Lobachevskii Journal of Mathematics Vol. 7, 2000, 3–14 \begin{matrix}c \\ ◯ \end{matrix}A. I. Fedotov

A. I. Fedotov

ON CONVERGENCE OF THE POLYNOMIAL COLLOCATION

METHOD FOR SINGULAR INTEGRAL EQUATIONS AND PERIODIC PSEUDODIFFERENTIAL EQUATIONS

(submitted by A.M.Elizarov)

Abstract. We prove the convergence of polynomial collocation method for periodic singular integral, pseudodifferential and the systems of pseudodifferential equations in Sobolev spaces H^{s} via the equivalence between the collocation and modified Galerkin methods. The boundness of the Lagrange interpolation operator in this spaces when s > 1/2 allows to obtain the optimal error estimate for the approximate solution i.e. it has the same rate as the best approximation of the exact solution by the polynomials.

Introduction

Arnold and Wendland in [1] proposed the original technique of justification the spline collocation methods for pseudodifferential equations in Sobolev spaces. The justification is based on the equivalence between spline collocation and modified Galerkin-Petrov methods and justification of the last one by reducing it to the standart Galerkin method. In [2] - [7] this approach was used for justification of spline collocation methods for the various classes of singular integral and pseudodifferential eqations. It was shown that that strong ellipticy of the equation is sufficient and in some cases (see [8] - [10]) necessary condition for the convergence of spline collocation method. Note that earlier in [11] Golberg used the analogy between collocation and Galerkin methods as a criterion of optimal nodes choice.

Here the approach of the article [1] is used for justification of the polynomial collocation method for the singular integral, periodic pseudodifferential and the systems of the pseudodifferential equations in Sobolev spaces. The results show that polynomial collocation method converges for wider class of equations than spline collocation method. Namely it converges for all elliptic i.e. uniquely solvable equations while spline collocation method converges only for strong elliptic ones. Moreover, the rate of convergence of polynomial collocation method grows up with the growing of the smoothness of exact solution infinetely while for spline collocation method this rate is bounded by the order of used splines. Finally, the calculation schemes of polynomial collocation methods are easier than of spline collocation methods for the wide range of equations including singular integral and integro-differential equations, because the singular integrals of polynomials could be calculated explicitly.

1991 Mathematical Subject Classification. 42A10, 45E05, 47G30, 65R20, 45F15, 45J05, 65E05, 65N30, 65N35..

Key words and phrases. singular integral equations, periodic pseudodifferential equations, Galerkin method, collocation method..

1. Collocation with trigonometric polynomials

Consider the linear equation

Au = f, A : H^{s+β} → H^{s}, s, β ∈ R,  (1)

where f is known and u is desired unknown 2π-periodic, complex-valued functions, H^{s} denotes Sobolev space of order s i.e. the closure of all smooth 2π-periodic complex-valued functions of a real variable t with respect to the norm

∥f∥_{s} = ∥f∥_{H^{s}} = {| f̂(0) |^{2} + ∑_{0≠k∈Z} | f̂(k) |^{2}| k |^{2s}}^{1/2},

where

f̂(k) = (2π)^{−1} ∫_{−π}^{π} f(t)ē_{k}(t)dt, k ∈ Z,

are the complex Fourier coefficents of the function f ∈ H^{s},

e_{k}(t) = exp(ikt), k ∈ Z,

are trigonometric monomials and β ∈ R is the order of the operator A. For the following we’ll asume that s > 1/2 providing the embedding of H_{s} in C.

Denote the space of trigonometric polynomials of degree n by

T_{n} = span{e_{k}(t) : | k |≤ n}, n ∈ N_{0}.

Here N_{0} = {0, 1, ...} is a set of natural numbers including zero. When zero is excluded we write N = {1, 2, ...}. Since dim T_{n} = 2n + 1, define the equallyspaced collocation points by

t_{k} = 2πk/(2n + 1), | k |≤ n,  (2)

and seek u_{n}^{∗} ∈ T_{n} satisfying

(Au_{n}^{∗})(t_{k}) = f(t_{k}), | k |≤ n.  (3)

When it exists, the function u_{n}^{∗} is said to be a trigonometric collocation solution of (1).

