**COLLOCATION FOR SINGULAR INTEGRAL EQUATIONS WITH FIXED**
**SINGULARITIES OF PARTICULAR MELLIN TYPE**^{∗}

PETER JUNGHANNS^{†}, ROBERT KAISER^{†},ANDGIUSEPPE MASTROIANNI^{‡}

**Abstract. This paper is concerned with the stability of collocation methods for Cauchy singular integral equa-**
tions with fixed singularities on the interval[−1,1]. The operator in these equations is supposed to be of the form
aI+bS+B^{±}with piecewise continuous functionsaandb. The operatorSis the Cauchy singular integral operator
andB^{±}is a finite sum of integral operators with fixed singularities at the points±1of special kind. The collo-
cation methods search for approximate solutions of the formν(x)p^{n}(x)orµ(x)p^{n}(x)with Chebyshev weights
ν(x) =q

1+x

1−xorµ(x) =q

1−x

1+x, respectively, and collocation with respect to Chebyshev nodes of first and third
or fourth kind is considered. For the stability of collocation methods in a weightedL^{2}-space, we derive necessary
and sufficient conditions.

**Key words. collocation method, stability,**C^{∗}-algebra, notched half plane problem
**AMS subject classifications. 65R20, 45E05**

**1. Introduction. Polynomial collocation methods for singular integral equations with**
fixed singularities are studied, for example, in [1,11,17]. In [11], the stability of a poly-
nomial collocation method is investigated for a class of Cauchy singular integral equations
with additional fixed singularities of Mellin convolution type. The papers [1,17] are more
concerned with the computational aspects of these methods. While [17] deals with integral
equations of the form

u(x) +b(x) Z 1

−1

h 1 +x

1 +y

u(y)dy 1 +y +

Z 1

−1

h0(x, y)u(y)dy=f(x), −1< x <1,
whereh: (0,∞)−→ C,b, f : [−1,1]−→ C, andh_{0} : [−1,1]^{2} −→Care given (contin-
uous) functions, the paper [1] deals with the effective realization of polynomial collocation
methods for the equation (see [1, (1.8)])

1 π

Z 1

−1

1

y−x− 1

2 +y+x+ 6(1 +x)

(2 +y+x)^{2} − 4(1 +x)^{2}
(2 +y+x)^{3}

u(y)dy=f(x),

−1< x <1, (1.1)

associated with the so-called notched half plane problem; see also [14, Section 37a] and
[2, Section 14]; we also refer to [1, Remark 2.6]. In particular, if the right-hand sidef(x)
of (1.1) is a constant function, then the solutionu(x)has a singularity of the form(1−x)^{−}^{1}^{2} at
the endpoint1of the integration interval. More detailed, the function√

1−x u(x)is bounded
and satisfies certain smoothness properties; cf. [2, Theorem 14.1]. In [11], singularities of the
solutions are considered which can be represented by a Jacobi weight the exponents of which
are in the interval(−^{1}_{4},^{3}_{4}). Hence, the stability results given in [11] are not applicable to the
equation (1.1) if we want to represent the asymptotic behaviour(1−x)^{−}^{1}^{2} of the solution at
the right endpoint of the integration interval.

∗Received April 14, 2014. Accepted June 5, 2014. Published online on August 1, 2014. Recommended by L. Reichel.

†Department of Mathematics, Chemnitz University of Technology, Reichenhainer Str. 39, D-09126 Chemnitz, Germany ({peter.junghanns,robert.kaiser}@mathematik.tu-chemnitz.de).

‡Dipartimento di Matematica, Universit`a della Basilicata, Via dell’Ateneo Lucano, 85100 Potenza, Italy (mastroianni.csafta@unibas.it).

190

In the present paper, we investigate the stability of collocation methods applied to a class of Cauchy singular integral equations with additional fixed singularities of Mellin type (of special form) covering equation (1.1) of the notched half plane problem, where the solu- tionu(x)can be represented in the form

(1.2) u(x) =

r1 +x

1−xu0(x) or u(x) =

r1−x 1 +xu0(x)

with sufficiently regular functionsu0(x). Of course, for the problem (1.1), this asymptotic behavior is not the best one, and further investigations are necessary. Let us also mention that other exponents in the weights are of interest depending on the concrete problem; see, for example, [2, Theorem 15.1] or [5, Section 2]. In [11], the stability of the collocation methods is proved by using respective results for Cauchy singular integral equations (cf. [12,13]) and a representation of the Mellin operators by Bochner integrals. Since the kernels of Mellin operators under consideration in the present paper do not satisfy all assumptions made in [11], we develop here necessary and sufficient conditions for the stability of these methods in a more direct manner taking advantage of the special structure of the Mellin kernels occurring for example in (1.1).

The paper is organized as follows. In Section2we introduce the class of integral equa- tions under consideration and describe the polynomial collocation methods we want to apply.

In Section3.1an algebra of operator sequences is defined for which the stability of these oper- ator sequences is equivalent to its invertibility modulo a suitable ideal and the invertibility of four limit operators associated to the operator sequence. The fact that the operator sequences of our collocation methods belong to this algebra is the topic of Section3.2, where also the respective four limit operators are presented. Section3.3discusses the invertibility of these limit operators and prepares the proof of the main result on the stability of the collocation methods, which is presented in Section4. Section5shows how to deal with the first type of singularities in (1.2) since the previous results are concerned with the second type in (1.2).

In Section6we discuss some numerical aspects of the investigated collocation methods and present numerical results for their application to the notched half plane problem (1.1) together with a discussion of the numerical results already presented in [1]. The final Sections7and8 give the technical proofs for the results of Section3.2and of Lemma4.8, respectively.

