Quadrature Methods for Singular Integral Equations of Mellin Type Based on the Zeros of Classical Jacobi Polynomials

: In this paper we formulate necessary conditions for the stability of certain quadrature methods for Mellin type singular integral equations on an interval. These methods are based on the zeros of classical Jacobi polynomials, not only on the Chebyshev nodes. The method is considered as an element of a special C ∗ -algebra such that the stability of this method can be reformulated as an invertibility problem of this element. At the end, the mentioned necessary conditions are invertibility properties of certain linear operators in Hilbert spaces. Moreover, for the proofs we need deep results on the zero distribution of the Jacobi polynomials.


Introduction
The present paper is part of the efforts done during the last three decades to establish necessary and sufficient conditions for the stability of numerical methods for singular integral equations by using so called C * -algebra techniques.The integral equations under consideration contain strong singular integral operators of Cauchy and Mellin type.In general, they are of the form a(x)u(x) + b(x) where the functions a, b, c ± : [−1, 1] −→ C , f : (−1, 1) −→ C , H ± : R + −→ C as well as K : (−1, 1) × (−1, 1) −→ C are given and u : (−1, 1) −→ C is looked for.As usual, R and C denote the sets of real and complex numbers, respectively.Moreover, by R + = {t ∈ R : t > 0} we refer to the set of positive real numbers.The minimal conditions on the given functions are the piecewise continuity of the coefficient functions a, b , and c ± as well as the continuity of the kernel functions H ± (t) and K(x, y) .Moreover, the righthand side f should belong to a Hilbert space L 2 α,β .Equation ( 1) is considered in this space L 2  α,β and written shortly as For real numbers α, β > −1 , the Hilbert space L 2 α,β := L 2 α,β (−1, 1) is defined by the inner product f , g α,β = where v α,β (x) = (1 − x) α (1 + x) β is a classical Jacobi weight.Hence the norm in L 2 α,β is given by f α,β = f , f α,β .We call a function a : [−1, 1] −→ C piecewise continuous if it is continuous at ±1 , the one-sided limits a(x ± 0) exist for all x ∈ (−1, 1) , and at least one of them coincides with the function value a(x) .The set of these piecewise continuous functions is denoted by PC := PC[−1 , 1] .For a continuous function H : (0, ∞) −→ C, i.e., H ∈ C(R + ) , and a continuous function K : (−1, 1) × (−1, 1) −→ C the Mellin-type operator M H and the integral operator K are defined by and respectively.Moreover, S denotes the Cauchy singular integral operator defined in the sense of a principal value integral as Furthermore, the operator of multiplication by a bounded function a : α,β itself denotes the identity operator.If B : L 2 α,β −→ L 2 α,β is another operator, then we use the abbreviation aB for the product of the multiplication operator aI and the operator B, i.e., aB := aI B .
The numerical methods for the approximate solution of Equation (1), which are of interest here, are collocation and collocation-quadrature methods, which we will describe later on in more detail.Since we know that solutions of (1) usually contain singularities at the endpoints of the integration interval, in these methods we look for an approximate solution u n (x) to u(x) of the form u n (x) = v ρ 0 ,τ 0 (x)p n (x) = (1 − x) ρ 0 (1 + x) τ 0 p n (x) , (5) where ρ 0 , τ 0 > −1 are real numbers and p n (x) is a polynomial of degree less than n .In case of a collocation method we choose a sequence of collocation points where n ∈ N-the set of positive integers, and try to determine u n (x) with the help of the conditions (Au n )(x jn ) = f n (x jn ) , j = 1, . . ., n , (7) where f n ∈ v ρ 0 ,τ 0 P n is an approximation to f , P n denotes the space of algebraic polynomials of degree less than n , and v ρ 0 ,τ 0 P n is considered as a subspace of L 2 α,β (i.e., equipped with the norm of L 2 α,β ).To realize a so called collocation-quadrature method, in a first step we approximate the integral operators M ± H and K with the help of a quadrature method of interpolation type where x − x kn x jn − x kn , k = 1, . . ., n , are the fundamental Lagrange interpolation polynomials with respect to the nodes x kn .Thus, the Mellin type operator is approximated by and the Fredholm integral operator K by λ kn ω(x kn ) K(x, x kn )u(x kn ) .
