Shamir-Duduchava Factorization of Elliptic Symbols
Tony Hill

TL;DR
This paper revisits and refines the Shamir-Duduchava factorization theory for elliptic symbols, providing new proofs and clarifications that enhance understanding of matrix-valued elliptic symbol factorization.
Contribution
It offers a new detailed proof of sub-algebra factorization conditions and completes the proof of a key elliptic matrix function factorization result.
Findings
Established sufficient conditions for right standard factorization in Wiener algebra
Provided a complete proof of elliptic matrix-valued function factorization
Improved understanding of elliptic operator inversion in half-spaces
Abstract
This paper considers the factorization of elliptic symbols which can be represented by matrix-valued functions. Our starting point is a \textit{Fundamental Factorization Theorem}, due to Budjanu and Gohberg. We critically examine the work of Shamir, together with some corrections and improvements as proposed by Duduchava. As an integral part of this work, we give a new and detailed proof that certain sub-algebras of the Wiener algebra on the real line satisfy a sufficient condition for right standard factorization. Moreover, assuming only the Fundamental Factorization Theorem, we provide a complete proof of an important result from Shargorodsky, on the factorization of an elliptic homogeneous matrix-valued function, useful in the context of the inversion of elliptic systems of multidimensional singular integral operators in a half-space.
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Taxonomy
TopicsAlgebraic and Geometric Analysis · Mathematical Analysis and Transform Methods · Holomorphic and Operator Theory
Shamir-Duduchava factorization of elliptic symbols
Dedicated to Roland Duduchava on the occasion of his 70th birthday.
Tony Hill
Department of Natural and Mathematical Sciences
King’s College London
May 2015
Abstract
This paper considers the factorization of elliptic symbols which can be represented by matrix-valued functions. Our starting point is a Fundamental Factorization Theorem, due to Budjanu and Gohberg [2]. We critically examine the work of Shamir [15], together with some corrections and improvements as proposed by Duduchava [6]. As an integral part of this work, we give a new and detailed proof that certain sub-algebras of the Wiener algebra on the real line satisfy a sufficient condition for right standard factorization. Moreover, assuming only the Fundamental Factorization Theorem, we provide a complete proof of an important result from Shargorodsky [16], on the factorization of an elliptic homogeneous matrix-valued function, useful in the context of the inversion of elliptic systems of multidimensional singular integral operators in a half-space.
Contents
1 Introduction
This paper considers the factorization of elliptic symbols which can be represented by matrix-valued functions. Our starting point is a Fundamental Factorization Theorem, due to Budjanu and Gohberg [2]. We critically examine the work of Shamir [15], together with some corrections and improvements as proposed by Duduchava [6]. We shall call the combined efforts of these two latter authors the Shamir-Duduchava factorization method.
One important application of the Shamir-Duduchava factorization method has been given by Shargorodsky [16]. Our primary goal is to provide, in a single place, a complete proof of Shargorodsky’s result on the factorization of a matrix-valued elliptic symbol, assuming only the Fundamental Factorization Theorem. As an integral part of this work, we will give a new and detailed proof that certain sub-algebras of the Wiener algebra on the real line satisfy a sufficient condition for right standard factorization.
2 Background
Let denote a simple closed smooth contour dividing the complex plane into two regions and , where for a bounded contour we identify with the domain contained within . We shall be especially interested in the case where , the one point compactification of the real line. In this situation, of course, are simply the upper and lower half-planes respectively. We let denote the union .
2.1 Factorization
Suppose we are given a nonsingular matrix-valued function A(\zeta)=\big{(}a_{jk}(\zeta))^{N}_{j,k=1}, then we define a right standard factorization or simply the factorization as a representation of the form
[TABLE]
where is strictly diagonal with non-zero elements for . The exponents are integers and are certain fixed points chosen in respectively. (In passing, we note that if , it is customary to take .) are square matrices that are analytic in and continuous in . Moreover, the determinant of is nonzero on .
As one would expect, interchanging the matrices and in gives rise to a left standard factorization. In either a right or a left factorization, the integers are uniquely determined (see [9]) by the matrix . Further, if the matrix admits a factorization for a pair of points , then it admits the a factorization of the same type for any pair of points , in that the right or left indices, denoted by , are independent of the points .