2. Some preliminaries

It’s known that being equiped with the inner product

< f, g >_{s}= f̂(0) · ¯ ĝ(0) + ∑_{0≠k∈Z} f̂(k) · ¯ ĝ(k) | k |^{2s},

H^{s} becomes the Hilbert space and H^{0} = L_{2}.

Denote the polynomial of the best approximation to x ∈ H^{s} of degree ≤ n and the corresponding best approximation

(S_{n}x)(t) = ∑_{|k|≤n} x̂(k)e_{k}(t), E_{n}(x)_{s} = inf_{x_{n}∈T_{n}} ∥x − x_{n}∥_{s} = ∥x − S_{n}x∥_{s},

where (S_{n}x)(t) is the n-th partial sum of the Fourier series of x.

Lemma 1. . If x ∈ H^{p} and s ≤ p, then

E_{n}(x)_{s} ≤ cn^{s−p}E_{n}(x)_{p}.

Proof. Indeed

E_{n}(x)_{s} = ∥x−S_{n}x∥_{s} = { ∑_{|k|>n} | x̂(k) |^{2}| k |^{2s}}^{1/2} = { ∑_{|k|>n} | x̂(k) |^{2}| k |^{2p}| k |^{2(s−p)}}^{1/2} ≤

≤ (n + 1)^{s−p}E_{n}(x)_{p} ≤ cn^{s−p}E_{n}(x)_{p}.□

Let L_{n} be the usual polynomial Lagrangian interpolation operator that assigns to every function x ∈ H^{s} the polynomial

(L_{n}x)(t) = (2n + 1)^{−1} ∑_{|k|≤n} x(t_{k})\frac{sin((2n + 1)(t − t_{k})/2)}{sin((t − t_{k})/2)}

coinciding with x in the nodes (2).

Lemma 2. . The operator L_{n} is bounded and the following estimate is valid

∥L_{n}∥_{H^{s}→H^{s}} ≤ \sqrt{1 + ζ(2s)}, n = 1, 2, ...,

where ζ(t) - is the Riemann’s ζ-function bounded and decreasing for t > 1.

Proof. Let’s rewrite Lagrange polynomial L_{n}x in a form

(L_{n}x)(t) = ∑_{|k|≤n} x̂(k)^{(n)}e_{k}(t),

where

x̂(k)^{(n)} = (2n + 1)^{−1} ∑_{|l|≤n} x(t_{l})ē_{k}(t_{l}), | k |≤ n,

are Fourier-Lagrange coefficients of the function x with respect to the nodes (2). Substituting x by it’s Fourier series expansion and omiting summunds equal to zero we will obtain

x̂(k)^{(n)} = (2n + 1)^{−1} ∑_{|l|≤n} ē_{k}(t_{l}) ∑_{m∈Z} x̂(m)e_{m}(t_{l}) = ∑_{μ∈Z} x̂(k + μ(2n + 1)).

Now, denoting for the convenience k′ = k + δ_{0k} (δ_{ij} is the Kronecker symbol) estimate with the help of Hölder inequality the norm of the polynomial L_{n}x

∥L_{n}x∥_{s}^{2} = ∑_{|k|≤n} | x̂(k)^{(n)} |^{2}| k′ |^{2s}= ∑_{|k|≤n} | ∑_{μ∈Z} x̂(k + μ(2n + 1)) |^{2}| k′ |^{2s}=

= ∑_{|k|≤n} | ∑_{μ∈Z} | (k +μ(2n+1))′ |^{−s} x̂(k +μ(2n+1)) | (k +μ(2n+1))′ |^{s}|^{2}| k′ |^{2s}≤

≤ ∑_{|k|≤n}(∑_{μ∈Z} | x̂(k+μ(2n+1)) |^{2}| (k+μ(2n+1))′ |^{2s} ∑_{μ∈Z} | k′/(k+μ(2n+1))′ |^{2s}) ≤

≤ max_{|k|≤n}{∑_{μ∈Z} | k′/(k + μ(2n + 1))′ |^{2s}}∥x∥_{s}^{2}.

The chain of the inequalities

max_{|k|≤n}{∑_{μ∈Z} | k′/(k + μ(2n + 1))′ |^{2s}} ≤ 1 + ∑_{μ∈N}((2μ + 1)^{−2s} + (2μ − 1)^{−2s}) ≤

≤ 1 + ∑_{μ∈N} μ^{−2s} = 1 + ζ(2s)

concludes the proof. □

Let

Jx = (2π)^{−1} ∫_{−π}^{π} x(t)dt and J_{n}x = (2n + 1)^{−1} ∑_{|k|≤n} x(t_{k})

be the Riemann’s integral of the function x ∈ H^{s} and quadrature rule for its approximate calculation correspondently.