In the second step, we again use the nodes x kn as collocation points and try to determine u n (x) by solving Note that, in the collocation-quadrature method (9), the quadrature rule (8) is not applied to (Su n )(x) .Both the collocation method (7) and the collocation-quadrature method (9) can be written as an operator equation where A n : v ρ 0 ,τ 0 P n −→ v ρ 0 ,τ 0 P n is a linear operator (cf.( 44) and ( 45)).The definition of the stability of the method (10) or, in other words, of the stability of the sequence (A n ) = (A n ) ∞ n=1 of the operators A n , includes the unique solvability of (10) for all sufficiently large n and the uniform boundedness of the inverse operators A −1 n : v ρ 0 ,τ 0 P n −→ v ρ 0 ,τ 0 P n (see Definition 1).Now, the application of C * -algebra techniques is based on the idea to consider the sequence (A n ) as an element of a suitable C * -algebra and to translate stability into in- vertibility modulo zero sequences of this element (see Section 4).To find necessary and sufficient conditions for the stability of the sequence (A n ) it is necessary to segue to certain C * -subalgebras and quotient algebras (cf.Proposition 2).In Table 1 we give an overview on the efforts done in the literature during the last 25 years to equations of type (1) or (2), where we ignore the Fredholm integral operator K .Table 1.Cases already considered in the literature.
There exists a series of papers (see, for example, refs.[17][18][19][20][21]) devoted to the application of the Nyström method to Fredholm integral equations of the form with non-compact integral operators K 1 , for which the operators M ± H are respective examples.Thereby, Equation ( 11) is studied in spaces of continuous or weighted continuous functions.However, since the idea of proving stability and convergence of the Nyström method is essentially based on the concept of collectively compact operator sequences, which works only for compact operators K 1 + K 2 , in the mentioned papers there is assumed that the norm of the operator K 1 is less than 1 and that this is true uniformly also for the approximating operators K 1n .Then, for I + K 1 and I + K 1n one can use the Neumann series argument.In the present paper, we are not constrained to apply such a condition on the norm of an operator.

Properties of Integral Operators with Mellin Kernels
Let us start with collecting some statements on integral operators of interest here and already proved in the literature.Lemma 1 ([22], Proposition 3.13).Let β ∈ (−1, 1) and H ∈ C(R + ) .Moreover, we assume that there are real numbers p, q with p < q such that 1+β 2 ∈ (p, q) and such that lim t→+0 t p k(t) = 0 and lim t→∞ t q k(t) = 0 .
Then, for all α ∈ (−1, 1) , the integral operator The following corollary is an immediate consequence of the previous lemma.
If the function is continuous and bounded, then the operator For ρ ∈ R , we introduce the weighted L 2 -space L 2 ρ := L 2 ρ (R + ) defined by the norm ) is a bounded function on (−1, 1).
For n ∈ N 0 and H ∈ C n (R + ) , we define the operators Let R = R(−1, 1) and C = C(−1, 1) denote the sets of all functions f : (−1, 1) −→ C being bounded and Riemann integrable as well as continuous on each closed subinterval of (−1, 1), respectively.For S ∈ {R, C} and ψ, χ ∈ R with ψ, χ ≥ 0 , by S b ψ,χ we refer to the set of all functions f ∈ S , for which the function v ψ,χ f is bounded on (−1, 1) .If we introduce the norm ψ,χ , .ψ,χ,∞ becomes a Banach space.Moreover, by S ψ,χ we denote the set of all functions f ∈ S(−1, 1) , for which the finite limits exist, and by S ψ,χ the subspace of S ψ,χ of those functions f ∈ S(−1, 1) , for which the limits in (12) are equal to zero if ψ > 0 or χ > 0, respectively.The spaces S ψ,χ and S ψ,χ are closed subspaces of S b ψ,χ and, consequently, also Banach spaces.Finally, for ψ 0 , χ 0 > 0 , set Note that Moreover, assume the map Applying the Arzela-Ascoli theorem delivers the assertion.