2.2 Banach algebras of continuous functions
We let denote a Banach algebra of continuous functions on which includes the set of all rational functions not having any poles on . Further we insist that is inverse closed in the sense that if and does not vanish anywhere on , then . Of course, , where is the Banach algebra of all continuous functions on , with the usual supremum norm.
Consider the region . We let denote the set of all rational functions not having any poles in this domain and denote the closure of in with respect to the norm of . It is easy to see that is a subalgebra of consisting of those functions that have analytic continuation to and which are continuous on . We can now define . Again, it is straightforward to show that is a subalgebra of . (Similar definitions of and follow by considering the region .)
2.3 Splitting algebras
It turns out that the ability to factorize a given matrix is intimately linked to the ability to express as a direct sum of two subalgebras - one containing analytic functions defined on and the other analytic functions on . To ensure uniqueness of this partition we let denote the subalgebra of consisting of all functions that vanish at the chosen point . We now say that a Banach algebra if we can write
[TABLE]
The prototypical example of a splitting algebra is the Wiener algebra, , of all functions defined on , the unit circle , of the form
[TABLE]
with the norm . The Banach algebras have a simple characterisation. For example, consists of all functions in that can be expanded as an absolutely converging series in nonnegative powers of . However, the algebra does not split. For more details see [2].
2.4 R-algebras
We say that a Banach algebra of complex-valued functions continuous on is an R-algebra if the set of all rational functions with poles not lying on is contained in and this set is dense, with respect to the norm of . In passing, we note that any R-algebra of continuous functions is inverse closed. (See, for example Chapter 2, Section 3, p. 44 [4].) Following Theorem 5.1, p. 20 [3], we have:
Theorem 2.1**.**
(Fundamental Factorization Theorem.) Let be an arbitrary splitting -algebra. Then every nonsingular matrix-valued function admits a right standard factorization with factors in the subalgebras .
2.5 Wiener algebras on the real line
Let denote the usual convolution algebra of Lebesgue integrable functions on the real line. For any we define the Fourier Transform of as the function , or , given by
[TABLE]
We let denote the algebra of continuous functions on which vanish at . It is well known, (see, for example, Chapter 9, Theorem 9.6, p. 182 [13]), that if , then
[TABLE]
The Wiener algebra is the set of all functions of the form where and is a constant. The norm on is given by
[TABLE]
Supose . Then since (see, for example, Chapter 9, Theorem 9.2, p. 179 [13]), it is straightforward to show that is a Banach algebra.
We will also consider certain subalgebras of the Wiener algebra . For we define to be the set of functions such that
[TABLE]
where is the th order derivative. (Of course, is simply .) We shall show that is a Banach algebra and, moreover, is a splitting -algebra.
2.6 Homogeneity, differentiability and ellipticity
Suppose for some integer . It will be convenient to write where . We assume that has the usual Euclidean norm, and we let denote the set .
We further suppose that is an matrix-valued function defined on , which is homogeneous of degree [math]. In addition, we will assume the elements of the matrix belong to , for some non-negative integer , where denotes the set of times continuously differentiable functions on the domain . Finally, we assume that is elliptic, in that
[TABLE]
2.7 The matrices and
We will be particularly interested in the behaviour of as .
Our approach is effectively to fix , and thereby consider factorization in the one-dimensional (scalar) variable . Since is homogeneous of degree zero,
[TABLE]
for fixed . We define
[TABLE]
2.8 The matrices
It is a standard result that any can be expressed in Jordan Canonical Form
[TABLE]
where the Jordan block is a matrix of order with eigenvalue on every diagonal entry, on the super-diagonal and [math] elsewhere. The matrix is invertible and
[TABLE]
The Jordan matrix, , is unique up to the ordering of the blocks .
Let to be the matrix given by
[TABLE]
We now define
[TABLE]
where . By construction, is a lower triangular matrix whose block structure and diagonal elements are identical to those of .
A routine inspection of the equation
[TABLE]
shows that the eigenspace associated with the eigenvalue has dimension one. Therefore, see p. 191 [5], the matrix is similar to the Jordan block for . Thus is similar to , and we have
[TABLE]
for some nonsingular matrix . Hence we can write
[TABLE]
For any and positive integer , it is easy to show that the matrix-valued functions satisfy
[TABLE]
In particular, taking gives
[TABLE]
In the analysis that follows we will be using the logarithm function on the complex plane. Unless specifically stated to the contrary, we will always take the principal branch of the logarithm, Log , defined by
[TABLE]
for any non-zero . In other words, we assume that the discontinuity in occurs across the negative real axis.