Lemma 3. . Let x ∈ H^{s}. Then

x(t_{l}) = 0, | l |≤ n,  (4)

if and only if

< x − e_{k}J(xē_{k}) + e_{k}J_{n}(xē_{k}), e_{k} >_{s}= 0, | k |≤ n.  (5)

Proof. Denote r_{k} = J(xē_{k}) − J_{n}(xē_{k}), | k |≤ n the residuals of the quadrature sums and form a polynomial

x_{n}(t) = ∑_{|k|≤n} r_{k}e_{k}(t).  (6)

We can rewrite (5) simply as < x − x_{n}, e_{k} >_{s}= 0, | k |≤ n. Assume that (4) are valid, then

< x − x_{n}, e_{k} >_{s}=< x, e_{k} >_{s} − < x_{n}, e_{k} >_{s}= J_{n}(xē_{k}) | k′ |^{2s}= 0, | k |≤ n.

Now assume that (5) are valid, i.e

(2n + 1)^{−1} | k′ |^{2s} ∑_{|l|≤n} x(t_{l})ē_{k}(t_{l}) = 0, | k |≤ n.

As {(e_{k}(t) : | k |≤ n} are linear independent polynomials, then the vectors

{(e_{k}(t_{−n}), e_{k}(t_{−n+1}), ..., e_{k}(t_{−n})) : | k |≤ n}

are also linear independent and the homogeneous system of equations (5) has the unique solution x(t_{l}) = 0, | l |≤ n. □

Denote by R_{n} the operator

R_{n}(x) = ∑_{|k|≤n} r_{k}e_{k}(t), R_{n} : H^{s} → H^{s},

which assigns the polynomial (6) to every function x ∈ H^{s}. The estimation of its norm gives the following Lemma 4. . For every x ∈ H^{s}

∥R_{n}x∥_{s} ≤ cE_{n}(x)_{s}, n ∈ N_{0}.

Proof. The coefficients of the polynomial R_{n}x

r_{k} = J(xē_{k}) − J_{n}(xē_{k}) = J(xē_{k}) − J(ē_{k}L_{n}x) = J((x − L_{n}x)ē_{k}), | k |≤ n,

are the first 2n + 1 Fourier coefficients of the function (x − L_{n}x)(t), so with the help of Lemma 2 we’ll obtain

∥R_{n}x∥_{s} = ∥ ∑_{|k|≤n} r_{k}e_{k}∥_{s} = ∥S_{n}(x−L_{n}x)∥_{s} = ∥L_{n}(S_{n}x−x)∥_{s} ≤ \sqrt{1 + ζ(2s)}E_{n}(x)_{s}. □

Lemmas 3 and 4 play a crucial role in the following account. The first one allows to prove the equivalence of the collocation and modified Galerkin methods. The second allows to estimate the rate of convergence of the last one.

3. Singular integrals equtions

Suppose that (1) is the singular integral equation

A ≡ aRu + bQu + Ku = f, A : H^{s} → H^{s},  (7)

where P and Q are the complementary projection operators defined by

(P u)(t) = ∑_{l≥0} û(l)e_{l}(t), (Qu)(t) = ∑_{l≤−1} û(l)e_{l}(t),

and where K : H^{s} → H^{s} is a compact linear operator. For simplicity, assume that the coefficients a and b belong to C^{∞}, then

A : H^{s} → H^{s}

is a bounded linear operator. If the coefficients satisfy

| a(t) |^{2} + | b(t) |^{2}≠ 0 for all t ∈ R,

then A is said to be elliptic (or non-degenerate).

The function t → a(t) parametrizes a smooth, closed curve in the complex plane, whose winding number (about the origin) is denoted by

W (a) = (1/2π)[arg a(t)]_{t=−π}^{π}.

Also, the kernel, image, cokernel and index of A : H^{s} → H^{s} are denoted by

ker (A) = {u ∈ H^{s} : Au = 0},
im (A) = {f ∈ H^{s} : f = Au for some u ∈ H^{s}},
coker (A) = H^{s}/\overline{im (A)},

ind (A) = dim ker (A) − dim coker (A).