For z ∈ C and a measurable function f : (0, ∞) −→ C , for which t z−1 f (t) is integrable on each compact subinterval of (0, ∞) , the Mellin transform f (z) is defined as if this limit exists.Moreover, for ξ ∈ R and p < q , let Γ ξ = {z ∈ C : Re z = ξ} and Lemma 5. Let p, q ∈ R with p < q and f ∈ L 2 2ξ−1 ∩ C(R + ) for every ξ ∈ (p, q) .Then (a) the Mellin transform f belongs to the space C 0 (Γ ξ ) for every ξ ∈ (p, q) and is holomorphic in the strip Γ p,q .
For a function H ∈ C(R + ) and a real number ξ , we formulate the following conditions: (A 0 ) There exist real numbers p and q with p < q such that H ∈ L 2 2p−1 (R + ) ∩ L 2 2q−1 (R + ) , ξ ∈ (p, q) , and sup (1 + |z|) 1+τ H (z) : z ∈ Γ p 0 ,q 0 < ∞ for all intervals [p 0 , q 0 ] ⊂ (p, q) and some τ = τ(p 0 , q 0 ) > 0 .(A 1 ) There exist real numbers p and q with p < q such that H ∈ for all intervals [p 0 , q 0 ] ⊂ (p, q) and some τ = τ(p 0 , q 0 ) > 0 .We set where c ± ∈ L ∞ (−1, 1) and H ± ∈ C(R + ) .The following lemma is an application of [10] (Theorem 4.12) to the operator in (16).16) is Fredholm if and only if the closed curve does not contain the point 0 , where In this case, the Fredholm index of A is equal to the negative winding number of the curve Γ A , where the orientation of Γ A is due to the above given parametrizations of Γ ± A .
By p α,β n (x) we refer to the normalized (with respect to the inner product ., .α,β ) polynomials with positive leading coefficient.If we set then (see [24] (Chapter I, (4.10)) and from (17) we get where we have with certain real numbers A n , B n , and C n .Together with (20) this yields Let . With the help of the relations (18) and Γ(z + 1) = zΓ(z) we get Furthermore, Using the orthogonality properties of p α,β n (x) and ( 19) we obtain, for n > 1 , and, for n = 1 , as well as Hence, relation (22) can be written in the form where and In what follows, by C we will denote a positive constant, which can assume different values at different places, and we will write C = C(x, n, . . . . Then (cf.[25] (Theorem 5)) Since Jacobi weights are so-called doubling weights (see, for example, [26] (Section 3.2.1,Exercise 3.2.4),we also have (see [27] (Theorem 1)) which can equivalently be written as (see [28] (Theorem 3.2) and cf.[26] (Exercise 3.2.25)) Note that, due to (28), and Hence, for k = 1, . . ., n , and as well as which implies, due to (28), Proof.In view of (26) we have and it remains to take into account , and consider a system n for k = 2, . . ., n and n ∈ N .Moreover, let m be a fixed positive integer.Then there exists a positive constant (36) holds true for all Q ∈ P mn , n ∈ N , where P n denotes the set of all algebraic polynomials of degree less than n .

Corollary 2. Assume 2γ
where ϑ n+1,n = π and ϑ 0n = 0 .Moreover, let m be a fixed positive integer.Then there exists a positive constant . Now, we consider the following system of nodes If we apply Lemma 9 to this system, we immediately arrive at our assertion.
