**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^{−1}SµI+
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 weightedL^{2}-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)^{δ}with 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 weightedL^{2}-spaceL^{2}_{ν},
and whereu∈L^{2}_{ν} stands for the unknown solution. The Hilbert spaceL^{2}_{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^{−1}SµI+K)u=f.

(1.2)
HereaI :L^{2}

ν −→L^{2}

ν denotes the multiplication operator defined by(au)(x) :=a(x)u(x),
the operatorS:L^{2}

ν −→L^{2}

ν is the CSIO given by (Su)(x) := 1

πi Z 1

−1

u(y)
y−x dy,
and K : L^{2}

ν −→ L^{2}

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:L^{2}_{ν}−→L^{2}_{ν}is continuous, i.e.S∈ L(L^{2}_{ν})(see [3]).

Letσ(x) = (1−x^{2})^{−}^{1}^{2} andϕ(x) = (1−x^{2})^{1}^{2} 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}_{jn}are 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^{1}^{4}^{−}^{α}^{2}^{,}^{1}^{4}^{−}^{β}^{2} .We write the above method in operator form as

Anun=Mnf , un∈imLn. (1.3)

HereLndenotes the orthogonal projection ofL^{2}

νonto thendimensional trial spaceimLnof
all polynomials of degree less thannmultiplied byϑ .ByMn=M_{n}^{τ}we 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 theL^{2}_{ν}-norm. We call the method (1.3)
convergent if, for any right-hand sidef ∈ L^{2}_{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 ofL^{2}_{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(L^{2}_{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 : A^{0} −→ L(L^{2}_{ν}),
whereA^{0} denotes a C^{∗}-algebra of operator sequences including all sequences{Mn(aI +
bµ^{−1}SµI)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} ∈ A^{0}.

**2. A**C^{∗}**-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^{−1}ϕ=v^{1}^{4}^{−}^{α}^{2}^{,}^{1}^{4}^{−}^{β}^{2} .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 inL^{2}_{ν},the orthogonal projectionLn

takes the form

Lnu=

n−1X

k=0

hu,uekiνuek.

The interpolation operatorMn=M_{n}^{τ}can be written asM_{n}^{τ}=ϑL^{τ}_{n}ϑ^{−1}I ,whereL^{τ}_{n}denotes
the polynomial interpolation operator with respect to the nodesxjn=x^{τ}_{jn}, j = 1, . . . , n .By

`^{2}we 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^{1}^{4}^{+}^{α}^{2}^{,}^{1}^{4}^{+}^{β}^{2}(x^{σ}_{kn}), ω_{kn}^{ϕ} :=

r π

n+ 1v^{1}^{4}^{+}^{α}^{2}^{,}^{1}^{4}^{+}^{β}^{2}(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 ∈
L^{1}(−1,1)*and let*j ∈N*be fixed. Then for each polynomial*q*with*degq≤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 on*n*and*q*and where*x^{µ}_{kn}*and*

λ^{µ}_{kn}=
Z 1

−1

`^{µ}_{kn}(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 weight*µ ,
*respectively.*

LetQ^{µ}_{n}denote the Gaussian quadrature rule with respect to the weightµ ,
Q^{µ}nf =

Xn k=1

λ^{µ}_{kn}f(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.*If*f ∈R*satisfies*

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

n→∞Q^{µ}_{n}f =
Z 1

−1

f(x)µ(x)dx .*If even*

|f(x)| ≤const (1−x)^{ε−}^{1+γ}^{2} (1 +x)^{ε−}^{1+δ}^{2} , −1< x <1,
*then* lim

n→∞kf−L^{µ}nfkµ = 0.

COROLLARY*2.3. Let*f ∈R*and, for some*ε >0,

|f(x)| ≤const (1−x)^{ε−}^{1+α}^{2} (1 +x)^{ε−}^{1+β}^{2} , −1< x <1.
*Then* lim

n→∞kf−M_{n}^{τ}fk^{ν} = 0*for*τ =σ*and*τ =ϕ .
*Proof. Introduce the quadrature rule*

Qnf = Z 1

−1

(L^{σ}_{n}f)(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−x^{2})σ(x)dx= π

n[ϕ(x^{σ}_{kn})]^{2}
forn >2.Consequently,

Qnf = π n

Xn k=1

[ϕ(x^{σ}_{kn})]^{2}f(x^{σ}_{kn}).