Theorem 1. . Suppose that for the equation (7) the following is held:

A1 the singular integral operator A is elliptic with W (a) = W (b) and ker (A) = {0},

A2 there is an ε > 0 such that K : H^{s} → H^{s+ε} is bounded,

A3 s > 1/2.

Then for all n sufficiently large, there exists a unique collocation solution u_{n}^{∗} which converges to the exact solution u^{∗} and

∥u^{∗} − u_{n}^{∗}∥_{s} ≤ cE_{n}(u^{∗})_{s}.

Corollary 1. . If in the Theorem 1 u^{∗} ∈ H^{s+p}, then

∥u^{∗} − u_{n}^{∗}∥_{s} ≤ cn^{−p}E_{n}(u^{∗})_{s+p}.  (8)

Proof of the Theorem 1. The index Theorem for singular integral operators (see [12]) states that A is a Fredholm operator if and only if A is elliptic, in which case

ind (A) = W (a) − W (b).

Define the pair of spaces

C_{+}^{∞} = {f ∈ C^{∞} : f̂(l) = 0 for all l ≤ 0},
C_{−}^{∞} = {f ∈ C^{∞} : f̂(l) = 0 for all l ≥ 0}.

It is not difficult to verify that the following are equivalent:

1. f ∈ C_{+}^{∞}.

2. For all N > 0, f̂(l) = O(l^{−N}) as l → ∞.

3. The function f ∈ C_{+}^{∞} admits an analytic continuation into the upper half plane, which is bounded and 2π-periodic (i.e. f(z + 2π) = f(z) for Imz ≥ 0).

There are analogous characterization for C_{−}^{∞}, and as a consequence one has the well known factorization property [13, p. 191], [14, p. 78].

Lemma 5. . Let a ∈ C^{∞} satisfy a(t) ≠ 0 for all t ∈ R. If W (a) = κ then there exist functions a_{±} ∈ C_{±}^{∞} such that

1. a(t) = a_{+}(t)e^{iκt}a_{−}(t) for all t ∈ R.

2. 1/a_{±} ∈ C_{±}^{∞}.

We’ll also need the following identities (see [14, p. 71]):

S_{n}(a_{+}P + a_{−}Q)S_{n} = S_{n}(a_{+}P + a_{−}Q),  (9)
S_{n}(P a_{−} + Qa_{+})S_{n} = (P a_{−} + Qa_{+})S_{n},

for any a_{±} ∈ C_{±}^{∞}.

According to the Theorem 1

W (b^{−1}a) = W (a) − W (b) = 0,

so due to Lemma 5 there exists a factorization

b^{−1}a = ρ_{+}ρ_{−}, ρ_{±} ∈ C_{±}^{∞}.

Define the operators

M = bρ_{+}, N = P ρ_{−} + Qρ_{+}^{−1},

then, using the fact that P + Q = I, it is easy to see that

M^{−1} = ρ_{+}^{−1}b^{−1}, N^{−1} = P ρ_{−}^{−1} + Qρ_{+}.

Let [·, ·] be the usual commutator bracket, and define

T = M^{−1}K + [ρ_{−}, P] + [ρ_{+}^{−1}, Q],

then elementary algebra gives

A = M(N + T).

This representation in other notations appeared in the monographs of N.I. Muskhelishvili [15] and F.D. Gakhov [16]. For the justification of the approximate methods for singular integral equations it was used for the first time by V.V. Ivanov [17].

Consider the equation

Bu ≡ (N + T)u = y, y = M^{−1}f, B : H^{s} → H^{s},  (10)

which is equivalent to the equation (1). Equivalence here means that the equations (1) and (10) are either both solvable or not, and their solutions coincide. Moreover, McLean and Wendland [18, p. 367] have shown that the operator T : H^{s} → H^{s+ε} is bounded for the same s and ε as the operator K.

Analogously, consider the system of equations

(Bu_{n})(t_{k}) = y(t_{k}), | k |≤ n,

which is equivalent to the system of equations (3). Lemma 3 allows us to rewrite the system of equations (3) as the system of equtions of modified Galerkon-Petrov method

< Bu_{n} − y − R_{n}(Bu_{n} − y), e_{l} >_{s}= 0, | l |≤ n,

or in the form of operator equation of projection method

S_{n}Bu_{n} = S_{n}(y + R_{n}(Bu_{n} − y)).  (11)

With the help of (9) and consequential identities

S_{n}NS_{n} = NS_{n}, S_{n}Nu_{n} = S_{n}NS_{n}u_{n} = NS_{n}u_{n} = Nu_{n},  (12)

left hand side of the equation (11) could be represented as

B_{n}u_{n} = S_{n}Bu_{n} = S_{n}(N + T)u_{n} = (N + S_{n}T)u_{n}.

