The operator in these equations is supposed to be of the form aI+bS+B±with piecewise continuous functionsaandb

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)pⁿ(x)orµ(x)pⁿ(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 weightedL²-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, andh₀ : [−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)⁻¹² 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(−¹₄,³₄). 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)⁻¹² 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−1u(y)dy (y+x+ 2)^k +

m+

X

k=1

β_k⁺ πi

Z 1

−1

(1−x)^k−1u(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 functions¹, the right-hand side func- tionfis assumed to belong to the weightedL²-spaceL²_ν, andu∈L²_νstands for the unknown solution. The inner product in the Hilbert spaceL²_νis 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:L²_ν →L²_ν, u7→ 1

πi Z 1

−1

u(y) y− ·dy

be the Cauchy singular integral operator,aI :L²_ν →L²_ν,u7→ aube the operator of multi- plication bya, and

B^±k :L²_ν −→L²_ν, u7→ 1 πi

Z 1

−1

(1∓ ·)^k−1u(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⁺_kBk⁺

! u=f.

It is a well known fact that the single operators aI, S, and Bk^± are bounded in L²_ν; see [2, Theorem 1.16 and Remark 8.3]. This means that these operators belong to the Banach algebraL(L²_ν)of all bounded and linear operatorsA:L²_ν −→ L²_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−x²,ϕ(x) = √

1−x²,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+¹₂)s

√πcos^s₂ , Pn(coss) =sin(n+¹₂)s

√πsin₂^s . The zerosx^τ_jnofp^τ_n(x)are given by

x^σ_jn= cosj−¹2

n π, x^ϕ_jn= cos jπ

n+ 1, x^ν_jn= cosj−¹2

n+¹₂π, x^µ_jn= cos jπ n+¹₂, forj = 1,· · · , n. We introduce the Lagrange interpolation operator L^τn defined for every functionf : (−1,1)→Cby

L^τnf = 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+¹₂ , λ^µ_jn=π(1−x^µ_jn) n+¹₂ .

The collocation method seeks an approximationun∈L²_νof 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_ndenotes 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:L²_ν →L²_ν, u7→

nX−1 j=0

hu,pejiνpej

and the weighted interpolation operatorsM^τn:=µL^τnµ⁻¹I, then the collocation system (2.3) can be written as an operator equation

(2.4) A^τn:=M^τnALⁿun=M^τnf, un ∈imLⁿ,

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^τn are invertible for all sufficiently largen∈Nand if the norms(A^τn)⁻¹Ln

L(L²

ν)are uniformly bounded. If the collocation method is stable, then the strong convergence ofA^τnLⁿtoA ∈ L(L²_ν)as well as the convergenceM^τnf −→f inL²_νimply the convergence of the approximationsunto the exact solutionuinL²_ν. This can be seen from the estimate

kLⁿu−unkν =A⁻n¹Lⁿ(AⁿLⁿu− Aⁿun)_ν

≤A⁻n¹Lⁿ

L(L2

ν)(kAⁿLⁿu− Aukν+kf− M^τnfkν),

which also shows that, for getting convergence rates, one has to estimate the errorsLnu−u andAnLnu− Auwith the solution uand the errorM^τnf −f with the right-hand side f. The technique, which we use to prove stability, includes the proof of strong convergence A^τnLⁿ−→ A; cf. the definition of the algebra F in Section 3.1. For M^τnf −→ 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. Byℓ² 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

Wⁿ :L²_ν −→L²_ν, u7→

nX−1 j=0

hu,pen−1−ji^νpej, Pⁿ :ℓ²−→ℓ², (ξj)_j=0^∞ 7→(ξ0,· · ·, ξn−1,0, . . .), and, forτ ∈ {σ, µ},

