A method is described for effecting the explicit Wiener-Hopf factorisation of a class of (2 x 2)-matrices. The class is determined such that the factorisation problem can be reduced to a matrix Hilbert problem which involves an upper or lower triangular matrix. Then the matrix Hilbert problem can be further reduced to three scalar Hilbert problems on a half-line, which are solvable in the standard manner

Rawlins, AD

SIAM Journal on Applied Mathematics

English

Brunel University Research Archive

Matrix Wiener-Hopf-Hilbert FactorisationAuthor(s): A. D. RawlinsSource: SIAM Journal on Applied Mathematics, Vol. 46, No. 1 (Feb., 1986), pp. 133-139Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2101492Accessed: 09/12/2009 10:37Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and youmay use content in the JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/action/showPublisher?publisherCode=siam.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to SIAM Journal on Applied Mathematics.http://www.jstor.orgSIAM J APPL. MATH. ? 1986 Society for Industrial and Applied Mathematics Vol. 46, No. 1, February 1986 011 MATRIX WIENER-HOPF-HILBERT FACTORISATION* A. D. RAWLINSt Abstract. A method is described for effecting the explicit Wiener-Hopf factorisation of a class of (2 x 2)-matrices. The class is determined such that the factorisation problem can be reduced to a matrix Hilbert problem which involves an upper or lower triangular matrix. Then the matrix Hilbert problem can be further reduced to three scalar Hilbert problems on a half-line, which are solvable in the standard manner. Key words. Wiener-Hopf-Hilbert method, diffraction, matrix factorisation AMS(MOS) subject classifications. 30A88, 78, 15A21 1. Introduction. In a recent paper by Rawlins and Williams [1] (see also Rawlins [2]), it was shown how a class of (2 x 2)-matrices could be explicitly factorised. In this paper a different class of matrices is constructively factorised. By using the idea of Rawlins [3] and evaluating the matrix to be factorised on both sides of an assumed branch cut that commonly arises in diffraction problems, the problem of factorisation reduces to a matrix Hilbert problem along the branch cut. In the work of Rawlins and Williams [1], and Rawlins [2], the form of the original matrix was chosen so that the matrix Hilbert problem was reducible to two uncoupled scalar Hilbert problems. These could be solved without difficulty by the well-known methods given in Muskhelishvili's book on singular integral equations [4]. The reduction to these two scalar Hilbert problems required that the two diagonal element of the matrix involved in the Hilbert problem were zero. However, it is known, see Gohberg and Krein [5], that upper and lower triangular matrix Hilbert problems can also be solved explicitly. Thus we need only require one off-diagonal element of the matrix Hilbert problem to vanish, in order to effect a Wiener-Hopf factorisation of the original matrix. It is conjectured that the technique of matrix factorisation of the present class of (2 x 2)-matrices may go some way towards an eventual solution of some hitherto intractable diffraction problems. The truth of this conjecture hinges on the growth at infinity of the factor matrices. The consideration of such behaviour is more appropriate in the actual application of the method to diffraction problem and is therefore, omitted here. We mention that the type of matrix factorised in this paper does not fall into the class considered by Daniele [6], Rawlins [7]. Jones [8] has devised a method for the Wiener-Hopf factorisation of a special type of (2 x 2)-matrix, that ensures that the Wiener-Hopf factors commute. In addition, the factors of various matrices whose Wiener-Hopf factors do not commute