Therefore from the invertibility of the operator N + T : H^{s} → H^{s} and the convergence

∥(T − S_{n}T)u_{n}∥_{z} = E_{n}(T u)_{s} ≤ cn^{−ε}E_{n}(T u)_{s+ε} ≤ cn^{−ε}∥u∥_{s}

it follows that there is such a number n_{1} ∈ N that for all n ≥ n_{1} the perturbed operator B_{n} = N + S_{n}T has an inverse satisfying the uniform bound

∥B_{n}^{−1}∥_{H^{s}→H^{s}} ≤ c.

To estimate the rate of convergence of the approximate solutions u_{n}^{∗} ∈ T_{n} to the exact solution u^{∗} ∈ H^{s} of the eqution (1) we denote by ū_{n} = S_{n}u^{∗} ∈ T_{n} the polynomial of the best approximation of u^{∗} and taking into account that ū_{n} = B_{n}^{−1}S_{n}Bū_{n} write

u^{∗} − u_{n}^{∗} = u^{∗} − ū_{n} + B_{n}^{−1}S_{n}Bū_{n} − B_{n}^{−1}S_{n}(y + R_{n}(Bu_{n}^{∗} − y)) =
= u^{∗} − ū_{n} + B_{n}^{−1}S_{n}Bū_{n} − B_{n}^{−1}S_{n}Bu^{∗} − B_{n}^{−1}S_{n}R_{n}(Bu_{n}^{∗} − y) =
= (I − B_{n}^{−1}S_{n}B)(u^{∗} − ū_{n}) + B_{n}^{−1}S_{n}R_{n}(Bu^{∗} − Bu_{n}^{∗}).

The boundness of the operators B_{n}^{−1}, S_{n}, B and the obvious identity S_{n}R_{n} = R_{n} allow us to obtain the estimation

∥u^{∗} − u_{n}^{∗}∥_{s} ≤ c∥u^{∗} − ū_{n}∥_{s} + c∥R_{n}(Bu^{∗} − Bu_{n}^{∗})∥_{s} ≤  (13)
≤ cE_{n}(u^{∗})_{s} + c∥R_{n}(Nu^{∗} − Nu_{n}^{∗})∥_{s} + c∥R_{n}(T u^{∗} − T u_{n}^{∗})∥_{s}.

With the help of the Lemma 4 the third summand of the right hand side can be estimated as follows:

∥R_{n}(T u^{∗}−T u_{n}^{∗})∥_{s} ≤ cE_{n}(T (u^{∗}−u_{n}^{∗}))_{s} ≤ cn^{−ε}E_{n}(T (u^{∗}−u_{n}^{∗}))_{s+ε} ≤ cn^{−ε}∥u^{∗}−u_{n}^{∗}∥_{s}.

To estimate the second summand we will use once more Lemma 4, and the identities (12)

∥R_{n}(Nu^{∗} − Nu_{n}^{∗})∥_{s} ≤ c∥N(u^{∗} − u_{n}^{∗}) − S_{n}N(u^{∗} − u_{n}^{∗})∥_{s} ≤
≤ c∥N(u^{∗} − u_{n}^{∗}) − N(ū_{n} − u_{n}^{∗}) + S_{n}N(ū_{n} − u_{n}^{∗}) − S_{n}N(u^{∗} − u_{n}^{∗})∥_{s} ≤
≤ c∥N(u^{∗} − ū_{n})∥_{s} + c∥S_{n}N(u^{∗} − ū_{n})∥_{s} ≤ cE(u^{∗})_{s}.

Hence, the estimation (13) will take the form

∥u^{∗} − u_{n}^{∗}∥_{s} ≤ cE_{n}(u^{∗})_{s} + cn^{−ε}∥u^{∗} − u_{n}^{∗}∥_{s}.