In the particular case x kn = x γ,δ kn and γ 0 = δ 0 = 0 , from (35) and Corollary 2 we get for all Q ∈ P mn and with C = C(n, Q) .
for all Q ∈ P n , n ∈ N , where the second inequality holds true without condition (a).

The Algebra alg T (PC)
By alg T (PC) we denote the smallest C * -subalgebra of the algebra L( 2) of all linear and bounded operators on the Hilbert space 2 generated by the Toeplitz matrices with piecewise continuous generating functions 2) generated by all Toeplitz matrices T( f ) with piecewise continuous generating function f : T −→ C .It is well known (see Chapter 16 in [30]) that there exists an isometrical isomorphism smb from the quotient algebra L T ( 2 )/K( 2 ) (K( 2 )-the ideal in L( 2) of compact operators) onto the algebra (C(M), .∞ ) of all complex valued and continuous functions on the compact space M = T × [0, 1] , where the topology on M is defined by the neighborhoods (cf.Theorem 16.1 in [30]) Proposition 1 ([30], Theorem 16.2, [31], Theorem 4.97).The mapping smb has the following properties: where smb R := smb(R) .(b) If R ∈ alg T (PC) is Fredholm, then the index is equal to the negative winding number of the closed curve where the orientation of Γ R is due to the above given parametrization.
, then, for every s > 0 , the matrix defines an operator M ∈ L( 2 ) , which belongs to the algebra alg T (PC) , and its symbol is given by

The Collocation-Quadrature Method
We consider the integral Equation (cf.( 16)) where , and the kernel function K(x, y) of the integral operator K (cf.( 4)) is supposed to be continuous on (−1, 1) × (−1, 1) .In order to get approximate solutions, we use a polynomial collocation-quadrature method.To introduce that method, we need some further notations.Let n ∈ N and γ, δ, ρ, τ > −1 be real numbers.For u : (−1, 1) −→ C, the Lagrange interpolation operator L γ,δ n is defined by To the integral operators M ± H and K , we associate the quadrature operators respectively.For certain ρ, τ > −1 , the collocation-quadrature method seeks for approximations u n ∈ L 2 α,β of the form to the exact solution of (40) by solving where P n stands for the set of all algebraic polynomials of degree less than n and the functions f n : (−1, 1) −→ C are continuous and satisfy ϑ −1 f n ∈ P n as well as lim We set n=0 forms a complete orthonormal system in L 2 α,β .Using the weighted fundamental Lagrange interpolation polynomials we can write u n as If we introduce the Fourier projections and the weighted Lagrange interpolation operator then the collocation system (42) can be written as Note also that, with the introduced notations, the assertion of Corollary 3 remains true for It is well known that, in the investigation of numerical methods for operator equations, the stability of the respective operator sequences plays an essential role.
Definition 1.We call the sequence

If the method is stable and if
α,β -convergence of the solution u n of (44) to the (unique) solution u ∈ L 2 α,β of (40).This can be seen from the estimate

C * -Algebra Framework
In order to investigate the stability of the collocation-quadrature method, we use specific C * -algebra techniques.With the help of those tools, we are able to transform our stability problem into an invertibility problem in an appropriate C * -algebra.The sequence (A n ) is considered as an element of such a C * -algebra.To define that algebra, we need some operators and spaces.
Let n ∈ N and Let the operators F n ∈ L(im P n ) be given by Moreover, for t ∈ T := {1, 2, 3} , we define E (t) n by

E
(1) where n are invertible with Moreover, Whence, in view of (46), the following lemma is proved.