**2. The integral equation and a collocation method. Here we consider the Cauchy**
singular integral equation with fixed singularities of the form

a(x)u(x) +b(x) πi

Z 1

−1

u(y) y−xdy+

m−

X

k=1

β^{−}_{k}
πi

Z 1

−1

(1 +x)^{k}^{−}^{1}u(y)dy
(y+x+ 2)^{k}
+

m+

X

k=1

β_{k}^{+}
πi

Z 1

−1

(1−x)^{k}^{−}^{1}u(y)dy

(y+x−2)^{k} =f(x), −1< x <1,
(2.1)

with givenβ_{k}^{±} ∈ Cand nonnegative integersm_{±}. In this equation, the coefficient func-
tionsa, bbelong to the setPCof piecewise continuous functions^{1}, the right-hand side func-
tionfis assumed to belong to the weightedL^{2}-spaceL^{2}_{ν}, andu∈L^{2}_{ν}stands for the unknown
solution. The inner product in the Hilbert spaceL^{2}_{ν}is given by

hu, viν :=

Z 1

−1

u(y)v(y)ν(y)dy,

1We call a functiona: [−1,1]→Cpiecewise continuous if it is continuous at±1,if the one-sided limits a(x±0)exist for allx∈(−1,1),and at least one of them coincides witha(x).

whereν(x) =q

1+x

1−xis the Chebyshev weight of third kind. Let
S:L^{2}_{ν} →L^{2}_{ν}, u7→ 1

πi Z 1

−1

u(y) y− ·dy

be the Cauchy singular integral operator,aI :L^{2}_{ν} →L^{2}_{ν},u7→ aube the operator of multi-
plication bya, and

B^{±}k :L^{2}_{ν} −→L^{2}_{ν}, u7→ 1
πi

Z 1

−1

(1∓ ·)^{k}^{−}^{1}u(y)dy
(y+· ∓2)^{k}

be the integral operators with a fixed singularity at±1. We write equation (2.1) in the form Au:= aI+bS+

m−

X

k=1

β_{k}^{−}Bk^{−}+

m+

X

k=1

β^{+}_{k}Bk^{+}

! u=f.

It is a well known fact that the single operators aI, S, and Bk^{±} are bounded in L^{2}_{ν};
see [2, Theorem 1.16 and Remark 8.3]. This means that these operators belong to the Banach
algebraL(L^{2}_{ν})of all bounded and linear operatorsA:L^{2}_{ν} −→ L^{2}_{ν}. In order to get approx-
imate solutions of the integral equation, we use a polynomial collocation method. For this
we need some further notions. Letσ(x) = √^{1}

1−x^{2},ϕ(x) = √

1−x^{2},andµ(x) = q

1−x 1+x

be the Chebyshev weights of first, second, and fourth kind, respectively. For n ≥ 0 and
τ∈ {σ, ϕ, ν, µ}, we denote byp^{τ}_{n}(x)the corresponding normalized Chebychev polynomials
of degreenwith respect to the weightτ(x)and with positive leading coefficient, which we
abbreviate byTn(x) =p^{σ}_{n}(x),Un(x) =p^{ϕ}_{n}(x),Rn(x) =p^{ν}_{n}(x), andPn(x) = p^{µ}_{n}(x). We
know that

T0(x) = 1

√π, Tn(coss) = r2

πcosns, n≥1, s∈(0, π), and, forn≥0, s∈(0, π),

Un(coss) =

√2 sin(n+ 1)s

√πsins , Rn(coss) = cos(n+^{1}_{2})s

√πcos^{s}_{2} , Pn(coss) =sin(n+^{1}_{2})s

√πsin_{2}^{s} .
The zerosx^{τ}_{jn}ofp^{τ}_{n}(x)are given by

x^{σ}_{jn}= cosj−^{1}2

n π, x^{ϕ}_{jn}= cos jπ

n+ 1, x^{ν}_{jn}= cosj−^{1}2

n+^{1}_{2}π, x^{µ}_{jn}= cos jπ
n+^{1}_{2},
forj = 1,· · · , n. We introduce the Lagrange interpolation operator L^{τ}n defined for every
functionf : (−1,1)→Cby

L^{τ}nf =
Xn
j=1

f(x^{τ}_{jn})ℓ^{τ}_{jn}, ℓ^{τ}_{jn}(x) = p^{τ}_{n}(x)

(x−x^{τ}_{jn})(p^{τ}_{n})^{′}(x^{τ}_{jn}) =
Yn
k=1,k6=j

x−x^{τ}_{kn}
x^{τ}_{jn}−x^{τ}_{kn}.

We remark that the respective Christoffel numbersλ^{τ}_{jn}=
Z 1

−1

ℓ^{τ}_{jn}(x)τ(x)dxare equal to

λ^{σ}_{jn}=π

n, λ^{ϕ}_{jn}=π

1−(x^{ϕ}_{jn})^{2}

n+ 1 , λ^{ν}_{jn}= π(1 +x^{ν}_{jn})

n+^{1}_{2} , λ^{µ}_{jn}=π(1−x^{µ}_{jn})
n+^{1}_{2} .