For any , we now define the complex-valued functions
[TABLE]
Then
[TABLE]
Corresponding to the block decomposition in (4), we set
[TABLE]
We note, in passing, that in the special case that , then .
Following [15], we now give a simple test for membership of for continuously differentiable functions.
Lemma 2.2**.**
Let and suppose the function has the property that, for some ,
[TABLE]
then .
Proof.
We follow the approach given in [15]. For , we define
[TABLE]
Our goal is to show that .
Differentiating with respect to ,
[TABLE]
Then, by hypothesis, and are continuous. Moreover as , we have and . Hence .
On applying the Fourier transform () to the function , we obtain . But, using the Cauchy-Schwarz inequality,
[TABLE]
Hence, is absolutely integrable everywhere outside a neighbourhood of zero. On the other hand, for small , from Theorem 127, p. 173 [17], and hence is absolutely integrable inside .
Thus, . We now define a new function . Then, by construction, and taking the Fourier transform () of we obtain
[TABLE]
Now , and hence, . This completes the proof of the lemma. ∎
2.9 Key theorem from Shamir
The next theorem, see Appendix pp. 122-123 [15], considers some properties of a certain matrix-valued function derived from an elliptic homogeneous matrix-valued function of degree zero. Together with Theorem 2.1, it will provide the starting point for proving our second result.
Theorem 2.3**.**
Suppose that is a matrix-valued function which is homogeneous of degree [math] and elliptic. Suppose that the Jordan form of has blocks of size for . Let the matrix , and the constant invertible matrix be as in equation (5). Then, for fixed ,
[TABLE]
Further let , where
[TABLE]
*and define .
Then, for fixed ,
[TABLE]
and
[TABLE]
Proof.
A detailed proof of this theorem is given in Appendix A.
∎
Remark 2.4**.**
Note that in (11), the definition of includes a multiplicative factor of (-1) not given in [15].
Remark 2.5**.**
Since we are assuming that for every non-zero we have , it follows immediately that
[TABLE]
and hence
[TABLE]
3 Statement of results
Theorem 3.1**.**
For is a splitting R-algebra.
Our second result considers the factorization of an elliptic matrix-valued function of degree , and it confirms the isotropic case of Lemma 1.9, p. 60 [16].
Theorem 3.2**.**
Let . Suppose that is a matrix-valued function which is homogeneous of degree and elliptic. Then, for fixed ,
[TABLE]
admits the factorization
[TABLE]
where and are homogeneous matrix-valued functions of order [math] that, for fixed , satisfy estimates of the form
[TABLE]
*Further, they have analytic extensions, with respect to , in the lower half plane and the upper half plane respectively.
* is a lower triangular matrix with elements*
[TABLE]
on its diagonal. Its off-diagonal terms are homogeneous of degree [math], and they satisfy an estimate of the form (14). The integer
[TABLE]
depends continuously on . The partial sums , are upper semicontinuous;
[TABLE]
* are the eigenvalues of the matrix to which there correspond Jordan blocks of dimension .*
4 Proof of the first result
The objective of this section is to prove Theorem 3.1. Let denote the characteristic functions of respectively.
Lemma 4.1**.**
The Wiener algebra is an R-algebra.
Proof.
An abbreviated proof of this lemma is given in Chapter 2, Section 4, pp. 62-63 [4]. A more detailed proof is included here, both for completeness and to introduce some analysis that will be useful when considering the subalgebras for .
We begin by showing that contains all rational functions with poles off . Firstly, we note the identities
[TABLE]
[TABLE]
where the functions and . Secondly, since all functions in are bounded at infinity, any rational function in must be such that the degree of the numerator must be less than or equal to the degree of the denominator. (In particular, non-constant polynomial functions are not included in .) Finally, the fact that contains all rational functions with poles off now follows directly, because is an algebra and we have the usual partial fraction decomposition over .
We now wish to show that rational functions with poles off are dense in . Suppose is arbitrary and is rational. By definition, we can write and , where and . Let denote the set of smooth functions with compact support in . Then, is dense in and
[TABLE]
Of course, the approximations to and , by and respectively, are independent but similar. Hence, to prove that is an R-algebra, it is enough for us to show that we can approximate , where , arbitrarily closely in the norm by a function such that is rational and has no poles in the upper half plane.