Lemma 13.For r, t ∈ T with r = t , the operators E r,t n := E n converge together with their adjoints weakly to the zero Proof.Since the operators E r,t n are uniformly bounded, it suffices to verify the convergence on a dense subset.At first, we consider the operator E 2,1 n = V n L n .Let k, m ∈ N 0 be arbitrary but fixed and n > max{m, k} as well as Taking into account ρ > −1, we conclude the weak convergence of E 2,1 n to the zero operator.Similarly, using (32) instead of (31), we get Hence, E 3,1 n converges weakly to the zero operator.Fix k ∈ N 0 and p(x) = ϑ(x)(1 − x 2 )p(x) with p(x) being a polynomial.Then, for all sufficiently large n ∈ N , . In virtue of [32] (2.2) there is a constant Note that, due to (35), Consequently, n converges weakly to zero.Analogously, we get the same for E 1,3 n .Let s, t ∈ {2, 3} with s = t .The weak convergence of E s,t n follows by the relations n also E r,t n * converges weakly to the zero operator, which follows immediately from the definition of the weak convergence.The lemma is proved.
For all what follows we assume that ρ − γ, τ − δ ∈ − 1 2 , 3  2 .By F we denote the set of all sequences (A n ) := (A n ) ∞ n=1 of linear operators A n : im L n → im L n for which the strong limits exist for all t ∈ T .If we provide F with the algebraic operations , and the supremum norm one can easily check, that F becomes a C * -algebra with the identity element (L n ) .
Corollary 4. Let T t ∈ L(X t ) , t ∈ T be compact operators, i.e., T t ∈ K(X t ) .Then the sequences n belong to F , where Proof.This is due to the compactness of T t : X t −→ X t and the weak convergence of E r,t n as well as E r,t n * to the zero operator if r = t (see Corollary 4).
Proof.Of course W t : F −→ L X (t) , t ∈ T are * -homomorphisms.The relation W t L(X (t) ) = 1 follows from the fact that * -homomorphism are bounded by 1 and that The convergence together with Corollary 4 deliver that Proposition 2 ([33,34], Theorem 10.33).The set J 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.

The Limit Operators of the Collocation-Quadrature Methods
In this section, under certain conditions, we prove that the sequence (A n ) of the collocation quadrature method, defined in (45), belongs to the algebra F from the previous section.We do this by determine the limit operators W t (A n ) , t = 1, 2, 3 , and proving that also the sequences of the respective adjoint operators converge strongly.
The following corollary is an immediate consequence of the previous lemma and concerned with the case be a continuous function.Then Proof.Fix ψ 0 , χ 0 such that ψ < ψ 0 < 1+α 2 and χ < χ 0 < 1+β 2 .By assumption f x ∈ C ψ 0 ,χ 0 for all x ∈ [−1, 1] , where f x (y) := f (x, y) .Moreover, lim Suppose the assertion of the lemma is not true.Then, there are an ε > 0 and a sequence Hence, for every k ∈ N , there is an ≥ ε, and we can assume that Due to our assumptions we have 2 ) and the Banach-Steinhaus theorem, Moreover, there is an For k ≥ k 0 , we get the contradiction and the lemma is proved.
Moreover, let the map Proof.By Corollary 6 we have Define (50) Furthermore, for γ, δ > −1 , by S γ,δ n we refer to the Fourier operator given by From [36] (Theorem 1) we infer the following lemma.
Lemma 18.Let α, β, γ, δ, > −1 .Then, there is a constant C = C(n, f ) such that, for all n ∈ N and f ∈ L 2 α,β , the inequality holds true if and only if Corollary 7. Let a ∈ PC and α, β, γ, δ, γ 0 , are satisfied, then L γ,δ n aS Proof.The set P of all algebraic polynomials is dense in L 2 α,β .Moreover, for every p ∈ P , we have L γ,δ n aS γ 0 ,δ 0 n p = L γ,δ n ap for all sufficiently large n and, by Lemma 15, L γ,δ n ap −→ ap in L 2 α,β .Additionally, in view of Corollary 3 and Lemma 18, for f ∈ L 2 α,β we can Hence, the operators L γ,δ n aS , n ∈ N , are uniformly bounded.Now, the Banach-Steinhaus theorem gives the assertion.