The collocation method seeks an approximationun∈L^{2}_{ν}of the form

(2.2) un(x) =µ(x)pn(x), pn∈P_{n},

to the exact solution ofAu=fby solving

(2.3) (Aun)(x^{τ}_{kn}) =f(x^{τ}_{kn}), k= 1,2, . . . , n,

whereP_{n}denotes the set of all algebraic polynomials of degree less thann∈N. We set
e

pn(x) :=µ(x)Pn(x), n= 0,1,2, . . . Using the Lagrange basis

ℓe^{τ}_{kn}(x) =µ(x)ℓ^{τ}_{kn}(x)

µ(x^{τ}_{kn}) , k= 1, . . . , n,
inµP_{n}, we can writeunas

un =

nX−1 j=0

αjnpej = Xn k=1

ξknℓe^{τ}_{kn}.
If we introduce the Fourier projections

Ln:L^{2}_{ν} →L^{2}_{ν}, u7→

nX−1 j=0

hu,pejiνpej

and the weighted interpolation operatorsM^{τ}n:=µL^{τ}nµ^{−}^{1}I, then the collocation system (2.3)
can be written as an operator equation

(2.4) A^{τ}n:=M^{τ}nAL^{n}un=M^{τ}nf, un ∈imL^{n},

whereimdenotes the range of an operator. For the relation between the approximate solution
and the exact solution, we have to investigate the stability of the collocation method. We
call the collocation method stable if the approximation operators A^{τ}n are invertible for all
sufficiently largen∈Nand if the norms(A^{τ}n)^{−}^{1}Ln

L(L^{2}

ν)are uniformly bounded. If the
collocation method is stable, then the strong convergence ofA^{τ}nL^{n}toA ∈ L(L^{2}_{ν})as well as
the convergenceM^{τ}nf −→f inL^{2}_{ν}imply the convergence of the approximationsunto the
exact solutionuinL^{2}_{ν}. This can be seen from the estimate

kL^{n}u−unkν =A^{−}n^{1}L^{n}(A^{n}L^{n}u− A^{n}un)_{ν}

≤A^{−}n^{1}L^{n}

L(L2

ν)(kA^{n}L^{n}u− Aukν+kf− M^{τ}nfkν),

which also shows that, for getting convergence rates, one has to estimate the errorsLnu−u
andAnLnu− Auwith the solution uand the errorM^{τ}nf −f with the right-hand side f.
The technique, which we use to prove stability, includes the proof of strong convergence
A^{τ}nL^{n}−→ A; cf. the definition of the algebra F in Section 3.1. For M^{τ}nf −→ f, see
Lemma7.2. But the focus of the present paper is the stability of the methods under consid-
eration. Proving convergence rates by using certain smoothness properties of the right-hand
sidef and of the solutionuis a further task; cf., for example, [17, Section 5]. The main
result of our paper on the stability of the collocation methods (2.4) applied to the integral
equation (2.1) is given in Theorem4.11.

Of course, by making the ansatz (2.2), we are only concerned with the second representa- tion of the solution in (1.2). How to use the corresponding results for the other representation in (1.2) is shown in Section5.

**3. The stability of the collocation methods.**

**3.1. The Banach algebra framework for the stability of operator sequences. In what**
follows, the operator sequence, for which we want to prove stability, is considered as an
element of a Banach algebra. For the definition of this algebra, we need some spaces as
well as some useful operators. Byℓ^{2} we denote the Hilbert space of all square summable
sequencesξ= (ξj)_{j=0}^{∞},ξj ∈C,with the inner product

hξ, ηi= X∞ j=0

ξjηj. Additionally, we define the following operators

W^{n} :L^{2}_{ν} −→L^{2}_{ν}, u7→

nX−1 j=0

hu,pen−1−ji^{ν}pej,
P^{n} :ℓ^{2}−→ℓ^{2}, (ξj)_{j=0}^{∞} 7→(ξ0,· · ·, ξn−1,0, . . .),
and, forτ ∈ {σ, µ},

Vn^{τ} : imLn −→imPn, u7→

ω_{n}^{τ}p

1 +x^{τ}_{1n}u(x^{τ}_{1n}), . . . , ω_{n}^{τ}p

1 +x^{τ}_{nn}u(x^{τ}_{nn}),0, . . .
,
Ven^{τ} : imLn −→imPn, u7→

ω_{n}^{τ}p

1 +x^{τ}_{nn}u(x^{τ}_{nn}), . . . , ω_{n}^{τ}p

1 +x^{τ}_{1n}u(x^{τ}_{1n}),0, . . .
,
whereω^{σ}_{n}=p_{π}

n andω_{n}^{µ}=q _{π}

n+^{1}_{2}. LetT ={1,2,3,4}and set

X^{(1)}=X^{(2)}=L^{2}_{ν}, X^{(3)}=X^{(4)}=ℓ^{2}, L^{(1)}n =L^{(2)}n =L^{n}, L^{(3)}n =L^{(4)}n =P^{n},
and defineE^{n}^{(t)}: imL^{n} −→X^{(t)}n := imL^{(t)}^{n} fort∈T by

En^{(1)}=Ln, En^{(2)} =Wn, En^{(3)}=Vn^{τ}, En^{(4)}=Ven^{τ}.

Here and at other places, we use the notionL^{n},W^{n},. . .instead ofL^{n}|^{im}Ln,W^{n}|^{im}Ln,. . . ,
respectively. All operatorsEn^{(t)}, t∈T, are invertible with inverses

En^{(1)}

−1

=En^{(1)},
En^{(2)}

−1

=En^{(2)},
En^{(3)}

−1

= (Vn^{τ})^{−}^{1},
En^{(4)}

−1

= (eV_{n}^{τ})^{−}^{1},
where, forξ∈imPn,

(Vn^{τ})^{−}^{1}ξ= (ω_{n}^{τ})^{−}^{1}
Xn
k=1

p 1

1 +x^{τ}_{kn}ξk−1ℓe^{τ}_{kn}
and

(Ve_{n}^{τ})^{−}^{1}ξ= (ω_{n}^{τ})^{−}^{1}
Xn
k=1

p 1

1 +x^{τ}_{kn}ξn−kℓe^{τ}_{kn}.

