. Thus,
(MnσaLn)∗=1
2(2Ln−Ln−1)Mnσa(Ln+Ln−1), (3.7)
whence we have the strong convergence of(MnσaLn)∗toaI inL2ν.
We verify the convergence ofWnMnσaWneumfor each fixedm≥0.Letn > m .With the help of (3.6), the identity
e
un−1−m(xσkn) = ϑ(xσkn)
ϕ(xσkn)ϕ(xσkn)Un−1−m(xσkn)
= 1
ρ(xσkn) r2
πsin(n−m)(2k−1)π (3.8) 2n
=(−1)k+1
ρ(xσkn) γmTm(xσkn),
and the formula for the Fourier coefficients of the interpolating polynomialLσnf , Lσnf =
n−1X
j=0
e
ασjn(f)Tj with αeσjn(f) =π n
Xn k=1
f(xσkn)Tj(xσkn), (3.9)
we get, using Lemma2.2, WnMnσaWnuem=
n−1X
j=0
ασn−1−j,n(aeun−1−m)euj
=
n−1X
j=0
εn−1−j,n
π n
Xn k=1
a(xσkn)eun−1−m(xσkn)ν(xσkn)ϕ(xσkn)eun−1−j(xσkn)uej
=
n−1X
j=0
εn−1−j,n
π n
Xn k=1
a(xσkn)γmTm(xσkn)γjTj(xσkn)euj
=
n−1X
j=0
π n
Xn k=1
a(xσkn)(Jνeum)(xσknTj(xσkn)Jν−1Tj
=Jν−1LσnaJνeum−→Jν−1aJνuem in L2ν. Thus,
WnMnσaWn =Jν−1LσnaJνLn −→Jν−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}=
iJν−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,
kMnσρ−1Sρunk2ν ≤2Qσn|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 ,
Mnσρ−1Sρuem=iMnσρ−1Tm+1→iρ−1Tm+1=ρ−1Sρuem in L2ν. Whence, the strong convergence of{An}toAis proved.
The well-known Poincar`e-Bertrand commutation formula implies that, foru∈L2ν 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∈L2ν,
hMnσρ−1SρLnu,eujiν =hLσnϕ−1SρLnu, ϕ2Ujiσ
=π n
Xn k=1
(SρLnu)(xσkn)ϕ(xσkn)Uj(xσkn) =hSρLnu, LσnϕUjiσ
=hSρLnu, σν−1LσnϕUjiν =hρLnu, ν−1SσLσnϕUjiν
=hu, LnϑSσLσnρeujiν
and, using relations (2.5) and (2.6),
hMnσρ−1SρLnu,uen−1iν=hLσnϕ−1SρLnu, ϕ2Un−1iσ
= π 2n
Xn k=1
(SρLnu)(xσkn)
ϕ(xσkn) Tn−1(xσkn)
= 1
2hSρLnu, LσnϕUn−1iσ
= 1
2hu, LnϑSσLσnρeun−1iν. Hence, in view of (3.1)
(Mnσρ−1SρLn)∗=1
2LnϑSσLσnρ(Ln+Ln−1) =1
2ϑSσLσnρ(Ln+Ln−1). Using Lemma2.2, we obtain the strong convergence of(Mnσρ−1SρLn)∗toϑSϑ−1I .
In view of (3.1), (3.2), (3.10), (2.5), and Lemma2.2, we have, forn > m+ 1, WnMnσρ−1SρWneum=WnMnσρ−1Sρeun−1−m
=iWnMnσρ−1Tn−m
= i
2WnMnσρ−1ϑ−1(eun−m−eun−m−2)
=−i
2WnMnσϕ−1Wn(eum+1−uem−1)
=−i
2Jν−1Lσnϕ−1Jν(eum+1−uem−1)