Since the nodesx^{σ}_{kn}of the quadrature ruleQnare the zeros of2Tn(x) =Un(x)−Un−2(x),
the estimate

Z 1

−1|(L^{σ}_{n}f)(x)|^{2}ϕ(x)dx≤2Qn|f|^{2}
(2.1)

holds true (see [2, Hilfssatz 2.4,§III.2]). As an immediate consequence we obtain
kM_{n}^{σ}fk^{2}ν=kL^{σ}_{n}ϑ^{−1}fk^{2}ϕ≤ 2π

n Xn k=1

|ϑ^{−1}(x^{σ}_{kn})ϕ(x^{σ}_{kn})f(x^{σ}_{kn})|^{2}= 2Q^{σ}_{n}|ϑ^{−1}ϕf|^{2}.
(2.2)

Now let >0be arbitrary andpbe a polynomial such thatkϑp−fk^{ν} < .Forn >degp
we havekM_{n}^{σ}f−fk^{2}ν≤2

kM_{n}^{σ}(ϑp−f)k^{2}ν+kϑp−fk^{2}ν

.Since, in view of Lemma2.2,

n→∞lim Q^{σ}_{n}|ϑ^{−1}ϕ(ϑp−f)|^{2}=kϑ^{−1}ϕ(ϑp−f)k^{2}σ =kϑp−fk^{2}ν,we get in view of (2.2) that
lim sup

n→∞ kM_{n}^{σ}f−fk^{2}ν <6^{2}.

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)})^{−1}L^{(ω)}_{n} , ω∈T :={1,2,3,4},

in some Hilbert spacesX_{ω}.Here,L^{(ω)}n :X_{ω}−→X_{ω}are projections andEn^{(ω)}: imLn−→

imL^{(ω)}n are certain operators defined by
X_{1}:=X_{2}:=L^{2}

ν, X_{3}:=X_{4}:=`^{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^{ν}euk,

V_{n}^{τ}u:={ω_{1n}^{τ} u(x^{τ}_{1n}), . . . , ω^{τ}_{nn}u(x^{τ}_{nn}),0,0, . . .},
Ven^{τ}u:={ωnn^{τ} u(x^{τ}nn), . . . , ω^{τ}1nu(x^{τ}1n),0,0, . . .}.
Immediately from the definitions, we conclude that

(E_{n}^{(1)})^{−1}=Ln, (E_{n}^{(2)})^{−1}=Wn,

(E_{n,τ}^{(3)})^{−1}ξ=
Xn
k=1

ξk−1

ω_{kn}^{τ} e`^{τ}_{kn}, (E_{n,τ}^{(4)})^{−1}ξ=
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}^{−1}Pn =VnVe_{n}^{−1}Pn=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.

LEMMA*2.4. Let*Vn =V_{n}^{σ}*and*Ven=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}^{−1}ξ, u

ν =

* _{n}
X

k=1

ξk−1

ω^{σ}_{kn}ϑ(x^{σ}_{kn})ϑ`^{σ}_{kn}, ϑUj

+

ν

=

* _{n}
X

k=1

ξk−1

ω_{kn}^{σ} ϑ(x^{σ}_{kn})`^{σ}_{kn}, ϕ^{2}Uj

+

σ

, and, forj= 0, . . . , n−2,we obtain

V_{n}^{−1}ξ,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}_{kn}ϑ(x^{σ}_{kn})Uj(x^{σ}_{kn}) =hξ, Vnueji`2.

Forj=n−1,using the relation
(1−x^{2})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^{σ}kn) =−Tn−1(x^{σ}kn), n >1,
(2.6)

we get, forn >1,

V_{n}^{−1}ξ,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}_{kn}ϑ(x^{σ}_{kn})Tn−1(x^{σ}_{kn})

= π 2n

Xn k=1

ξk−1

ω^{σ}_{kn}ϑ(x^{σ}_{kn})[ϕ(x^{σ}_{kn})]^{2}Un−1(x^{σ}_{kn})

=1 2

Xn k=1

ξk−1ω^{σ}_{kn}ϑ(x^{σ}_{kn})Un−1(x^{σ}_{kn})

=1

2hξ, Vneun−1i`2 .
LEMMA *2.5. The sequences*n

E^{(ω}n ^{1}^{)} En^{(ω}^{2}^{)}−1

L^{(ω}n^{2}^{)}

o

*converge weakly to zero for all*
*indices*ω1, ω2∈T*with*ω16=ω2.

*Proof. The proof for the case*τ =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}^{σ}uk^{2}l^{2} =π
n

Xn k=1

ϕ^{2}(x^{σ}_{kn})|pn(x^{σ}_{kn})|^{2}

≤const Z 1

−1

ϑ(x)pn(x) ϑ(x)

2

[ϕ(x)]^{2}σ(x)dx= constkuk^{2}ν

and

(V_{n}^{σ})^{−1}ξ^{2}

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}= 2kξk^{2}`^{2}.