Vn^τ : imLn −→imPn, u7→

ω_n^τp

1 +x^τ_1nu(x^τ_1n), . . . , ω_n^τp

1 +x^τ_nnu(x^τ_nn),0, . . . , Ven^τ : imLn −→imPn, u7→

ω_n^τp

1 +x^τ_nnu(x^τ_nn), . . . , ω_n^τp

1 +x^τ_1nu(x^τ_1n),0, . . . , whereω^σ_n=p_π

n andω_n^µ=q _π

n+¹₂. LetT ={1,2,3,4}and set

X⁽¹⁾=X⁽²⁾=L²_ν, X⁽³⁾=X⁽⁴⁾=ℓ², L⁽¹⁾n =L⁽²⁾n =Lⁿ, L⁽³⁾n =L⁽⁴⁾n =Pⁿ, and defineE^n(t): imLⁿ −→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 notionLⁿ,Wⁿ,. . .instead ofLⁿ|^imLn,Wⁿ|^imLn,. . . , 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)⁻¹ξ= (ω_n^τ)⁻¹ Xn k=1

p 1

1 +x^τ_knξk−1ℓe^τ_kn and

(Ve_n^τ)⁻¹ξ= (ω_n^τ)⁻¹ Xn k=1

p 1

1 +x^τ_knξn−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(Aⁿ) := lim

n→∞En^(t)Aⁿ 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≥1kAnLnkL(L²

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^tEn^(t)+Cⁿ

! ,

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 setJforms a two-sided closed ideal in theC^∗-algebraF. Moreover, a sequence(Aⁿ)∈Fis stable if and only if the operatorsW^t(Aⁿ) : X^(t) → X^(t),t ∈ T,and the coset(Aⁿ) +J ∈ F/Jare invertible.

3.2. The collocation sequence as an element of the Banach algebraF. For the inves- tigation of the stability of the collocation method(A^τn) = (M^τnALn), 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:L²_ν→L²_ν, u7→

X∞ j=0

hu,pejiνRj, (3.1)

J²:L²_ν→L²_ν, u7→

X∞ j=0

hu,peji^ν√

1−x Uj,

J³:L²_ν→L²_ν, u7→

X∞ j=0

hu,peji^ν 1

√1 +xTj, and the shift operator

(3.2) V :L²_ν →L²_ν, u7→

X∞ j=0

hu,pejiνpej+1.

The adjoint operatorsJ1^∗,J2^∗,J3^∗,V^∗:L²_ν→L²_νare given by J1^∗u=J1⁻¹u=

X∞ j=0

hu, Rjiνpej, J2^∗u=J2⁻¹u= X∞ j=0

u,√

1−x Uj

νepj

and

J3^∗u=J3⁻¹u= X∞ j=0

u, 1

√1 +xTj

ν

e

pj, V^∗u= X∞ j=0

hu,pej+1i_νpej.

Finally, we denote byI= [δjk]^∞_j,k=0the identity inℓ²and byS,e S^τ:ℓ²−→ℓ²the 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^τn=M^τnALn. Then, forτ ∈ {σ, µ}, we have(A^τn)∈Fand

W¹(A^τn) =A, W²(A^τn) =





J1⁻¹(aJ1+ibI) :τ=σ, J2⁻¹(aJ2−ibJ3V) :τ=µ, W³(A^τn) =a(1)I+b(1)S^τ, W⁴(A^τn) =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 toW⁴(A^σn) =a(−1)I−b(−1)eSand not toW⁴(Aⁿ) =a(−1)I−b(−1)S^σas formulated in [10, Proposition 3.5].

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

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

PROPOSITION 3.4. Leta, b ∈ PC, A = aI +bS +

m−

X

k=1

β⁻_kBk⁻ +

m+

X

k=1

β_k⁺B⁺k, and A^τn=M^τnALn. Then, forτ ∈ {σ, µ}, we have(A^τn)∈Fand

W¹(A^τn) =A, W²(A^τn) =

( J1⁻¹(aJ¹+ibI) :τ=σ, J2⁻¹(aJ²−ibJ³V) :τ=µ, W³(A^τn) =a(1)I+b(1)S^τ+A^τ+K^τ, W⁴(A^τn) =a(−1)I−b(−1)eS+A+K,

where the operatorsA,A^τ∈ L(ℓ²)are defined as A=

m−

X

k0=1

β_k⁻₀

2h⁻_k

0

(j+¹₂)² (k+¹₂)²

j+¹₂ (k+¹₂)²

∞ j,k=0

, (3.3)

A^σ=

m+

X

k0=1

β_k⁺

0

2h⁺_k

0

(j+¹₂)² (k+¹₂)²

1 k+¹₂

∞ 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 whereK,K^τ :ℓ²−→ℓ²are 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⁻_kBk⁻+

mP+

k=1

β_k⁺Bk⁺. For this, we need the Mellin transform

b y(z) :=

Z _∞

0

y(x)x^z−1dx

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)^kbh_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.