For , we let and define
[TABLE]
Since has compact support, is identically zero in some interval , where . Thus, by construction, .
Hence, given any , we can choose a Bernstein polynomial, see [12], , of degree , such that
[TABLE]
We let and observe, therefore, that our proposed approximant to is .
Of course, the Fourier transform of is a rational function with no poles in the upper half-plane, since for we have
[TABLE]
Finally, we take and then
[TABLE]
This completes the proof that is an R-algebra.
∎
Remark 4.2**.**
Suppose now that . From the proof of the above lemma, we can show that . (See section 2.2.) Indeed, applying inequality (2) we have
[TABLE]
*Since , we immediately have , because is the closure of with respect to the supremum norm. It follows in an exactly similar way that .
Lemma 4.3**.**
The Wiener algebra splits.
Proof.
An abbreviated proof of this lemma is given in Chapter 2, Section 4, p. 63 [4]. A more detailed proof is included here for completeness.
Our method of proof is a direct construction. Suppose then, since , we have
[TABLE]
where , and is chosen such that
[TABLE]
But since , we have . Moreover, from Remark 4.2, we have and thus
[TABLE]
In other words, we have the required decomposition, and thus
[TABLE]
where . This completes the proof that splits.
∎
Remark 4.4**.**
For any , we now define three integral operators:
[TABLE]
For more details see [7] and [8]. Each of these operators is bounded on . Moreover, see Chapter II Section 5, pp. 70-71 [7],
[TABLE]
*But since is dense in , each of the singular integral operators can be extended, by continuity, to a bounded operator on .
Finally, we have the well-known formulae
[TABLE]
Lemma 4.5**.**
For is a Banach algebra with a norm that is equivalent to the norm
[TABLE]
Proof.
The proof that is a Banach algebra is straightforward. However, as an illustration, we will prove that given , the product and , for some constant that only depends on .
The existence of a norm equivalent to and such that is then guaranteed by Theorem 10.2, p. 246 [14].
Suppose . Then, for any integer satisfying ,
[TABLE]
We assume that is a Banach algebra and therefore, and . Hence, as required.
By definition,
[TABLE]
where the strictly positive constant only depends on the integer . This completes the proof of the lemma.
∎
We now show that splits. To do this, we will need two intermediate lemmas.
Lemma 4.6**.**
Suppose and . Then .
Proof.
From Chapter I, Section 4.4, p. 31 [8], we have
[TABLE]
But, from Remark 4.4 we have respectively, and so
[TABLE]
∎
Lemma 4.7**.**
Suppose and . Let denotes the commutator of and . Then .
Proof.
Suppose . Then
[TABLE]
Finally, we note that and hence . This completes the proof of the lemma.
∎
Lemma 4.8**.**
For the algebra splits.
Proof.
Suppose for some nonnegative integer . Since , it is enough to consider the case where . Moreover, by Remarks 4.2 and 4.4, we can write
[TABLE]
Thus, to complete the proof, we must show that . That is, we must prove that for we have .
We now proceed by induction on . Our inductive hypothesis is that for any , we have . We have previously proved this result for . Suppose that the inductive hypothesis holds for .
From Lemma 4.6,
[TABLE]
But since , applying the inductive hypothesis gives
[TABLE]
Hence, using Lemma 4.7 (applied to ) gives
[TABLE]
This completes the proof by induction. So, finally, for , we have and thus, for , the algebra splits.
∎
Our final objective in this section is to show that is an R-algebra for , noting that in Lemma 4.1, we have proved this result for the special case , corresponding to .
In Appendix B we show that the Fourier transforms of smooth functions with compact support and which are zero in a neighbourhood of , are dense in the space . Then, proceeding analogously to Lemma 4.1, it is enough for us to show that we can approximate , where and is zero near [math], arbitrarily closely in the norm by a function that is rational and has no poles in the upper half plane.
As previously, for , we set and define
[TABLE]
Since has compact support, is identically zero in some interval , where . Thus, by construction, .
Remark 4.9**.**
The motivation for choosing the Bernstein polynomial, , see [12], as the approximant to in Lemma 4.1, is that we can simultaneously choose such that for
[TABLE]
Given that , we can consider in terms of , as given by the equation . The following lemma expresses the derivatives of in terms of the derivatives of .
Lemma 4.10**.**
[TABLE]
Proof.
Note that, by definition, and . We use proof by induction on .