Analogously we get the uniform boundedness of the sequencesn
Ve_{n}^{σ}o

andn

(Ve_{n}^{σ})^{−1}o
.
COROLLARY*2.6. The sequences*

n

E^{(ω}n ^{1}^{)}−∗

En^{(ω}^{2}^{)}∗

L^{(ω}n^{2}^{)}

o

*converge weakly to zero*
*for all indices*ω1, ω2∈T *with*ω16=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)})^{−1}L^{(ω)}_{n} −→Wω{An},

(2.7)

E_{n}^{(ω)}An(E_{n}^{(ω)})^{−1}L^{(ω)}_{n} ∗

−→Wω{An}^{∗}

holds inX_{ω}in 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 all*ω ∈ T *and all compact operators*Tω ∈ K(X_{ω}),*the se-*
*quences*{A^{(ω)}n } = n

(En^{(ω)})^{−1}L^{(ω)}n TωEn^{(ω)}

o

*belong to*F,*and for* ω1 6= ω2, *we get the*
*strong limits*

E_{n}^{(ω}^{1}^{)}A^{(ω}_{n}^{2}^{)}(E_{n}^{(ω}^{1}^{)})^{−1}L^{(ω}_{n}^{1}^{)}−→0,

E_{n}^{(ω}^{1}^{)}A^{(ω}_{n}^{2}^{)}(E_{n}^{(ω}^{1}^{)})^{−1}L^{(ω}_{n} ^{1}^{)}∗

−→0.

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

L(X_{ω}), ω∈T ,*are *-homomorphisms.*

*Proof. Of course, the mappings*Wω : F −→ L(X_{ω}), ω ∈ T ,are homomorphisms.

Hence, it suffices to show that the operator sequences{En^{(ω)}A^{∗}_{n}(En^{(ω)})^{−1}L^{(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:L^{2}

ν−→

L^{2}

ν,and Corollary2.7, we get
VnA^{∗}_{n}V_{n}^{−1}Pn

= 1 2

Vn(2Ln−WnL1Wn)An(Ln+WnL1Wn)V_{n}^{−1}Pn

∗

= Pn+V_{n}^{−1}WnL1WnV_{n}^{−1}Pn∗

VnAnV_{n}^{−1}Pn∗1

2 2Pn−V_{n}^{−1}WnL1WnVnPn∗

−→(W3{An})^{∗}.

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

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

ω=1

n(E_{n}^{(ω)})^{−1}L^{(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(L^{2}

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 of*F.
*A sequence*{An} ∈ F*is stable if and only if the operators*Wω{An}:X_{ω}−→X_{ω}, ω∈T ,
*are invertible and if the coset*{An}+J *is invertible in*F/J .

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

the subsetJ^{2}⊂ F^{2}of all sequences of the form
X2

ω=1

n

(E_{n}^{(ω)})^{−1}L^{(ω)}_{n} TωE_{n}^{(ω)}o

+{Cn}

whereTω ∈ K(X_{ω})and where{Cn}is in the idealN ⊂ F.Obviously, the setJ^{2}forms a
two-sided closed ideal ofF^{2},andF ⊂ F^{2},J^{2}⊂ 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 function*k(x, y)/ρ(y),*where*ρ= √νϕ =ϑ^{−1}ϕ ,*is con-*
*tinuous on*[−1,1]×[−1,1]*and that*K *is the integral operator with kernel*k(x, y).*Then*
{MnKLn} ∈ J^{2}⊂ J.*Moreover, if the approximations*Kn∈ L(imLn)*are defined by*

Kn= (E_{n}^{(3)})^{−1}
e

ω_{n}^{τ}k(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,

*where*ωe_{n}^{σ} =π/n*and*eω_{n}^{ϕ}=π/(n+ 1),*then the operator norm of*Kn−LnK|^{im}^{L}^{n}*tends*
*to zero and*{Kn}*is in*J^{2}.

*Proof. Consider the case*τ =σ .Since
Z 1

−1

`^{σ}_{kn}(y)ϕ(y)dy=
Z 1

−1

`^{σ}_{kn}(y)ϕ^{2}(y)σ(y)dy= π

n[ϕ(x^{σ}_{kn})]^{2},
the operatorsKncan be written asM_{n}^{σ}K_{n},where

(K_{n}un)(x) =
Z 1

−1

ϕ(y)L^{σ}_{n}[k(x,·)ϕ^{−1}un](y)dy .