were also determined by Jones [8]. It is possible that by appropriate pre- and post-multiplying a matrix (which is susceptible to Jones' method) by appropriate analytic matrices the Wiener-Hopf factorisation can be carried out for the matrices considered here by his approach. However, the result obtained here seems to be different from that of Jones [8], and it is not clear to me how one could prove the equivalence of the two results. The difference is apparent in the scalar factorisation problem. In Jones [8] the classical approach by Cauchy's theorem leads to a solution for the factors expressed in terms of Cauchy integrals along a line parallel to the real axis in the strip of analyticity. On the other hand, the approach used here through the Hilbert problem leads to a solution involving Cauchy * Received by the editors August 28, 1984, and in revised form April 8, 1985. t Department of Mathematics and Statistics, Brunel University, Uxbridge, Middlesex UB8 3PH, England. 133 134 A. D. RAWLINS integrals along a branch cut, i.e. along a half line. The strip of analyticity is not strictly necessary in the present approach. This would indicate that the present method would be suitable for problems without dissipation. Pioneering and important work on Wiener-Hopf-Hilbert factorisation of matrices has been carried out by Hurd [9] and his coworkers. Jones [10] has extended the class of (2x2)-matrices whose factors commute to a class of (n x n)-matrices whose factors commute. In ? 2 of the paper a general matrix will be considered, and its general form is appropriately specified in order that the Wiener-Hopf factorisation problem reduces to a triangular matrix Hilbert problem. In ? 3 this class of matrices will be constructively factorised by solving appropriate Hilbert problems. In ? 4 some remarks are made vis-a-vis direct Wiener-Hopf factorisation of the present class of matrices. 2. Determination of the class of matrices whose Wiener-Hopf factorisation reduces to a triangular matrix Hilbert problem on a half-line. Considerthe general (2 x 2)-matrix. A(a) (all(a) a12(a)\ a2l(a) a22(a) where the elements aij (a), i,j = 1, 2 are functions of the complex variable a. These functions will be assumed to have only branch point singularities; specifically we shall assume that the branch points arise through the function y = ,a2 - k2, where k has positive real and imaginary parts, and the branch cuts C and C' lie along the half-lines C: a = -k-8, C': a = k+ 8? 0 ' O. The elements aij(a) are also assumed to be analytic functions in the cut a-plane; and det A(a) $ 0 within the strip -ki < Im (a) < ki where ki denotes the imaginary part of k. The occurrence of a complex k with Im (k) > 0 is traditional in Wiener-Hopf-type problems and is needed to have a common strip of analyticity; in the final solution the complex k is removed by taking the limit as Im k 4 0. The Wiener-Hopf factorisation problem requires the determination of (2x2)- matrices U(a) and L(a), whose elements are analytic for Im (a) > -Iki and Im (a) < ki respectively, such that (1) A(a) = U(a)L-l(a) U(a) and L(a) are also required to be nonsingular in their respective regions of analyticity. Obviously any matrix A'(a) which can be expressed in the form (2) A'(a) = Cu(a)A(a)BL(a), where BL(a) and Cu(a) are matrices whose elements are analytic functions of a for Im (a) < ki and Im (a)> -ki respectively, can also be factorised if A(a) can be factorised. In order to effect the factorisation, it will be assumed that U(a) is analytic except along the branch cut C through a = -k, whilst L(a) is analytic except along the branch cut C' through a = k. Evaluation of equation (1) on both sides of the cut C(C: a = -k -8 80- O) through a = -k gives, on using the suffices + to denote values