Suppose . Then and hence
[TABLE]
completing the first step of the inductive proof.
Now suppose the result is true for . Then, by the inductive hypothesis
[TABLE]
Hence
[TABLE]
Therefore
[TABLE]
proving the result for . This completes the proof by induction.
∎
Motivated by Lemma 4.10, for , we now define:
[TABLE]
Hence, we can write
[TABLE]
In exactly the same way, given and we define . Hence, and
[TABLE]
Analogously, for we define:
[TABLE]
Hence, we can similarly write
[TABLE]
and we can now express our approximations in terms of the variable .
Remark 4.11**.**
Using equations (15) and (16), we can now reformulate the Bernstein polynomial, , approximations to and its derivatives as
[TABLE]
and for
[TABLE]
Lemma 4.12**.**
For is an R-algebra.
Proof.
Our proposed approximant to is . From Appendix B, to show convergence to in , it is sufficient to show convergence to and in for all .
Of course, one important consequence of the fact that our smooth function is zero in a neighbourhood of [math] is that it implies that is also smooth.
We have already seen in Lemma 4.1 that
[TABLE]
Similarly, for we have
[TABLE]
Suppose that . Clearly, there exist constants , that only depend on , such that
[TABLE]
where , and for , and , and for .
Hence, for
[TABLE]
Therefore, is an R-algebra, as required. ∎
5 Proof of the second result
The objective of this section is to prove Theorem 3.2. In determining certain asymptotic estimates for matrices arising during factorization, we follow the approach of Duduchava [6]. (For full details, see Appendix C.)
Proof.
We begin by defining
[TABLE]
For fixed , we set
[TABLE]
From Theorem 2.3, for fixed ,
[TABLE]
Moreover, from Lemmas 4.8 and 4.12, is a splitting R-algebra. Hence, by Theorem 2.1, the matrix admits a right standard factorization.
Therefore, we can write
[TABLE]
where the factors , and have analytic extensions, with respect to , to the lower half-plane and the upper half-plane respectively.
Moreover, see p. 37 [10], since , there exist factors such that
[TABLE]
We now define
[TABLE]
Hence, as diagonal matrices commute,
[TABLE]
From equation (21)
[TABLE]
Suppose . Then, from Lemma C.4,
[TABLE]
Given this result for the off-diagonal terms of and equations (20) and (21), we have
[TABLE]
Further, if we set
[TABLE]
then we can write
[TABLE]
Now, by definition,
[TABLE]
where we now define
[TABLE]
Remark 5.1**.**
*We have already noted that the factors have analytic extensions, with respect to , to the lower half-plane and the upper half-plane respectively. From definitions (21) and (22), it is immediately clear that this property is also shared by the factors and .
Remark 5.2**.**
From Lemma C.6
[TABLE]
In particular, each element of the matrices satisfies a condition of the form:
[TABLE]
Finally, we have the required factorization, namely
[TABLE]
where
[TABLE]
Remark 5.3**.**
*Note that, by construction, the matrix-valued functions commute with the diagonal matrix . [To see this, choose an arbitrary block . On this block, acts like a scalar, since the relevant components of the vector are all equal to ].
By Remark 5.3, equation (10) which defines , together with the properties of the blocks (see equations (6) and (7)), we can write
[TABLE]
Remark 5.4**.**
We note that, by definition,
[TABLE]
*Hence, functions of or are homogeneous in the variable .
It remains to consider the sum and partial sums of the factorization indices. For fixed , our final factorization, see equation (5), is
[TABLE]
Hence, since are constant matrices and , we have
[TABLE]
Now is a lower triangular matrix, and hence its determinant is the product of the entries on its main diagonal. Thus
[TABLE]
Therefore, see Chapter II Section 6, p. 88 [7].
[TABLE]
From equations (5) and (25) we have
[TABLE]
The remaining assertions in Theorem 3.2, concerning the continuity of the sum and the semicontinuity of the partial sums of the factorization indices, can be found in Theorem 3.1 p. 113 [15]. ∎
\appendixpage\addappheadtotoc
Appendix A Proof of key theorem from Shamir
We shall give a proof of Theorem 2.3 in two steps:
[TABLE]
Our overall approach will be to reduce the general case to the simpler case.
We begin by establishing some simple decay estimates.
Lemma A.1**.**
Suppose that is a matrix-valued function which is homogeneous of degree [math]. Then, for fixed ,
[TABLE]
*where these estimates are uniform for .