Obviously, due to the Arzela-Ascoli theorem the operatorK : L^{2}_{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_{n}Ln −KLnk^{L(}^{L}^{2}ν,C[−1,1]) = 0. To this end, we introduce operators
e

K_{n} : imLn−→C[−1,1]by
(Ke_{n}un)(x) =

Z 1

−1

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

e

K_{n}uej=

L^{σ}_{n}[k(x,·)ρ^{−1}], ϕ^{2}Uj

σ=

L^{σ}_{n}[k(x,·)ρ^{−1}Uj], ϕ^{2}

σ =K_{n}uej,
and, in view of relations (2.5), (2.6),

2Ke_{n}uen−1=

L^{σ}_{n}[k(x,·)ρ^{−1}],2ϕ^{2}Un−1

σ

=

L^{σ}_{n}[k(x,·)ρ^{−1}], Tn−1−Tn+1

σ

=

L^{σ}_{n}[k(x,·)ρ^{−1}Un−1], ϕ^{2}

σ

=K_{n}uen−1.
Consequently,K_{n}Ln=Ke_{n}(2Ln−Ln−1).

Now, we deal with lim

n→∞kKe_{n}Ln−KLnk^{L(}^{L}^{2}ν,C[−1,1]).We take an arbitraryu ∈ L^{2}_{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_{n}Lnu−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^{σ}n[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^{ϕ}kLnuk^{ν}

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

n→∞kk(·,·)ρ^{−1}−kn(·,·)k^{∞}= 0,we obtain

n→∞lim kK_{n}Ln−KLnk^{L(}^{L}^{2}ν,C[−1,1])

≤ lim

n→∞kKe_{n}Ln−KLnkL(L2

ν,C[−1,1])k2Ln−Ln−1kL(L2 ν)

+ lim

n→∞kK(Ln−Ln−1)k^{L(}^{L}^{2}ν,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(L^{2}_{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^{−1}Sµwith a special weightµ=ρ(see
Lemma2.10), and forµ^{−1}Sµwith a generalµ .

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

SϕUn=iTn+1, Sϕ^{−1}Tn=−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(L^{2}_{ν},L^{2}_{σ}), u7→

X∞ n=0

γnhu,euniνTn, (3.3)

J_{ν}^{−1}∈ L(L^{2}_{σ},L^{2}_{ν}), u7→

X∞ n=0

1

γnhu, Tniσeun, (3.4)

V ∈ L(L^{2}_{ν}), u7→

X∞ n=0

hu,euniνeun+1, (3.5)

withγnas in (2.5), and their adjoint operators
J_{ν}^{∗}∈ L(L^{2}_{σ},L^{2}_{ν}), u7→

X∞ n=0

γnhu, Tniσeun,

J_{ν}^{−∗} ∈ L(L^{2}_{ν},L^{2}_{σ}), u7→

X∞ n=0

1

γnhu,euniνTn,
V^{∗}∈ L(L^{2}_{ν}), u7→

X∞ n=0

hu,uen+1iνuen.

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

REMARK*3.1. If*ξ, η∈`^{2}, ξ^{n} ={ξ_{k}^{n}},|ξ_{k}^{n}| ≤ |ηk|*for all*n > n0,*and if* lim

n→∞ξ^{n}_{k} =ξk

*for all*k= 0,1,2, ...,*then* lim

n→∞kξ^{n}−ξk`^{2} = 0.

LEMMA *3.2. Let*a ∈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}^{−1}aJν , τ =σ ,
aI =A , τ=ϕ .

*Proof. The proof in case of*τ =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}^{σ}aLn)^{∗}and ofWnM_{n}^{σ}aWn.

We writeM_{n}^{σ}f =

n−1X

j=0

α^{σ}_{jn}(f)uejand get, forj= 0,1, . . . , n−2,

α^{σ}_{jn}(f) =hM_{n}^{σ}f,eujiν=hL^{σ}_{n}ϑ^{−1}f, ϕ^{2}Ujiσ

= π n

Xn k=1

f(x^{σ}_{kn})

ϑ(x^{σ}_{kn})[ϕ(x^{σ}_{kn})]^{2}Uj(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}^{σ}f,eun−1iν =hL^{σ}_{n}ϑ^{−1}f, ϕ^{2}Un−1iσ

= 1

2hL^{σ}_{n}ϑ^{−1}f, 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})]^{2}Un−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 ∈L^{2}_{ν},

hM_{n}^{σ}aLnu, viν=

n−1X

j=0

hv,uejiν n−1X

l=0

hu,ueliνhM_{n}^{σ}auel,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}^{σ}a(Ln+Ln−1)viν.
Thus,

(M_{n}^{σ}aLn)^{∗}=1

2(2Ln−Ln−1)M_{n}^{σ}a(Ln+Ln−1),
(3.7)

whence we have the strong convergence of(Mn^{σ}aLn)^{∗}toaI inL^{2}_{ν}.