evaluated on the upper and lower sides of C, (3) A+(a) = U+(a)L71(a), (4) A_(a) = U_(a)L-1(a), (L(a) is analytic except along the branch cut C' through a = k and therefore takes the same values on both sides of C). Eliminating L(a) between (3) and (4) gives the matrix Hilbert problem: (5) U+(a) = G(a)U_(a), a E C, MATRIX WIENER-HOPF-HILBERT FACTORISATION 135 where G(a) = A+(a)A11(a). More explicitly ( g1I(a) g12(a)\ _ (a+a-2- a -a a + ah -a+ aT2\ g21(a) g22(a) det A_(a) a2+1a22- a22a21 a22a+ - a21a 2 The problem (5) reduces to an upper or lower traingular matrix Hilbert problem along C, if the condition gl2(a)-0 or g21(a)= 0 is satisfied. That is at(a)a-(a) = a+(a)a-(a), i = 1,j = 2 or i = 2,j = 1, a E C; or ignoring the trivial case of a (a) 0, and assuming aij(a) $ 0 on C, (6) aii(a) aii (a)> 0a aEC We shall assume that aij(a) can have isolated zeros in the cut a-plane, (but not on C), and consequently ajj(a)/aij(a) can have isolated poles in the cut a-plane. Then provided ajj(a)/ajj(a)=O(jk+a1j-Z), 0 c ,u <1 as a-*e-k, a solution of (6) follows immediately (see Muskhelishvili [4, ? 15]) as (7) aii(a) = ajj(a)F1(a). Here the function Fi(a) is analytic except along the branch cut C' through a = k,' and except for poles at the zeros of aij(a); the multiplicity of these poles is not greater than the multiplicity of the corresponding zeros. Thus since we can express Fi(a) = r1(a)/sj(a) where ri(a) and si(a) are analytic except possibly along the branch cut C' then G(a) will be of upper or lower triangular form if: (8) (i) A(a) = a21(a)s2(a) a21(a)r2(a)) or a12(a)r1a) a2(a)sl(a)\ (9) (ii) A(a) =(a12(a)rl(a) a22(a) a2l (a ) a22(a ) and det A(a) $ 0 in the cut a-plane. We shall now carry out the explicit Wiener-Hopt factorisation of the matrix given in case (i) above. The procedure for factorising the matrix given in case (ii) will be completely analogous. 3. Wiener-Hopf factorisation of the matrix defined by (8). We assume the matrix A(a) has the form (8) and the same general properties as outlined in the first paragraph of ? 2. If we carry out the same evaluation on the branch cut C, as described in ? 2, the equation (5) reduces to the upper triangular matrix Hilbert problem (10) U+(a) = G(a)U_(a), aE C, 1 If A(a) = A(-a) for a in the cut plane, then it is not difficult to show (see Rawlins [3]) that F, (a) must also be analytic along the branch cut C'. 136 A. D. RAWLINS where ,(a) (ajj(a)r2(a) - a12(a)S2(a))' gll(1t) (all(a)r2(a) - a12(a)S2(a)) ( a'2(a)all(a) - all(a)a12(a) a21(a)(all(a)r2(a) - a12(a)S2(a)) a'1( a) g2l(st=0 g22Ct)= - (a)' (12) g( , U()=(Uii(a) U12(a)) (u21(a) U22(a)j Evaluating the matrix expression (10) and equating corresponding elements of the matrices on both sides of the equality sign gives the following equations: ( 13 ) ua(a)=gjj(a)u-(a)+g12(a)u2j(a),} a C, j = 1,2 u+(a) = 922(a) u -(a),J The four equations (13) can clearly be solved if the coupled system (14) vt(a) = gll(a)vL(a)+g12(a)vj(a), (15) V+(a)= g22(a)v v(a), can be solved. The equation (15) is a standard Hilbert problem whose fundamental solution is given directly by the methods of Muskhelishvili [4, Chap. 10]. Similarly we can determine the fundamental solution of the standard auxiliary Hilbert problem (16) v+(a) = gj1(a)v-(a), a E C, for v(a). Then the equation (14) can be written as (17) u+(a) - u(a) = g12(a)v-(a)/v (a), a E C, where u(a) = vi(a)/v(a). In the equation (17) the right-hand side is a known quantity and therefore we have a standard Hilbert problem whose fundamental solution is given by using the techniques described in Muskhelishvili [4, Chap. 10]. Suppose therefore we have found funda- mental solutions v20)(a), v(?)(a) and u(?)(a) of the equations (15), (16) and (17), respectively. To determine the general solution, we set v2(a) = v(?)