Proof.
Suppose that . Then since is homogeneous of degree [math],
[TABLE]
This completes the proof for .
For we can ignore the constant matrix , and we readily obtain
[TABLE]
Of course, estimates for the case follow in exactly the same way. This completes the proof of the lemma.
∎
Let us consider the first step. We are assuming that the invertible matrix is similar to diag. In this formulation the eigenvalues are listed according to their multiplicity and, of course, are all non-zero.
We now define by
[TABLE]
Remark A.2**.**
*The definition of , given by equation (27), includes a multiplicative factor not shown in [15]. As will be seen, this modification allows us to correct an error in the treatment of the discontinuity across the negative real axis. (See Lemma 4.2, [15].)
Lemma A.3**.**
Suppose that is a matrix-valued function which is homogeneous of degree [math] and elliptic. Suppose further that for some invertible constant matrix ,
[TABLE]
If , for and , then for fixed ,
[TABLE]
and
[TABLE]
Proof.
By hypothesis, we have . If we define , we may assume, without loss of generality, that
[TABLE]
We define a new matrix-valued function
[TABLE]
Then, for , we can write
[TABLE]
and similarly for , we have
[TABLE]
Since the matrices and are diagonal, we can write a typical element of the first summand as
[TABLE]
using Lemma A.1 and equation (13).
Now suppose and . Then , where are certain constants. Moreover, from Lemma A.1, as . Hence, for , we can write
[TABLE]
We now consider the second summand, which is diagonal as it is the product of diagonal matrices. For any and , it will be useful to factorize using the following identity:
[TABLE]
noting, as expected, that the decomposition on the right-hand side preserves the modulus and argument of the left-hand side.
For , the entry of the second summand is given by
[TABLE]
since the product of the first two terms is .
Similarly, for we have
[TABLE]
since for from equation (27).
So, for the second summand, combining the results for we have for
[TABLE]
So expanding the factors on the right-hand side in powers of we have
[TABLE]
Thus, on differentiating times with respect to , we obtain
[TABLE]
for . Combining estimates (A) and (32), we obtain
[TABLE]
From Lemma 2.2, we have . This completes the proof of the first step.
∎
With these preparations complete, we now turn to the general case. For convenience, we now restate Theorem 2.3.
Lemma A.4**.**
Suppose that is a matrix-valued function which is homogeneous of degree [math] and elliptic. Suppose that the Jordan form of has blocks of size for . Let , where
[TABLE]
Let . Then for fixed ,
[TABLE]
and
[TABLE]
Proof.
By hypothesis, and using equation (5), we have
[TABLE]
for some invertible matrix . If we define we may assume, without loss of generality, that
[TABLE]
Mimicing the approach in the first case, we define
[TABLE]
Moreover, we calculate
[TABLE]
We now define
[TABLE]
Now we can set , exactly as in the first case.
Remark A.5**.**
*When regarded as a function of , the matrix-valued functions are analytic in the regions Im and Im respectively. Note also that, by construction, the matrix-valued functions commute with the diagonal matrix . [To see this, choose an arbitrary block . On this block, acts like a scalar, since the relevant components of the vector are all equal to ].
As previously, see equation (29), we define a new matrix-valued function
[TABLE]
Now, since , using the established properties of we have
[TABLE]
On the other hand, since E_{-}=\text{diag }\big{[}\lambda_{1}B^{m_{1}}(1),\dots,\lambda_{l}B^{m_{l}}(1)\big{]},
[TABLE]
where is an block identity matrix.
So, as in Lemma A.3, we see that
[TABLE]
To show that satisfies estimates of the form given in equation (33), we follow exactly the approach taken in Lemma A.3.
[TABLE]
where
[TABLE]
The presence of the logarithmic terms in the matrices adds only a minor complication. For any fixed positive integer we have
[TABLE]
But since
[TABLE]
for any fixed integer and any , we can effectively repeat the proof of Lemma A.3 with any satisfying .
From Lemma 2.2, we have . This completes the proof of the general case.
∎
Appendix B Function approximation in
The goal in this appendix is prove that the Fourier transforms of smooth functions which have compact support and are zero in a neighbourhood of , are dense in the space . To show this, we use the standard approach of cut-off functions and convolution with a mollifier. In simple terms, this analysis is required because we are effectively working in a weighted Sobolev space. (See, for example, [1]).