Now we can introduce the algebra of operator sequences we are interested in. ByFwe denote
the set of all sequences(An)^{∞}_{n=1} =: (An)of linear operatorsAn : imLn −→imLn for
which the strong limits

W^{t}(A^{n}) := lim

n→∞En^{(t)}A^{n}
En^{(t)}

−1

L^{(t)}n , t∈T,
W^{t}(An)∗= lim

n→∞

En^{(t)}An

En^{(t)}

−1

L^{(t)}n

∗

, t∈T,

exist. If we provideFwith the supremum normk(An)kF:= sup_{n}_{≥}_{1}kAnLnkL(L^{2}

ν)and with
operations(An) + (Bn) := (An+Bn),(An)(Bn) := (AnBn)and(An)^{∗} := (A^{∗}n), one
can easily check thatFbecomes aC^{∗}-algebra with the identity element(Ln). Moreover, we
introduce the setJ⊂Fof all sequences of the form

X4 t=1

En^{(t)}

−1

L^{(t)}n T^{t}En^{(t)}+C^{n}

! ,

where the linear operators Tt : X^{(t)} −→ X^{(t)} are compact and kCnLnkL(L2

ν) −→ 0 asn→ ∞.

PROPOSITION3.1 (Lemma 2.1 in [10], Theorem 10.33 in [18,19]). The setJ*forms a*
*two-sided closed ideal in the*C^{∗}*-algebra*F. Moreover, a sequence(A^{n})∈F*is stable if and*
*only if the operators*W^{t}(A^{n}) : X^{(t)} → X^{(t)}*,*t ∈ T,*and the coset*(A^{n}) +J ∈ F/J*are*
*invertible.*

**3.2. The collocation sequence as an element of the Banach algebra**F. For the inves-
tigation of the stability of the collocation method(A^{τ}n) = (M^{τ}nALn), we have to show that
this sequence belongs to the algebraF, which means to prove the existence of the four limit
operatorsW^{t}(An). Regarding the multiplication operatoraIas well as the Cauchy singular
integral operator S, Proposition3.2below was proved in [10]. To describe the respective
limit operators we need some further notation. Define the isometries

J1:L^{2}_{ν}→L^{2}_{ν}, u7→

X∞ j=0

hu,pejiνRj, (3.1)

J^{2}:L^{2}_{ν}→L^{2}_{ν}, u7→

X∞ j=0

hu,peji^{ν}√

1−x Uj,

J^{3}:L^{2}_{ν}→L^{2}_{ν}, u7→

X∞ j=0

hu,peji^{ν} 1

√1 +xTj, and the shift operator

(3.2) V :L^{2}_{ν} →L^{2}_{ν}, u7→

X∞ j=0

hu,pejiνpej+1.

The adjoint operatorsJ1^{∗},J2^{∗},J3^{∗},V^{∗}:L^{2}_{ν}→L^{2}_{ν}are given by
J1^{∗}u=J1^{−}^{1}u=

X∞ j=0

hu, Rjiνpej, J2^{∗}u=J2^{−}^{1}u=
X∞
j=0

u,√

1−x Uj

νepj

and

J3^{∗}u=J3^{−}^{1}u=
X∞
j=0

u, 1

√1 +xTj

ν

e

pj, V^{∗}u=
X∞
j=0

hu,pej+1i_{ν}pej.

Finally, we denote byI= [δjk]^{∞}_{j,k=0}the identity inℓ^{2}and byS,e S^{τ}:ℓ^{2}−→ℓ^{2}the operators
defined by

e S=

1−(−1)^{j}^{−}^{k}

πi(j−k) +1−(−1)^{j+k+1}
πi(j+k+ 1)

∞ j,k=0

and

S^{τ} =

1−(−1)^{j}^{−}^{k}

πi(j−k) −1−(−1)^{j+k+1}
πi(j+k+ 1)

∞ j,k=0

:τ =σ,
1−(−1)^{j}^{−}^{k}

πi

1

j−k− 1 j+k+ 2

∞ j,k=0

:τ =µ.

The following proposition is already known.

PROPOSITION 3.2 (Proposition 3.5 in [10]). Let a, b ∈ PC, A = aI +bS, *and*
A^{τ}n=M^{τ}nALn*. Then, for*τ ∈ {σ, µ}*, we have*(A^{τ}n)∈F*and*

W^{1}(A^{τ}n) =A, W^{2}(A^{τ}n) =

J1^{−}^{1}(aJ1+ibI) :τ=σ,
J2^{−}^{1}(aJ2−ibJ3V) :τ=µ,
W^{3}(A^{τ}n) =a(1)I+b(1)S^{τ}, W^{4}(A^{τ}n) =a(−1)I−b(−1)eS.

REMARK3.3. We have to mention that in [10, p. 745, line 13] there is a sign error. One has

−

"

1−(−1)^{j}^{−}^{k}

2insin^{j}_{2n}^{−}^{k} π+1−(−1)^{j+k+1}
2insin^{j+k+1}_{2n} π

#n−1

j,k=0

instead of

−

"

1−(−1)^{j}^{−}^{k}

2insin^{j}_{2n}^{−}^{k} π−1−(−1)^{j+k+1}
2insin^{j+k+1}_{2n} π

#n−1 j,k=0

.

This leads toW^{4}(A^{σ}n) =a(−1)I−b(−1)eSand not toW^{4}(A^{n}) =a(−1)I−b(−1)S^{σ}as
formulated in [10, Proposition 3.5].

Having in mind Proposition3.2, our next aim is to show that the sequences M^{τ}nB^{±}kL^{n}
,
k∈N,belong toFand to determine their limit operatorsW^{j} M^{τ}nB^{±}kLn

. As a result, we can state the following proposition, the proof of which is given in Section7.