(a)v*(a), v(a) = v(O)(a)v*(a), and u(a) = u(O)(a) + u*(a); then we are led to the Hilbert problems [v2*(a)] = [v*(a)]f, [v*(a)]? = [v*(a)]p, and [U-(a)] [u*(a)] =2(a)[v2 2 which have a solution [Muskhelishvili [4, ? 15]) (18) V*(a) = P2(a), v*(a) = P2(a), u*(a) = PI(a), where P1(a), P2(a) are entire functions of a. Thus a suitably general solution of (14) and (15) is given by v2(a) = P2(a)v(Y)(a) and vi(a) = u(a)v(a) = (u(O)(a)+Pl(a))P2(a)v(0)(a). MATRIX WIENER-HOPF-HILBERT FACTORISATION 137 A suitably general solution of the equation (13) and consequently of (10) is therefore given by (( ((0(a) + P 1(a)) P21(a)v ( (a) (u ( (a)+ P12(a))P22(a)v(a)(a) (19) U(a)- P2,(a)v(?0)(a) P22(a)v(0)(a) I where det U(a) = P21(a) P22(a)v(0)(a)v(0)(a)(PII(a)- P12(a)), and Pij(a), ij = 1, 2 are entire functions. The choice of the entire functions P1%(a) is further restricted by the condition that U(a) is nonsingular; and the requirement that the corresponding matrix L(a) = A-(a)U(a) is nonsingular, and its elements should be analytic except along the branch cut C' through a = k. In particular, the elements of L(a) should not have poles at a= - k. For the applications we have in mind it is sufficient to let P21 = P22 = PI1 = -P12 = 1, giving ( (U ()(a)+ 1)v(?)(a) (u (O) (a)- 1)v()(a)) (20) ~ UO(a ()a ()a (20) det U'?)(a) = 2v(?)(a)v 2)(a). In a completely analogous way it can be shown that a Wiener-Hopf factorisation of the matrix defined by (9) is given by A(a) = U(a)L-l(a) where I PII(a)v(0)(a) P12(a)v(0)(a) (21) U(a) = I(O (O ( ) ( ) ( PI I ( a:) v (?)(a) )(u U(?)(a,) + P2, (a )) P12( a )V(?) (a)( U (0)(a) + P22(a ))) and Pij1(a), i, j = 1, 2 are entire functions, det U(a) = Pll(a)Pl2(a)v(?)(a)v(?)(a)(P22(a) - P21(a)); and v(?)(a), v(?)(a), and u(?)(a) are fundamental solutions of the standard Hilbert problems V (a) = gII(a)vj (a), v+(a) = g22(a)v (a), aEC. U+(a) - u(a) = 92I(a)v (a)1v(a), gij(a) are the elements of the lower triangular matrix G(a) = A+(a)A-I'(a). Imposition of the further restriction that U(a) and L(a) are nonsingular, and analytic everywhere except along the branch cuts C and C' respectively, dictates the choice PI 1 = P12 = P22 = -P21 = 1, giving (22) U ( (a) = )(a)(u (0a -1) v ( 0a)(au() +a)) The elements of U(a) have been constructed by assuming that the matrix L(a) in equation (2) is continuous across C and therefore L(a) defined by L(a) = A1(a)U(a), with the elements defined by one of the equations (19) to (22), should from the method of construction, be continuous across C. Equation (5) and the equation above gives L+(a) = A-l1(a)U?(a) = A+l(a)A+(a)A-1(a)U-(a) = L-(a), thus verifying explicitly that L(a) is indeed continuous across C. 138 A. D. RAWLINS 4. It was remarked in the introduction that the class of matrices (8) and (9) could possibly be factorised directly using Jones' technique [8] by expressing (8) or (9) in a suitable product form. One of the referees has indicated that this is possible by expressing them as the product of triangular matrices. Thus, for example, the matrix (8) can be written in the form A(a) _( a12(a)-al,(a)s(a)/r(a) all(a)/r(ca))( 0 1() 0 a2l(a) r(a) sa This product of matrices is now in the form of (2) where Cu(a) = 1 BL(a)- ? 1) The triangular matrix A'(a) = (a12(a)-ajj(a)s(a)1r(a) all(a)lr(a)) can be explicitly factorised since (a(a) b(a)) (a?