Lemma B.1**.**
Suppose and . Then
[TABLE]
In addition, if then ,
[TABLE]
Proof.
Since is an R-algebra and ,
[TABLE]
By definition, and it follows from (38) that
[TABLE]
Moreover, from (39)
[TABLE]
where are some constants. (Note that implies that .)
From Proposition 2.2.11, p. 100 [11]
[TABLE]
Hence, .
Let . Then, again from Proposition 2.2.11, p. 100 [11]
[TABLE]
and thus .
Finally, .
This completes the proof of the lemma.
∎
Suppose . Then, by definition,
[TABLE]
Hence
[TABLE]
Thus, to show convergence to in , it is sufficient to show convergence to and in for all .
Note that, for
[TABLE]
Hence, we have
[TABLE]
where is a constant that only depends on . So, an alternative sufficient condition for the convergence to in is the convergence to in for all .
We now define a cut-off function that is zero in a neighbourhood of , and is also equal to zero when is sufficiently large. Firstly, we define two smooth functions
[TABLE]
and
[TABLE]
Then, for , we define the smooth cut-off function by
[TABLE]
Notice that, by construction,
[TABLE]
In particular, for each , the function has compact support, and is identically zero in a neighbourhood of [math].
Therefore, for , the support of is contained in , where
[TABLE]
For any positive integer , we have and . Hence, for ,
[TABLE]
for certain constants that only depend on . Moreover, unless and unless .
Hence, for ,
[TABLE]
where is a (finite) constant that depends only on the smooth functions and the index .
Lemma B.2**.**
Suppose . Then as .
Proof.
Suppose . Then, by definition, and . It is immediately clear, from the definition of the norm, that and as . That is, as , as required.
Now suppose , and that . Then
[TABLE]
Further suppose that , and . Then
[TABLE]
We now show that, for ,
[TABLE]
since . Hence,
[TABLE]
That is, as , as required. ∎
Remark B.3**.**
*The significance of Lemma B.2 is that we can effectively assume, for the ensuing density arguments, that any function in has both compact support and is also identically zero in a neighbourhood of the origin. (To see this, simply approximate by .)
Following, for example [1], we now introduce the concept of a mollifier. Let be a nonnegative, real-valued function in satisfying the two conditions, if , and .
For , we define . Then and:
- (a)
if 2. (b)
.
As , the mollifier approaches the delta-function supported on . Formally, we define the convolution
[TABLE]
Suppose and has compact support. Then:
- (i)
2. (ii)
and 3. (iii)
If , then 4. (iv)
.
As a simple consequence of the above, we observe that if then
[TABLE]
and, thus, as in the (unweighted) Sobolev space .
Lemma B.4**.**
Suppose , and further that has compact support and is identically zero in a neighbourhood of . Then as .
Proof.
Suppose that has compact support and is identically zero in a neighbourhood of . Then, there exist positive real numbers and such that supp . Suppose . Then supp .
Now let be any function with supp , and . Then, for any integers such that , we have
[TABLE]
Therefore
[TABLE]
and summing over all we have
[TABLE]
where denotes the (unweighted) Sobolev norm, and is defined by equation (40).
Thus, since as , we have as . Finally, from equation (40), as , as required.
∎
Appendix C Matrix factor estimates from Duduchava
In this appendix, we follow the approach taken by Duduchava [6], and derive some asymptotic estimates for certain matrices arising during factorization.
Given , we have the factorization
[TABLE]
where , and have analytic extensions, with respect to , to the upper half-plane and lower half-plane respectively. Moreover, see p. 37 [10], since , there exist factors such that
[TABLE]
We now define
[TABLE]
We begin with two technical lemmas that will be useful later. Let denote the unit circle in the complex plane.
Lemma C.1**.**
Let . Suppose and as , for . Define
[TABLE]
*Then, for , where denotes the Hölder space of order . Moreover, .
Proof.
Choose any . Then , for some , and it is straightforward to show that
[TABLE]
Hence, for , we can write
[TABLE]
In particular, as so and we obtain . Now
[TABLE]
By hypothesis,
[TABLE]
Moreover, by direct calculation
[TABLE]
Combining these results,
[TABLE]
Now suppose . Then, by relabelling if necessary, we can suppose
[TABLE]
and we consider three cases:
- Case 1:
;
- Case 2:
;
- Case 3:
.