PROPOSITION *3.4. Let*a, b ∈ PC, A = aI +bS +

m−

X

k=1

β^{−}_{k}Bk^{−} +

m+

X

k=1

β_{k}^{+}B^{+}k*, and*
A^{τ}n=M^{τ}nALn*. Then, for*τ ∈ {σ, µ}*, we have*(A^{τ}n)∈F*and*

W^{1}(A^{τ}n) =A,
W^{2}(A^{τ}n) =

( J1^{−}^{1}(aJ^{1}+ibI) :τ=σ,
J2^{−}^{1}(aJ^{2}−ibJ^{3}V) :τ=µ,
W^{3}(A^{τ}n) =a(1)I+b(1)S^{τ}+A^{τ}+K^{τ},
W^{4}(A^{τ}n) =a(−1)I−b(−1)eS+A+K,

*where the operators*A,A^{τ}∈ L(ℓ^{2})*are defined as*
A=

m−

X

k0=1

β_{k}^{−}_{0}

2h^{−}_{k}

0

(j+^{1}_{2})^{2}
(k+^{1}_{2})^{2}

j+^{1}_{2}
(k+^{1}_{2})^{2}

∞ j,k=0

, (3.3)

A^{σ}=

m+

X

k0=1

β_{k}^{+}

0

2h^{+}_{k}

0

(j+^{1}_{2})^{2}
(k+^{1}_{2})^{2}

1
k+^{1}_{2}

∞ j,k=0

, (3.4)

A^{µ}=

m+

X

k0=1

β_{k}^{+}

0

2h^{+}_{k}

0

(j+ 1)^{2}
(k+ 1)^{2}

1 k+ 1

∞ j,k=0

, (3.5)

*with*

h^{±}_{k}(x) = (∓1)^{k}
πi

x^{k}^{−}^{1}

(1 +x)^{k}, x∈(0,∞), k∈N,
(3.6)

*and where*K,K^{τ} :ℓ^{2}−→ℓ^{2}*are compact operators.*

**3.3. The invertibility of the limit operators. In this section we consider the invert-**
ibility of the four limit operators. Due to Proposition3.1, this is necessary for the stability
of the collocation method. Thus, our main concern is devoted to necessary and sufficient
conditions for the invertibility of these limit operators. At first we consider the operator
A=aI+bS+

mP−

k=1

β^{−}_{k}Bk^{−}+

mP+

k=1

β_{k}^{+}Bk^{+}. For this, we need the Mellin transform

b y(z) :=

Z _{∞}

0

y(x)x^{z}^{−}^{1}dx

of a functiony: (0,∞)→C. With the help of the continuous functionsh^{±}_{k} : (0,∞)−→C
defined in (3.6), we can write the linear combination of the integral operatorsB^{±}k in (2.1) in
the form

m−

X

k=1

β_{k}^{−}(Bk^{−}u)(x) +

m+

X

k=1

β_{k}^{+}(B^{+}ku)(x)

=

m−

X

k=1

β_{k}^{−}
Z 1

−1

h^{−}_{k}

1 +x 1 +y

u(y) 1 +y dy+

m+

X

k=1

β_{k}^{+}
Z 1

−1

h^{+}_{k}

1−x 1−y

u(y) 1−ydy.

(3.7)

Forh^{±}_{k}(x),k ∈ N, the Mellin transform is given byhb^{±}_{k}(z) = (∓1)^{k}bh_{k}(z+k−1) with
h_{k}(x) = (1 +x)^{k}, and (see, for example, [4, 6.2.(6)])

b

h_{k}(z) = (−1)^{k}^{−}^{1}
z−1

k−1 π

sin(πz) is holomorphic in the strip0<Rez < k. This implies

b

h^{±}_{k}(β−it) =

β−it+k−2 k−1

(∓1)^{k}

sinh(π(iβ+t)), 1−k < β <1, t∈R.
We remark thathb^{±}_{k}(β−it)is analytic in the strip0< β <1for allk∈N. Due to (3.7) and
by [2, Theorem 9.1] (cf. also [3,7,8,16]), we can state the following proposition.

PROPOSITION*3.5. Let*a, b∈PC,A=aI+bS+

m−

X

k=1

β_{k}^{−}B^{−}k+

m+

X

k=1

β_{k}^{+}B^{+}k :L^{2}_{ν} −→L^{2}_{ν}*.*
*(a) The operator*A*is Fredholm if and only if:*

• *For any* x ∈ (−1,1), there holds a(x ± 0) + b(x ± 0) 6= 0 *and*
a(x±0)−b(x±0)6= 0 *as* *well* *as* a(±1) + b(±1) 6= 0 *and*
a(±1)−b(±1)6= 0.

• *If*a*or*b*has a jump at*x∈(−1,1), then there holds
λa(x+ 0) +b(x+ 0)

a(x+ 0)−b(x+ 0)+ (1−λ)a(x−0) +b(x−0)

a(x−0)−b(x−0) 6= 0, 0≤λ≤1.

• *For*x=±1, there holds
a(1) +b(1)icot ^{π}_{4} −iπξ

+

m+

X

k=1

β_{k}^{+}bh^{+}_{k}(^{1}_{4}−iξ)6= 0, − ∞< ξ <∞,
*and*

a(−1) +b(−1)icot ^{π}_{4} +iπξ
+

m−

X

k=1

β^{−}_{k}hb^{−}_{k}(^{3}_{4}−iξ)6= 0, − ∞< ξ <∞.
*(b) If*A*is Fredholm and if the coefficients*a*and*b*have finitely many jumps, then the*
*Fredholm index of*A : L^{2}_{ν} −→ L^{2}_{ν} *is equal to minus the winding number of the*
*closed continuous curve*Γ_{A}:= Γ_{−} ∪Γ1 ∪Γ^{′}_{1} ∪ . . . ∪ΓN ∪ Γ^{′}_{N} ∪ ΓN+1 ∪Γ+

*with the orientation given by the subsequent parametrization. Here,*N *stands for*
*the number of discontinuity points*xi, i= 1, . . . , N,*of the functions*a*and*b*chosen*
*such that*x0 :=−1 < x1 <· · · < xN < xN+1 := 1. Using thesexi*, the curves*
Γi, i= 1, . . . , N+ 1, andΓ^{′}_{i}, i= 1, . . . , N,*are given by*