PROPOSITION3.5. Leta, b∈PC,A=aI+bS+

m−

X

k=1

β_k⁻B⁻k+

m+

X

k=1

β_k⁺B⁺k :L²_ν −→L²_ν. (a) The operatorAis 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.

• Ifaorbhas a jump atx∈(−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.

• Forx=±1, there holds a(1) +b(1)icot ^π₄ −iπξ

+

m+

X

k=1

β_k⁺bh⁺_k(¹₄−iξ)6= 0, − ∞< ξ <∞, and

a(−1) +b(−1)icot ^π₄ +iπξ +

m−

X

k=1

β⁻_khb⁻_k(³₄−iξ)6= 0, − ∞< ξ <∞. (b) IfAis Fredholm and if the coefficientsaandbhave finitely many jumps, then the Fredholm index ofA : L²_ν −→ L²_ν is equal to minus the winding number of the closed continuous curveΓ_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 pointsxi, i= 1, . . . , N,of the functionsaandbchosen such thatx0 :=−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 curvesΓ_±, connecting the point1with one of the end points ofΓ1andΓN+1, are given by the formulas

Γ+:=

(a(1) +b(1)icot ^π₄−iπξ

+Pm+

k=1β_k⁺hb⁺_k(¹₄−iξ)

a(1)−b(1) : −∞ ≤ξ≤ ∞

)

and

Γ₋:=

(a(−1) +b(−1)icot ^π₄ +iπξ

+Pm−

k=1β_k⁻hb⁻_k(³₄−iξ)

a(−1)−b(−1) :

− ∞ ≤ξ≤ ∞ )

. (c) IfAis Fredholm and ifm₋= 0orm+= 0, thenAis 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 ofa(−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 ofa(1) = 0,m+= 3,βk⁺= 1.

The equalities

(a(1) +b(1)icot ^π₄ −iπξ

a(1)−b(1) : −∞ ≤ξ≤ ∞ )

=γr[c(1),1].

(3.9) and

(a(−1) +b(−1)icot ^π₄ +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 ³₄−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 operatorW². So we derive from [10, Lemma 4.4 and Corollary 4.5].

LEMMA3.6. LetA^τn =M^τn(aI+bS)Ln,τ∈ {σ, µ}.

(a) The operatorW²(A^σn) is invertible in L²_ν if and only if A = aI +bS has this property.

(b) IfaI+bS:L²_ν−→L²_νis invertible, then the invertibility ofW²(A^µn) :L²_ν−→L²_ν is equivalent to the condition|a(1)|>|b(1)|, which is again equivalent to the con- ditionRec(1)>0.

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

LEMMA 3.7. Leta, b ∈PC,τ ∈ {σ, µ},andA:=aI+bS : L²_ν → L²_ν, as well as A^τn :=M^τnALn. IfAis Fredholm, then the second limit operatorW²(A^σn) :L²_ν →L²_νis also Fredholm, where

(3.11) indW²(A^σn) =−indA.

IfA,W²(A^µn) :L²_ν −→L²_νare Fredholm, then (3.12) indW²(A^µn) =

−indA : Rec(1)>0,

−indA −1 : Re c(1)<0. Proof. LetindA=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]f¹

2(λ) :x=−1,

wheref_α(λ) = sinπαλ

sinπα e^{−iπα(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) = _2π¹ 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) = _2π¹ argd(x)satisfying the respective conditions (3.14). Since (cf. again the proof of [10, Lemma 4.4])

indW²(A^σn) = ind (bI −aS) and β(x) =−1

2 −α(x), we have

−indW²(A^σn) = 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 thatW²(A^µ_n) :L²_ν −→L²_νis Fredholm. From the proof of [10, Lemma 4.5], we have

−indW²(A^µn)∈Z∩

−α(1)−1

4,−α(1) +3 4

. In view of (3.15), we getκ∈ −α(1)−¹₄,−α(1) +³₄

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)−¹₄,−α(1) +³₄ if and only ifα(1) +¹₄ <windc(x, µ)< α(1) +³₄,i.e.,Rec(1)<0.