Choosing n_{2} ≥ n_{1} on condition that cn^{−ε} < 1/2 is valid for all n ≥ n_{2} and transposing the last summand of the estimation to the left hand side we obtain finally

∥u^{∗} − u_{n}^{∗}∥_{s} ≤ 2cE_{n}(u^{∗})_{s}. □  (14)

Lemma 1 and the estimation (14) allow to obtain the estimation (8) of the corollary 1.

Remark. As it was shown in [19, p. 135] when A1 is not satisfied, it may nevertheless be possible to obtain approximate solutions of Au = f by the following modification due to Mikhlin and Prössdorf [20, p. 442-443].

With κ = W (a) − W (b) ≠ 0 introduce the new operator

D = e_{−κ}aP + bQ + K(e_{−κ}P + Q),

whose coefficients have equal winding numbers. Now let v_{n} = ∑_{|l|≤n} ξ_{l}e_{l} denote the approximate solution of the equation Dv = f obtained by applying the collocation method. Then the sequence

u_{n} = ∑_{l=0}^{n} ξ_{l}e_{l−κ} + ∑_{l=−n}^{−1} ξ_{l}e_{l}

converges to a solution u of the original equation Au = f provided a solution u exists.

ON CONVERGENCE OF THE POLYNOMIAL COLLOCATION 11

4. Periodic pseudodifferential equations

We consider equations with operators

Bv ≡ (aP + bQ + K)Λ^{β}v = f  (15)

where the operator Λ^{β} denotes the Bessel potential operator of order β ∈ R, given by

Λ^{β}e_{l} =| l + δ_{0l} |^{β} e_{l}, l ∈ Z,

(δ_{ij} is the Kronecker symbol) and corresponding continuous extension. If β = 0, then B is just a singular integral operator.

From Agranovich’s Theorem [21], [22] it is known that any one-dimensional (classical) pseudodifferential operator of order β acting on periodic functions can be written in the form (15). In fact suppose B_{0} is the principal part of B, i.e.

(B_{0}u)(t) = ∑_{k∈Z} σ_{0}(t, k)û(k)e^{ikt},

where σ_{0}(t, k) - the principal symbol of B - is 2π-periodic in t, and positive homogeneous of degree β in k ≠ 0, i.e.,

σ_{0}(t, k) = \{ \begin{matrix}| k |^{β} σ_{0}(t, k/ | k |), k ≠ 0, \\ 1, k = 0. \end{matrix}

A simple calculation shows that with

a(t) = σ_{0}(t, +1), b(t) = σ_{0}(t, −1),  (16)

the principal part of B can be written as B_{0} = (aP + bQ)Λ^{β}. The following mapping property of Λ^{β} allows the extension of the former results to (15), see [23, p. 149].

Lemma 6. . For arbitrary β, s ∈ R the mapping

Λ^{β} : H^{s+β} → H^{s}

is an isomorphism.

Lemma 7. . Operator Λ^{β} commutes with S_{n}, with P and with Q.

Just as for the case of a singular integral operator (β = 0), we say that B is elliptic if the functions (16) satisfy

a(t) ≠ 0 and b(t) ≠ 0 for all t ∈ R.

Obviously, when B is elliptic, the mapping

B : H^{s+β} :→ H^{s}, β ∈ R,

is an isomorphism if and only if W (a) = W (b) and ker(B) = {0}.

Let us write A = aP + bQ + K, then the equation

Bv ≡ AΛ^{β}v = f  (17)

is equivalent to

Au = f with u = Λ^{β}v,

and the system of collocation method

(Bv_{n})(t_{k}) = f(t_{k}), | k |≤ n,  (18)

is equivalent to finding the polynomial

u_{n} = Λ^{β}v_{n} ∈ T_{n}  (19)

satisfying

(Au_{n})(t_{n}) = f(t_{n}), | k |≤ n.  (20)

Therefore, the solvability of the (18) is determined by the solvability of the (20) and (19). The solution of the last one consists in the calculation of the Fourier coefficients of the polynomial v_{n} according to the formulas

v̂_{n}(l) =| l |^{β} û_{n}(l), | l |≤ n.  (21)

Theorem 2. . Suppose that for the equation (17) the following is held:

B1 the pseudodifferential operator B is elliptic with W (a) = W (b) and ker (B) = {0},

B2 there is an ε > 0 such that K : H^{s+β} → H^{s+β+ε} is bounded for all s ∈ R, B3 s > 1/2. Then for all n sufficiently large, there exists a unique collocation solution v_{n}^{∗}

of (18), and

∥v^{∗} − v_{n}^{∗}∥_{s+β} ≤ cE_{n}(v^{∗})_{s+β}.