and such that the function Proof.First of all, we notice that K : L −→ 0 holds true, it suffices to verify the convergence We define r, s ∈ {0, 1} by and Recall that the application of the operator K n to a function u ∈ C(−1, 1) can be written as (see ( 41)) We define Let u n = ϑp n ∈ im L n , i.e., p n ∈ P n .Due to the algebraic accuracy of the Gaussian rule, in case of r + s ≤ 1 as well as in case of deg p n < n − 1 and r = s = 1 , we have In case of deg p n = n − 1 and r = s = 1 , we write p n = ε n p γ+1,δ+1 n−1 n−1 p n .We get, due to the previous considerations, with κ n = n+γ+δ+1 2n+γ+δ+1 , where we took into account relations ( 21) and ( 25) as well as the orthogonality properties of p γ,δ n (x) and p γ,δ n+1 (x) .Consequently, for u ∈ L 2 α,β , we have We show, that L n as well as L * n converge strongly in L 2 α,β to the identity operator.For the convergence of L n , it suffices to show that, in case r = s = 1 , ϑS γ+1,δ+1 by Lemma 18.In the present situation, the last conditions are equivalent to the conditions which are satisfied in case r = s = 1 .Since, for f , g ∈ L 2 α,β , strongly, which is equivalent to S γ+1,δ+1 n −→ I strongly in L 2 2γ+2−ρ,2δ+2−τ .Due to Lemma 18, this is again equivalent to (60).As a consequence of these considerations we have that We have Hence, in order to show relation (61) we can try to apply Lemma 16 for α 0 = 2γ + 2r − ρ and β 0 = 2δ + 2s − τ instead of α and β, respectively, as well as for Since (ρ, γ) ∈ Ω and we have α 0 > −1 and, analogously, β 0 > −1 .Moreover, such that α 0 − γ and, analogously, β 0 − δ belong to the interval − 1 2 , 3  2 .The conditions are equivalent to ψ < 1+α 2 and χ < 1−β 2 .This all together implies that the function is continuous on [−1, 1] 2 and that Lemma 16 is applicable to f (x, y) with ψ 0 and χ 0 instead of ψ and χ .Hence, For the proof of (57) it remains to refer to (49).
Analogously we can prove the following corollary.
be continuous functions which vanish in a neighbourhood of the point −1 and are identically 1 in a neighbourhood of the point 1 .Then, for (ρ, γ), (τ, δ) ∈ Ω , we have where the limit operators are given by In what follows we identify an element (ξ 0 , . . ., ξ n−1 , 0, . ..) ∈ im P n with the respective element ξ k n−1 k=0 ∈ C n and the linear operator A n : im P n −→ im P n with its matrix representation A n = a jk n j,k=1 ∈ C n×n , i.e., For example, we have the representation Let us formulate the following condition for a positive kernel function.
(B) For the function H : R + −→ R + , there are a positive constant c M and a real number Corollary 11.Let α, β ∈ (−1, 1) and H ∈ C(R + ) be a positive function, which satisfies condition (A 1 ) for ξ = 1+β 2 and condition (B).Then, for Proof.Due to Lemma 12 it suffices to prove the assertion for t = 2 .Furthermore, in view of Corollary 9 in combination with Lemma 5,(b), we have only to prove the uniform boundedness of the operators H χ n : im P n −→ im P n with (cf.also (62)) , where χ : [−1, 1] −→ [0, 1] is a continuous function, which vanishes in a neighbourhood of the point 1 and is identically 1 in a neighbourhood of the point −1 .We use the notation . Let us note that the entries h n,χ jk of the matrix H χ n are positive and, in view of the choice of the function χ : We conclude, by additionally using condition (B), if condition (B),(a) is in force, and As a consequence of these estimates we have where , d ∈ {a, b} , and J n = δ n−1−j,k n−1 j,k=0 .For d ∈ {a, b} , consider the function g d : R + −→ R with Due to our assumptions, there exist p, q ∈ R with p < q such that H ∈ L 2 2p−1 (R + ) ∩ L 2 2q−1 (R + ) and ξ = 1+β 2 ∈ (p, q) .From that, we derive the uniform boundedness of the operators H χ n : 2 −→ 2 is a consequence of Lemma 11, inequality (66) and the fact that the norm of the operators J n : 2 −→ 2 is equal to 1.