(a) a+(a)[b(a)/(c_(a)a+(a))]R) 0 c(a)l 0 Oc+(a)J *a_(aa) c_(a)[b(a)/(c_(a)a+(a))]L) 0 OC_(a) (a_a) a_(a)[b(a)/(c+(a)a_(a))]L 0 c_(a )J (a+(a) c+(a)[b(a)/(c+(a)a(a))]R) \ ?c+(a) J using the notation of Jones [8]. However the above factorisation is prima facie different from that obtained in the main text of this paper. The above results involves scalar factorisation along infinite lines in a strip of regularity, whereas the method used in this paper involves scalar factorisation along semi-infinite lines. It is also worth mentioning that the Wiener-Hopf type factorisation problem, using a strip of regularity, is subsumed in the more general Hilbert factorisation problem along a line segment, see Noble [12]. Conclusions. We have presented a method for factorising matrices which arise in diffraction problems. This could offer scope for deriving closed-form solutions to hitherto unsolved diffraction problems, see Rawlins [11]. The applicability of the present method to a given matrix A'(a) (whose elements, besides having the branch point singularities at a=?k, also have poles; and whose determinant vanishes or becomes infinite in the cut a plane) can be easily determined. If Al(a)[A'l(a)]-1 is triangular, then the present method can be used to factorise the matrix A'(a). One merely has to determine the Cu(a) and BL(a) of the expression (2) which ensures that the elements of A(a) have no poles and that det A(a) ?0. This can be effected without too much difficulty by inspection. Finally we mention that the (n x n) triangular matrix Hilbert problem can also be solved explicitly. Thus provided we can find the class of (n x n)-matrices that reduce to the (n x n) triangular matrix Hilbert problem on analytic evaluation about the branch cut C, we will have effected a Wiener-Hopf factorisation of this class of (n x n) - matrices. MATRIX WIENER-HOPF-HILBERT FACTORISATION 139 Acknowledgment. I would like to thank the referees for making helpful comments in the preparation of this paper. REFERENCES [1] A. D. RAWLINS AND W. E. WILLIAMS, Matrix Wiener-Hopf factorisation, Quart. J. Mech. Appl. Math., 34 (1981), pp. 1-8. [2] A. D. RAWLINS, The explicit Wiener-Hopffactorisation of a special matrix, Z. Angew. Math. Mech., 61 (1981), pp. 527-528. [3] - , A note on Wiener-Hopfmatrixfactorisation, Quart. J. Mech. Appl. Math., 38 (1985), pp. 433-437. [4] N. I. MUSKHELISHVILI, Singular Integral Equations, Noordhoff, Groningen, Holland, 1953. [5] I. C. GOHBERG AND M. G. KREIN, Systems of integral equations on a half line wit,h kernels depending on the difference of arguments, Amer. Math. Soc. Trans., Ser. 2, 14 (1960), pp. 201-287. [6] V. G. DANIELE, On thefactorisation of Wiener-Hopf matrices in problems solvable with Hurd's method, IEEE Trans. Antennas and Prop., AP-26 (1978), pp. 614-616. [7] A. D. RAWLINS, A note on thefactorisation of matrices occurring in Wiener-Hopfproblems, IEEE Trans. Antennas and Prop., AP-28 (1980), pp. 933-934. [8] D. S. JONES, Factorisation of a Wiener-Hopf matrix, J. Inst. Math. Appl., 32 (1984), pp. 211-220. [9] R. A. HURD, The Wiener-Hopf-Hilbert methodfor diffraction problems, Can. J. Phys., 54 (1976), pp. 775-780. [10] D. S. JONES, Commutative Wiener-Hopffactorisation of a matrix, Proc. Roy. Soc. Lon., A393 (1984), pp. 185-192. [11] A. D. RAWLINS, Brunel University Maths. Dept. internal report numbers TR/03/84 and TR/18/84. [12] B. NOBLE, Methods Based on the Wiener-Hopf Technique, Pergamon, Oxford, 1958. 

A note on thefactorisation of matrices occurring in Wiener-Hopfproblems,

A note on Wiener-Hopfmatrixfactorisation,

Commutative Wiener-Hopffactorisation of a matrix,

Factorisation of a Wiener-Hopf matrix,

Matrix Wiener-Hopf factorisation,

Methods Based on the Wiener-Hopf Technique, Pergamon,

On thefactorisation of Wiener-Hopf matrices in problems solvable with Hurd's method,

Singular Integral Equations,

Systems of integral equations on a half line wit,h kernels depending on the difference of arguments,

The explicit Wiener-Hopffactorisation of a special matrix,

The Wiener-Hopf-Hilbert methodfor diffraction problems,

Matrix Wiener-Hopf-Hilbert factorisation

Matrix Wiener-Hopf-Hilbert factorisation

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