Γi :=

a(y) +b(y)

a(y)−b(y) : xi−1< y < xi

,
Γ^{′}_{i} :=

λa(xi+ 0) +b(xi+ 0)

a(xi+ 0)−b(xi+ 0) + (1−λ)a(xi−0) +b(xi−0)

a(xi−0)−b(xi−0) : 0≤λ≤1

.
*The curves*Γ_{±}*, connecting the point*1*with one of the end points of*Γ1*and*ΓN+1*,*
*are given by the formulas*

Γ+:=

(a(1) +b(1)icot ^{π}_{4}−iπξ

+Pm+

k=1β_{k}^{+}hb^{+}_{k}(^{1}_{4}−iξ)

a(1)−b(1) : −∞ ≤ξ≤ ∞

)

*and*

Γ_{−}:=

(a(−1) +b(−1)icot ^{π}_{4} +iπξ

+Pm−

k=1β_{k}^{−}hb^{−}_{k}(^{3}_{4}−iξ)

a(−1)−b(−1) :

− ∞ ≤ξ≤ ∞ )

.
*(c) If*A*is Fredholm and if*m_{−}= 0*or*m+= 0, thenA*is one-sided invertible.*

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−3

−2.5

−2

−1.5

−1

−0.5 0

real

imag

FIG. 3.1.nP3
k=1bh^{−}k

3 4−iξ

:−∞ ≤ξ≤ ∞o
*.*

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

0 0.2 0.4 0.6 0.8 1 1.2

real

imag

FIG. 3.2.nP3
k=1bh^{+}k

1 4−iξ

:−∞ ≤ξ≤ ∞o
*.*

Letz1, z2 ∈ C. We denote by γℓ/r[z1, z2] the half circle line fromz1 toz2lying on the left, respectively, on the right of the segment[z1, z2]and byγ[z1, z2]the circle line with diameter[z1, z2]starting in z1 with clockwise orientation. For given functions a, b ∈ PC witha(x±0)−b(x±0)6= 0, x∈[−1,1],we define

(3.8) c(x) :=a(x) +b(x)

a(x)−b(x).

−2 −1 0 1 2

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5 0

real

imag

FIG. 3.3.Γ−*in case of*a(−1) = 0,m_{−}= 3,β_{k}^{−}= 1.

−1 −0.5 0 0.5 1

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8

real

imag

FIG. 3.4.Γ+*in case of*a(1) = 0,m+= 3,βk^{+}= 1.

The equalities

(a(1) +b(1)icot ^{π}_{4} −iπξ

a(1)−b(1) : −∞ ≤ξ≤ ∞ )

=γr[c(1),1].

(3.9) and

(a(−1) +b(−1)icot ^{π}_{4} +iπξ

a(−1)−b(−1) : −∞ ≤ξ≤ ∞ )

=γℓ[1, c(−1)]

(3.10)

can easily be shown. The curveΓ+ is a modified arc fromc(1)to1and the curve Γ_{−} is
a modified arc from1 toc(−1). For instance, Figures 3.1and3.2 display the images of
P3

k=1bh^{±}_{k} ^{3}_{4}−iξ

(i.e.,m_{±} = 3,β^{±}_{k} = 1) and Figures3.3and3.4the respective curvesΓ_{±}
in the casea(±1) = 0.

The above proposition enables us to give conditions for the invertibility of the second
limit operatorW^{2}. So we derive from [10, Lemma 4.4 and Corollary 4.5].

LEMMA*3.6. Let*A^{τ}n =M^{τ}n(aI+bS)Ln*,*τ∈ {σ, µ}*.*

*(a) The operator*W^{2}(A^{σ}n) *is invertible in* L^{2}_{ν} *if and only if* A = aI +bS *has this*
*property.*

*(b) If*aI+bS:L^{2}_{ν}−→L^{2}_{ν}*is invertible, then the invertibility of*W^{2}(A^{µ}n) :L^{2}_{ν}−→L^{2}_{ν}
*is equivalent to the condition*|a(1)|>|b(1)|*, which is again equivalent to the con-*
*dition*Rec(1)>0.

For the index calculation of the second limit operator, we can state the following lemma.

LEMMA *3.7. Let*a, b ∈PC,τ ∈ {σ, µ},*and*A:=aI+bS : L^{2}_{ν} → L^{2}_{ν}*, as well as*
A^{τ}n :=M^{τ}nALn*. If*A*is Fredholm, then the second limit operator*W^{2}(A^{σ}n) :L^{2}_{ν} →L^{2}_{ν}*is*
*also Fredholm, where*

(3.11) indW^{2}(A^{σ}n) =−indA.

*If*A,W^{2}(A^{µ}n) :L^{2}_{ν} −→L^{2}_{ν}*are Fredholm, then*
(3.12) indW^{2}(A^{µ}n) =

−indA : Rec(1)>0,

−indA −1 : Re c(1)<0.
*Proof. Let*indA=κ .Forλ∈[0,1], define

(3.13) c(x, λ) =

c(x−0)(1−λ) +c(x+ 0)λ :x∈(−1,1),
c(1) + [1−c(1)]f_{−}^{1}

2(λ) :x= +1,
1 + [c(−1)−1]f^{1}

2(λ) :x=−1,

wheref_{α}(λ) = sinπαλ

sinπα e^{−}^{i}^{πα(λ}^{−}^{1)}andc(x)is defined in (3.8). Note that, forz1, z2 ∈ C,
the image of the functionz1+ (z2−z1)f_{α}(λ),λ ∈ [0,1], describes the circular arc from
z1toz2 such that the straight line segment[z1, z2]is seen from the points of the arc under
the angleπ(1 +α), i.e., in caseα∈(−1,0), the arc lies on the right of the segment[z1, z2]
and, in caseα ∈ (0,1), on the left. By (3.9), (3.10), and Proposition3.5, it follows that
Γ_{A}={c(x, λ) : (x, λ)∈[−1,1]×[0,1]}. Moreover, we denote the winding number of this
curve with respect to the origin of the complex plane bywindc(x, λ). Due to the fact that
every piecewise continuous function can be approximated by a function with finitely many
jumps, we can assume that−1 < x1 <· · · < xN <1are the only discontinuities ofc(x).