Observe that the Fredholmness of the operatorA:=aI+bS:L²_ν→L²_νimplies 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)])

W²(A^µn) =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^τn). For that, we consider theC^∗-algebraL(ℓ²)of all linear and continuous operators inℓ². ByalgT(PC)we denote the smallestC^∗-subalgebra ofL(ℓ²)generated by the Toeplitz matrices

b gj−k ∞

j,k=0

with piecewise continuous generating functionsg(t) :=P

ℓ∈Zbgℓt^ℓdefined 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) mapsmbfrom algT(PC)into a set of complex valued functions defined onT×[0,1], which sends each R∈algT(PC)to the functionsmbR(t, λ), which is called symbol ofRand 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 onalgT(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 operatorR ∈ algT(PC)is Fredholm if and only if smb^R(t, λ) 6= 0for all (t, λ)∈T×[0,1].

(d) For any Fredholm operatorR∈algT(PC), the index ofRis the negative winding number of the closed curve

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

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

where the orientation ofΓR is given in a natural way by the parametrization ofT and[0,1].

(e) An operatorR∈algT(PC)is compact if and only if the symbolsmb^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^τn) 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.

LEMMA3.9. Letτ ∈ {σ, µ}andA^τn =M^τn(aI+bS)Lⁿ. The limit operatorsW^t(A^τn), t∈ {3,4},belong to the algebraalgT(PC),and their symbols are given by

smb_W³(A^τn)(t, λ) =a(1) +b(1)·



















1 : Imt >0,

−1 : Imt <0,

icot

π

4 +₄ⁱlog₁₋^λ_λ

:τ ∈ {σ, µ}, t= 1, icot

π

4 −₄ⁱlog₁₋^λ_λ

:τ =σ, t=−1,

−icot

π

4 +₄ⁱ log₁₋^λ_λ

:τ =µ, t=−1, and

smb_W⁴(A^τn)(t, λ) =a(−1)−b(−1)·













1 : Imt >0,

−1 : Imt <0,

−icot

π

4 −4ⁱlog₁₋^λ_λ

:t= 1,

−icot

π

4 +₄ⁱlog₁₋^λ_λ

:t=−1.

The respective closed curves (3.16) are

Γ_W³(A^σn)=γr[a(1) +b(1), a(1)−b(1)]∪γℓ[a(1)−b(1), a(1) +b(1)], Γ_W³_(A^µ_n)=γ[a(1) +b(1), a(1)−b(1)],

Γ_W⁴_(A^σ_n)= Γ_W⁴_(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^τn(aI+bS)Lⁿ),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^τn=M^τn(aI+bS)Ln,where τ∈ {σ, µ}.

(a) IfaI+bS:L²_ν−→L²_νis Fredholm, thenW³(A^σn)andW⁴(A^τn)are invertible.

(b) The operatorW³(A^µn)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 transformy(z)b of the functiony: (0,∞)−→Cis analytic in the strip

1

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

sup

1

2−ε<Rez<¹₂+ε

d^k

dz^ky(z)(1 +b |z|)^k

<∞, k= 0,1, . . .

Then,y: (0,∞)−→Cis infinitely differentiable, the operatorsM_±1,fM_±1∈ L(ℓ²)defined by

M₊₁:=

y

j+¹₂ k+¹₂

1 k+¹₂

∞ j,k=0

, Mf₊₁:=

y

j+ 1 k+ 1

1 k+ 1

∞ j,k=0

, and

M₋₁:=

(−1)^j−ky j+¹₂

k+¹₂ 1

k+¹₂ ∞

j,k=0

, Mf₋₁:=

(−1)^j−ky j+ 1

k+ 1 1

k+ 1 ∞

j,k=0

belong to the algebraalgT(PC), and their symbols are given by smb^M+1(t, λ) = smb_fM₊₁(t, λ) =

( yb

1

2 +_2πⁱ log₁₋^λ_λ

:t= 1,

0 :t∈T\{1},

and

smb^M−1(t, λ) = smb_fM₋₁(t, λ) = ( yb

1

2+_2πⁱ log₁₋^λ_λ

:t=−1,

0 :t∈T\{−1}.

Fork∈N, setg⁻_k(x) := 2h⁻_k(x²)xandg⁺_k(x) := 2h⁺_k(x²)such that b

g⁻_k(z) =bh⁻_k(^z+1₂ ) and bg⁺_k(z) =hb⁺_k(^z₂).