We verify the convergence ofWnM_{n}^{σ}aWneumfor 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}_{n}f ,
L^{σ}_{n}f =

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}^{σ}aWnuem=

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}_{kn}Tj(x^{σ}_{kn})J_{ν}^{−1}Tj

=J_{ν}^{−1}L^{σ}_{n}aJνeum−→J_{ν}^{−1}aJνuem in L^{2}_{ν}.
Thus,

WnM_{n}^{σ}aWn =J_{ν}^{−1}L^{σ}_{n}aJνLn −→J_{ν}^{−1}aJν in L^{2}

ν. (3.10)

LEMMA*3.3. Suppose*A =ρ^{−1}SρI ,*where*ρ=ϑ^{−1}ϕ=√νϕ ,*and*An =MnALn.
*Then*{An} ∈ F*and*

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

iJ_{ν}^{−1}ρV^{∗} , τ =σ ,

−A , τ=ϕ ,
*and*W3/4{An}=±S*with*

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 case*τ =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:L^{2}

σ−→

L^{2}

σ,we obtain, forun ∈imLn,

kM_{n}^{σ}ρ^{−1}Sρunk^{2}ν ≤2Q^{σ}_{n}|Sρun|^{2}

≤const Z 1

−1|(Sρun)(x)|^{2}σ(x)dx

≤constkρunk^{2}σ= constkunk^{2}ν,

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}^{σ}ρ^{−1}Sρuem=iM_{n}^{σ}ρ^{−1}Tm+1→iρ^{−1}Tm+1=ρ^{−1}Sρuem in L^{2}_{ν}.
Whence, the strong convergence of{An}toAis proved.

The well-known Poincar`e-Bertrand commutation formula implies that, foru∈L^{2}_{ν} and
v∈L^{2}

ν^{−1},

hSu, vi=hu, Svi,

whereh., .idenotes theL^{2}(−1,1)inner product without weight. Consequently, the adjoint
operator ofS :L^{2}

ν −→L^{2}

νis equal toν^{−1}Sν :L^{2}

ν −→L^{2}

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

hM_{n}^{σ}ρ^{−1}SρLnu,eujiν =hL^{σ}_{n}ϕ^{−1}SρLnu, ϕ^{2}Ujiσ

=π n

Xn k=1

(SρLnu)(x^{σ}_{kn})ϕ(x^{σ}_{kn})Uj(x^{σ}_{kn}) =hSρLnu, L^{σ}_{n}ϕUjiσ

=hSρLnu, σν^{−1}L^{σ}_{n}ϕUjiν =hρLnu, ν^{−1}SσL^{σ}_{n}ϕUjiν

=hu, LnϑSσL^{σ}_{n}ρeujiν

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

hM_{n}^{σ}ρ^{−1}SρLnu,uen−1iν=hL^{σ}_{n}ϕ^{−1}SρLnu, ϕ^{2}Un−1iσ

= π 2n

Xn k=1

(SρLnu)(x^{σ}_{kn})

ϕ(x^{σ}_{kn}) Tn−1(x^{σ}_{kn})

= 1

2hSρLnu, L^{σ}_{n}ϕUn−1iσ

= 1

2hu, LnϑSσL^{σ}_{n}ρeun−1iν.
Hence, in view of (3.1)

(M_{n}^{σ}ρ^{−1}SρLn)^{∗}=1

2LnϑSσL^{σ}_{n}ρ(Ln+Ln−1) =1

2ϑSσL^{σ}_{n}ρ(Ln+Ln−1).
Using Lemma2.2, we obtain the strong convergence of(M_{n}^{σ}ρ^{−1}SρLn)^{∗}toϑSϑ^{−1}I .

In view of (3.1), (3.2), (3.10), (2.5), and Lemma2.2, we have, forn > m+ 1,
WnM_{n}^{σ}ρ^{−1}SρWneum=WnM_{n}^{σ}ρ^{−1}Sρeun−1−m

=iWnM_{n}^{σ}ρ^{−1}Tn−m

= i

2WnM_{n}^{σ}ρ^{−1}ϑ^{−1}(eun−m−eun−m−2)

=−i

2WnM_{n}^{σ}ϕ^{−1}Wn(eum+1−uem−1)

=−i

2J_{ν}^{−1}L^{σ}_{n}ϕ^{−1}Jν(eum+1−uem−1)