Define the piecewise continuous argument functionα(x) = _{2π}^{1} argc(x)in such a way that
(3.14) |α(xk+ 0)−α(xk−0)|< 1

2, k= 1, . . . , N, and α(−1)∈

−3 4,1

4

. For the winding number, we derive

(3.15) windc(x, λ)∈Z∩

α(1)−1

4, α(1) +3 4

. Due to Proposition3.5, we haveκ=−windc(x, λ).

In caseτ =σ, setd(x) = b(x)−a(x)

b(x) +a(x) and defined(x, λ)analogously to (3.13). Then
(cf. the proof of [10, Lemma 4.4]),d(x, λ) 6= 0,∀(x, λ) ∈ [−1,1]×[0,1],if and only if
c(x, λ)6= 0,∀(x, λ)∈[−1,1]×[0,1]. Define the piecewise continuous argument function
β(x) = _{2π}^{1} argd(x)satisfying the respective conditions (3.14). Since (cf. again the proof of
[10, Lemma 4.4])

indW^{2}(A^{σ}n) = ind (bI −aS) and β(x) =−1

2 −α(x), we have

−indW^{2}(A^{σ}n) = windd(x, λ)∈Z∩

−3

4 −α(1),1 4 −α(1)

, proving, together with (3.15), the relation (3.11).

Let us turn to the caseτ=µ, and assume thatW^{2}(A^{µ}_{n}) :L^{2}_{ν} −→L^{2}_{ν}is Fredholm. From
the proof of [10, Lemma 4.5], we have

−indW^{2}(A^{µ}n)∈Z∩

−α(1)−1

4,−α(1) +3 4

.
In view of (3.15), we getκ∈ −α(1)−^{1}_{4},−α(1) +^{3}_{4}

if and only if α(1)−1

4 <windc(x, µ)< α(1) +1 4,

which is equivalent toRec(1)>0.Analogously,κ+ 1∈ −α(1)−^{1}_{4},−α(1) +^{3}_{4}
if and
only ifα(1) +^{1}_{4} <windc(x, µ)< α(1) +^{3}_{4},i.e.,Rec(1)<0.

Observe that the Fredholmness of the operatorA:=aI+bS:L^{2}_{ν}→L^{2}_{ν}implies that the
half circle lineγr[c(1),1]does not contain0, which impliesc(1)6∈ {iy : y≥0}. Moreover,
by the Fredholmness of (cf. [10, (4.4)])

W^{2}(A^{µ}n) =J2^{−}^{1}

√1 2

a(√

1 +x+ib√ 1−x

− ia√

1−x+b√ 1 +x

S ,

we get06∈γr

h 1 c(1),1i

, i.e.,c(1)6∈ {iy : y≤0}. Hence, (3.12) is proved.

We also need conditions for the Fredholmness of the operatorsW^{3/4}(A^{τ}n). For that, we
consider theC^{∗}-algebraL(ℓ^{2})of all linear and continuous operators inℓ^{2}. ByalgT(PC)we
denote the smallestC^{∗}-subalgebra ofL(ℓ^{2})generated by the Toeplitz matrices

b gj−k ∞

j,k=0

with piecewise continuous generating functionsg(t) :=P

ℓ∈Zbgℓt^{ℓ}defined on the unit circle
T:={t∈C : |t|= 1}and being continuous onT\ {±1}.

PROPOSITION 3.8 (Theorem 16.2 in [15]). There exists a (continuous) mapsmb*from*
algT(PC)*into a set of complex valued functions defined on*T×[0,1], which sends each
R∈algT(PC)*to the function*smbR(t, λ), which is called symbol ofR*and which satisfies*
*the following properties:*

*(a) For each fixed*(t, λ)∈T×[0,1], the mapalgT(PC)−→C*,*R7→smb^{R}(t, λ)*is*
*a multiplicative linear functional on*algT(PC).

*(b) For any* t 6= ±1, the value smb^{R}(t, λ) *is independent of* λ, and the function
t7→smb^{R}(t,0)*is continuous on*{t∈T : Imt >0}*and on*{t∈T : Imt <0}

*with the limits*

smb^{R}(1 + 0,0) := lim

t→+1,Imt>0smb^{R}(t,0) = smb^{R}(1,1),
smb^{R}(1−0,0) := lim

t→+1,Imt<0smb^{R}(t,0) = smb^{R}(1,0),
smb^{R}(−1 + 0,0) := lim

t→−1,Imt<0smb^{R}(t,0) = smb^{R}(−1,1),
smb^{R}(−1−0,0) := lim

t→−1,Imt>0smb^{R}(t,0) = smb^{R}(−1,0).

*(c) An operator*R ∈ algT(PC)*is Fredholm if and only if* smb^{R}(t, λ) 6= 0*for all*
(t, λ)∈T×[0,1].

*(d) For any Fredholm operator*R∈algT(PC), the index ofR*is the negative winding*
*number of the closed curve*

Γ^{R}: ={smb^{R}(e^{i}^{s},0) : 0< s < π} ∪ {smb^{R}(−1, s) : 0≤s≤1}

∪ {smb^{R}(−e^{i}^{s},0) : 0< s < π} ∪ {smb^{R}(1, s) : 0≤s≤1},
(3.16)

*where the orientation of*ΓR *is given in a natural way by the parametrization of*T
*and*[0,1].