We begin with Case 1, and apply the Mean Value Theorem to :
[TABLE]
where, due to constraints applicable in this case, and must have the same sign. Hence, from (45),
[TABLE]
For Case 2,
[TABLE]
But, in this case, and moreover,
[TABLE]
Therefore,
[TABLE]
Finally, turning to Case 3,
[TABLE]
∎
Lemma C.2**.**
Let and . Suppose as , for . Then as , for .
Proof.
By definition, for ,
[TABLE]
where, in the last step, we use the identity
[TABLE]
and repeated integration by parts.
For any we define the change of variable
[TABLE]
where, of course, . Note that, as we have . As previously, we define
[TABLE]
By Lemma C.1, with , for .
With this change of variable,
[TABLE]
But the operator is bounded on , and hence
[TABLE]
as . This completes the proof of the lemma.
∎
Our first task is to obtain some asymptotic estimates for the non-diagonal elements of , Due to the similarity of the calculations, it is enough to prove this result for the matrix . For brevity, we will ignore any constant terms that do affect the proof.
Lemma C.3**.**
Suppose with . Then
[TABLE]
for , and some .
Proof.
We begin by noting that, from the definition of ,
[TABLE]
Firstly, we suppose that . Then, in this case we can simply take
[TABLE]
so that . Since , the required result follows immediately. We note that, taking ,
[TABLE]
Secondly, suppose that and . From equation (42)
[TABLE]
Now consider b_{1}(t):=\bigg{[}\text{diag }\bigg{(}\dfrac{t+i}{t-i}\bigg{)}^{\kappa(\omega)}-I\bigg{]}A^{*}_{-}(\omega,t). We note that, as ,
[TABLE]
for . Hence, as ,
[TABLE]
We now consider the second term,
[TABLE]
where, by definition, b_{2}(t):=\bigg{(}\dfrac{t+i}{t-i}\bigg{)}^{\kappa(\omega)}A^{*}_{-}(\omega,t). Since we immediately have
[TABLE]
Moreover, from estimates (A) and (32)
[TABLE]
where is an arbitrarily small positive number that takes account of the logarithmic terms in the matrices used in the construction of . (See (37).) Using estimates (48) and (49)
[TABLE]
Let . By assumption, and hence we can choose any such that to ensure that . Moreover, , and hence .
Combining estimates (47) and (C)
[TABLE]
for , where .
For fixed , we can now apply Lemma C.2 with
[TABLE]
Let . Then
[TABLE]
So, finally
[TABLE]
for , and . ∎
Lemma C.4**.**
Suppose with . Then
[TABLE]
Proof.
The proof of the lemma follows directly from the estimates (46) and (51). (Of course, on using estimate (51) we take .)
∎
Remark C.5**.**
From equation (43), , and hence
[TABLE]
Thus, the proof of Lemma C.3 can readily be extended to obtain (c.f. (33))
[TABLE]
*for and .
Lemma C.6**.**
Suppose . Let
[TABLE]
Then
[TABLE]
Proof.
From Remark C.5
[TABLE]
for . Hence, from the definition of ,
[TABLE]
for and any such that . The required result now follows from Lemma 2.2. ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Adams R.A. (1975) Sobolev Spaces. Academic Press, Boston, MA.
- 2[2] Budjanu, M.S. and Gohberg, I.C. (1973) General Theorems on the factorization of matrix-valued functions. I. Fundamental Theorem , American Mathematical Society Translations, (2) Vol.14.
- 3[3] Budjanu, M.S. and Gohberg, I.C. (1973) General Theorems on the factorization of matrix-valued functions. II. Some tests and their consequences , American Mathematical Society Translations, (2) Vol.14.
- 4[4] Clancey, K. and Gohberg, I. (1981) Factorization of Matrix Functions and Singular Integral Operators , Birkhauser Verlag.
- 5[5] Dettman, J.W. (1987) Introduction to Linear Algebra and Differential Equations , Dover.
- 6[6] Duduchava, R. (1984) On multidimensional singular integral operators. I. The half-space case. , J. Operator Theory 11, pp. 41-76.
- 7[7] Eskin, G.I. (1980) Boundary Value Problems for Elliptic Pseudodifferential Equations , Volume 52, Translations of Mathematical Monographs, American Mathematical Society.
- 8[8] Gahov, F.D. (1966) Boundary value problems, First Edition , Pergamon Press, Oxford.