Analogously to the previous one, we can prove the following corollary.
Proof.First of all, we notice that only the verification of (66) is necessary.Moreover, it is well known that θ Hence, we can give up the usage of a cutting-off function χ , and the matrices , where σ ∈ {α, β} .Finally, we have only to recall Lemma 11.
Let s > −1 and J s be the Bessel function of the first kind and of order s .We have It is well-known, that J s has countable infinitely many positive simple zeros, which accumulate only at infinity.We denote these zeros in increasing order by ψ s,k , k = 1, 2, . . .By using Legendre's duplication formula we get Lemma 20 ([37], Theorem 4.1).For k ∈ N fixed, the nodes x The relation p Corollary 13.For k ∈ N fixed, we have Lemma 21.Let j, k ∈ N , and ζ 1 , ζ 2 be real numbers which satisfy Then, for n tending to infinity, we have Proof.The first relation is a consequence of and Lemma 20.By applying (67) we get the second one.
For an arbitrary a ∈ PC , let us compute the sequence of the adjoints of the operators L γ,δ n aL n : L 2 α,β −→ L 2 α,β .We define integers r, s ∈ {0, 1} as in (59) and obtain, for functions In case of r + s ≤ 1 , we can proceed as follows using the algebraic accuracy of the Gaussian rule If r = s = 1 we use relations ( 21) and ( 25) as well as the orthogonality properties of the polynomials p and , where κ n = n+γ+δ+1 2n+γ+δ+1 .Consequently, in case r = s = 1 we have It is easy to see that L γ,δ n aL n converges strongly to aI in L 2 α,β (see the beginning of the proof of Lemma 22 below).Let us discuss the convergence of the adjoint operators.At first, we consider the operators on the right hand side of (68).For γ 0 = γ + r and δ 0 = δ + s , the conditions (55) and (56 Hence, by definition (59) of the numbers r, s ∈ {0, 1} , we can apply Corollary 8 together with the strong convergence of L n −→ I in L 2 α,β and get the strong convergence for r, s ∈ {0, 1} .In particular, we have the strong convergence of L γ,δ n aL n * for r + s ≤ 1 .From (70) we also get the strong convergence of Thus, due to formula (69), to prove the strong convergence of the operators L γ,δ n aL n * in case r = s = 1 , it remains to show that the operators converge in L 2 α,β strongly to the zero operator.Up to now we know that the operators Thus, in view of relation (69), also the operators C n : L 2 α,β −→ L 2 α,β are uniformly bounded, and it suffices to show their convergence on a dense subset of L 2 α,β .Such a subset is the space v γ+1−ρ,δ+1−τ ϑP : P ∈ P because of the relations 2 ), and the density of P in L 2 α,β .These results can be used for the proof of the following lemma.Nevertheless, we will give a shorter proof of the strong convergence of the adjoint operators.
For H ∈ C(R + ) , we define the integral operator We have the following relation Furthermore, one can easily show that the adjoint operator of M − H : α,β are uniformly bounded for some γ, δ with (ρ, γ), (τ, δ) ∈ Ω , then we have the strong convergences Proof.Due to the Banach-Steinhaus theorem, it suffices to show the convergence on a dense subset of L 2 α,β .We consider functions g(x) of the form where p(x) is an arbitrary polynomial.By choosing χ ∈ p, : n ∈ N < ∞ .Consequently, since .