*(e) An operator*R∈algT(PC)*is compact if and only if the symbol*smb^{R}(t, λ)*van-*
*ishes for all*(t, λ)∈T×[0,1].

In what follows, we show that the limit operators W^{3/4}(A^{τ}n) belong to algT(PC)
and consider their symbols as well as the respective curves (3.16). Using the results of
[10, Section 4] and the relations

icot π

4 ± i

4 log λ 1−λ

=±(2λ−1) + 2ip

λ(1−λ), 0≤λ≤1, as well as

nicotπ 4 −iξ

:−∞ ≤ξ≤ ∞o

=γr[1,−1]

and n

icotπ 4 +iξ

:−∞ ≤ξ≤ ∞o

=γℓ[−1,1]

(cf. also (3.9), (3.10)), we get the following lemma.

LEMMA*3.9. Let*τ ∈ {σ, µ}*and*A^{τ}n =M^{τ}n(aI+bS)L^{n}*. The limit operators*W^{t}(A^{τ}n),
t∈ {3,4},*belong to the algebra*algT(PC),*and their symbols are given by*

smb_{W}^{3}(A^{τ}n)(t, λ) =a(1) +b(1)·

1 : Imt >0,

−1 : Imt <0,

icot

π

4 +_{4}^{i}log_{1}_{−}^{λ}_{λ}

:τ ∈ {σ, µ}, t= 1, icot

π

4 −_{4}^{i}log_{1}_{−}^{λ}_{λ}

:τ =σ, t=−1,

−icot

π

4 +_{4}^{i} log_{1}_{−}^{λ}_{λ}

:τ =µ, t=−1,
*and*

smb_{W}^{4}(A^{τ}n)(t, λ) =a(−1)−b(−1)·

1 : Imt >0,

−1 : Imt <0,

−icot

π

4 −4^{i}log_{1}_{−}^{λ}_{λ}

:t= 1,

−icot

π

4 +_{4}^{i}log_{1}_{−}^{λ}_{λ}

:t=−1.

*The respective closed curves (3.16) are*

Γ_{W}^{3}(A^{σ}n)=γr[a(1) +b(1), a(1)−b(1)]∪γℓ[a(1)−b(1), a(1) +b(1)],
Γ_{W}^{3}_{(}_{A}^{µ}_{n}_{)}=γ[a(1) +b(1), a(1)−b(1)],

Γ_{W}^{4}_{(}_{A}^{σ}_{n}_{)}= Γ_{W}^{4}_{(}_{A}^{µ}_{n}_{)}

=γℓ[a(−1)−b(−1), a(−1) +b(−1)]∪γr[a(−1) +b(−1), a(−1)−b(−1)].

We remark that the limit operatorsW^{t}(M^{τ}n(aI+bS)L^{n}),t= 3,4,are invertible if they
are Fredholm with index0[10, Corollary 4.9].

LEMMA3.10 (Lemma 4.2 and Lemma 4.6 in [10]). LetA^{τ}n=M^{τ}n(aI+bS)Ln,*where*
τ∈ {σ, µ}*.*

*(a) If*aI+bS:L^{2}_{ν}−→L^{2}_{ν}*is Fredholm, then*W^{3}(A^{σ}n)*and*W^{4}(A^{τ}n)*are invertible.*

*(b) The operator*W^{3}(A^{µ}n)*is invertible if and only if*|a(1)|>|b(1)|*.*

We turn to the limit operators ofB^{±}k and verify thatA,A^{σ},A^{µ} ∈algT(PC). For this
we recall the following lemma.

LEMMA 3.11 (Lemma 7.1 in [12] and Lemma 4.5 in [13]). Suppose that the Mellin
*transform*y(z)b *of the function*y: (0,∞)−→C*is analytic in the strip*

1

2 −ε <Rez < 1
2 +ε
*for some*ε >0*and that*

sup

1

2−ε<Rez<^{1}_{2}+ε

d^{k}

dz^{k}y(z)(1 +b |z|)^{k}

<∞, k= 0,1, . . .

*Then,*y: (0,∞)−→C*is infinitely differentiable, the operators*M_{±}1,fM_{±}1∈ L(ℓ^{2})*defined*
*by*

M_{+1}:=

y

j+^{1}_{2}
k+^{1}_{2}

1
k+^{1}_{2}

∞ j,k=0

, Mf_{+1}:=

y

j+ 1 k+ 1

1 k+ 1

∞ j,k=0

,
*and*

M_{−}_{1}:=

(−1)^{j}^{−}^{k}y
j+^{1}_{2}

k+^{1}_{2}
1

k+^{1}_{2}
∞

j,k=0

, Mf_{−}_{1}:=

(−1)^{j}^{−}^{k}y
j+ 1

k+ 1 1

k+ 1 ∞

j,k=0

*belong to the algebra*algT(PC), and their symbols are given by
smb^{M}+1(t, λ) = smb_{f}M_{+1}(t, λ) =

( yb

1

2 +_{2π}^{i} log_{1}_{−}^{λ}_{λ}

:t= 1,

0 :t∈T\{1},

*and*

smb^{M}−1(t, λ) = smb_{f}M_{−1}(t, λ) =
( yb

1

2+_{2π}^{i} log_{1}_{−}^{λ}_{λ}

:t=−1,

0 :t∈T\{−1}.

Fork∈N, setg^{−}_{k}(x) := 2h^{−}_{k}(x^{2})xandg^{+}_{k}(x) := 2h^{+}_{k}(x^{2})such that
b

g^{−}_{k}(z) =bh^{−}_{k}(^{z+1}_{2} ) and bg^{+}_{k}(z) =hb^{+}_{k}(^{z}_{2}).