Corollary 2. . If in the Theorem 2 v^{∗} ∈ H^{s+α}, α > β, then

∥v^{∗} − v_{n}^{∗}∥_{s+β} ≤ cn^{β−α}E_{n}(v^{∗})_{s+α}.

Proof of the Theorem 2. The solvability of the system of equations (18) follows immediately from the Theorem 1 and the identities (21). Theorem 1 also allows to obtain the error estimation

∥v^{∗} − v_{n}^{∗}∥_{s+β} = ∥Λ^{β}v^{∗} − Λ^{β}v_{n}^{∗}∥_{s} ≤ cE_{n}(Λ^{β}v^{∗})_{s} = cE_{n}(v^{∗})_{s+β}. □

The last estimation and the Lemma 1 yield the estimation of the Corollary 2.

5. Systems of periodic pseudodifferential equations

As in [19] (see also the bibliography there) we will extend the analysis of the previous paragraphs to systems of singular integral equations and, further, to elliptic systems of periodic pseudodifferential equations. As indicated in [24], and using the results of [21], [22], every system of periodic pseudodifferential equations can be written in the form

Bv ≡ (σ_{0}(t, +1)P + σ_{0}(t, −1)Q + K)Λ^{β}v = f,  (22)

where σ_{0} is the principal symbol of B, a matrix valued 2π-periodic by t function, and where β = (β_{1}, ..., β_{L}) ∈ R^{L} is a suitable vector of orders. Λ^{β} is defined by the diagonal matrix of operators

Λ^{β} = (δ_{ij}Λ_{i}^{β}), i, j = 1, ..., L,

(δ_{ij} is the Kronecker symbol), K satisfies assumption B2.

The trigonometric collocation method for (22) reads as to find the vector of polynomials v_{n} ∈ T_{n}^{L} satisfying

(Bv_{n})(t_{k}) = f(t_{k}), | k |≤ n.  (23)

This system of equations is equivalent to the system

(Au_{n})(t_{k}) = f(t_{k}), | k |≤ n,

with u_{n} = Λ^{β}v_{n} and

A = σ_{0}(t, +1)P + σ_{0}(t, −1)Q + K.

Theorem 3. . Suppose that for the equation (22) the following is held:

B1’ detσ_{0}(t, ±1) ≠ 0 for all t ∈ R and the left indices of σ_{0}(t, +1) and the right indices of σ_{0}(t, −1) and σ_{0}(t, −1)^{−1}σ_{0}(t, +1) are all equal to zero, ker(B) = {0},

B2’ there is an ε > 0 such that

K : H_{L}^{s+β} → H_{L}^{s+β+ε}, H_{L}^{s+β} = ∏_{j=1}^{L} H^{s+β_{j}}, ∥ · ∥_{s+β} = ∑_{j=1}^{L} ∥ · ∥_{s+β_{j}}

is bounded for all s ∈ R, B3’ s > 1/2. Then for all n sufficiently large, there exists a unique collocation solution v_{n}^{∗}

of (23), and

∥v^{∗} − v_{n}^{∗}∥_{s+β} ≤ cE_{n}(v^{∗})_{s+β}

Corollary 3. . If in the Theorem 3 v^{∗} ∈ H^{s+α}, α = (α_{1}, ..., α_{L}), α_{j} > β_{j}, j = 1, ..., L, then

∥v^{∗} − v_{n}^{∗}∥_{s+β} ≤ cn^{γ}E_{n}(v^{∗})_{s+α}, γ = max_{1≤j≤L}(β_{j} − α_{j}).

Proof of the Theorem 3. Assumption B1’ guarantees the existence of the matrix factorizations

σ_{0}(t, +1) = a_{+}a_{−}, σ_{0}(t, −1) = b_{−}b_{+},

with a_{±}, b_{±} ∈ (C_{±}^{∞})^{L×L}, having all the desired properties needed in §3 and §4. The rest of the proof follows as in the proofs of the Theorems 1 and 2 and their Corollaries if the function factorizations are replaced by matrix factorizations. □

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Received December, 29, 1999; Revised version November, 11, 2000