At first, we consider the term M − n,H g − M − H g 0,χ,∞ . For f : (−1, 1) −→ C, we denote by From [38] ((5.1.35))follows where ϕ(x) = √ 1 − x 2 and the constant does not depend on f and n .We have with With the help of Lemma 3, we get . Again by Lemma 3 we can estimate and get M − H g ∈ C b 0,χ , which implies, due to Corollary 6, Let us turn to the strong convergence of the adjoint operators.Let f ∈ L 2 α,β , define r, s ∈ {0, 1} as in (59), and take g from the dense subset X rs defined in (71), what means g = v γ+r−ρ,δ+s−τ ϑP with P ∈ P .We set and get, for all sufficiently large n , where In case of r + s ≤ 1 , we use the algebraic accuracy of the Gaussian rule and obtain . Thus, where we used the abbreviation N(x) = v −γ−1,−δ−1 ϑN − n,H v α,β g (x) and where, for the second sum in (77), we get For the first sum in (77) we use the relations and conclude with the help of ( 21) , which is equal to If we apply Corollary 8 with a ≡ 1 , γ 0 = γ + r , and δ 0 = δ + s , then we see that α,β strongly to the identity operator.Hence, it remains to show the convergence in L 2 α,β .For this, we can take g from the subset X 0 rs = ϑv γ+r−ρ,δ+s−τ v 2,2 P : P ∈ P of X rs , which is also dense in L , Choose . With the help of (80) we can estimate and again using Lemma 3 we obtain , and (80) delivers The lemma is proved.
The proof of the following lemma is analogous.
α,β are uniformly bounded for some γ, δ with (ρ, γ), (τ, δ) ∈ Ω , then we have the strong convergences The following lemma is a version of the dominated convergence theorem and will be useful in proofs of the strong convergence of operator sequences in the Hilbert space 2 .
For what follows we set Lemma 28.Let a ∈ PC and (ρ, γ), (τ, δ) ∈ Ω .Then the strong limits of the operators as well as exist, where W 2 ( L γ,δ n aL n ) = a(1)I and W 3 ( L γ,δ n aL n ) = a(−1)I as well as I : 2 −→ 2 is the identity operator.
Proof.We are only going to show the first two convergences.The proof of the other convergences can be done in the same way.In view of Lemmas 12 and 22 it suffices to show the convergence on a dense subset of the space 2 .Let e m = (δ j,m ) ∞ j=0 , m ∈ N 0 , For n > m + 1 , we have . Hence, , and the strong convergence of these adjoint operators follows as before.
We consider the function g ± : R + −→ R with Due to our assumption we have H ± ∈ L 2 2p ± −1 (R + ) ∩ L 2 2q ± −1 (R + ) and ξ ± ∈ (p ± , q ± ) .From that, we derive as well as In view of condition (b), the curves Γ W 2 (A n ) , Γ W 3 (A n ) do not contain the point 0 and their winding number is zero.From Proposition 1 we derive that W 2 (A n ) and W 3 (A n ) are Fredholm operators with vanishing index.Thus, condition (c) delivers the invertibility of those operators.
Of course, the advantage of the results in [7] is that there also the sufficiency of the stability conditions is proved and that in (1) also the case b(x) ≡ 0 is considered.These problems will be studied for the collocation-quadrature methods considered here in forthcoming papers.
Finally, we can conclude that we were able to prove necessary conditions for the stability of, in comparison with the existing literature, a wider class of collocation-quadrature methods based on the zeros of classical Jacobi polynomials.In this way we can enlarge the range of endpoint singularities of the solutions of singular integral equations of Mellin type, which we can represent in the respective approximate solutions.The questions on
Together with the compactness of the operator K : L 2 α,β −→ C η,ζ (see Lemma 4) and the strong convergence L * n = L n −→ I in L 2 α,β , this leads to the limit relation (48).Hence, L γ,δ n KL n = L n KL n + C n with (C n ) ∈ N and consequently, by definition, L γ,δ n KL n ∈ J .Corollary 4 yields (49).