REFERENCES

[AKH] N.I.AKlllEZER : The classical moment problem, Oliver and Boyd, Edinburgh,1965.

[ARO] N.ARONSZAJN: Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 , 1950,337-404.

[BGM] G.A.BAKER JR., P.GRA VES-MORRIS : Pade approximants, Addison­Wesley Pub!. Comp., Reading, Massachusetts, 1981.

[BOG] J. BOGNAR: Indefinite inner product spaces Springer verlag, Berlin, 1874.

[BOTl] A.BOTTCHER, B.SIL VERMANN : The asymptotic behavior of Toeplitz determinants for generating functions with zeros of integral orders, Math. Nachr., 102, 1981,79-105.

[BOT2] A.BOTTCHER, B.SILVERMANN : Invertibility and asymptotics of Toeplitz matrices, Akademie Verlag, Berlin, 1983.

[BGY] R.P.BRENT, F.G.GUSTAVSON, D.Y.Y.YUN : Fast solution of Toeplitz systems of equations and computation of Pade approximants, Journal of Algorithms, 1 , 1980,259-295.

[BRE] C.BREZINSKI : Pade type approximations and general orthogonal polynomials, Birkhauser Verlag, Basel, 1980.

[Bull A.BULTHEEL : The asymptotic behavior of Toeplitz determinants generated by the Laurent series of a meromorphic function, SIAM J. Algebraic and Discrete Math., 6 (4), 1985,624-629.

[BU2] A.BULTHEEL: Zeros of a rational function defined by its Laurent series. in Pade approximation and its applications, Bad Honnef 1983 , H.Werner, H.J. Bunger (eds.), Lect. Notes Math. 1071, Springer Verlag, Berlin, 1984, 34-48.

[BU3] A.BULTHEEL: Algorithms to compute the reflection coefficients of digital filters. in Numerische Methoden der Approximationstheorie 7 L. Collatz, G. Meinardus, H. Werner (eds.), ISNM 67, Birkhauser Verlag, 1984, 33-50.

258 REFERENCES

[BU4] A.BULTHEEL: Quotient-difference relations in connection with AR filtering. in Proc. EXXTD'83, Stuttgart, VDE Verlag, Berlin, 1983, 395-399.

[BU5] A.BULTHEEL: Recursive relations for block Hankel and Toeplitz systems. Part I: Direct recursions. Journ. Comp. Appl. Math., 10, 1984, 301-328. Part II: Dual recursions. Journ. Compo Appl. Math., 10, 1984, 329-328.

[BU6] A.BULTHEEL: Division algorithms for continued fractions and the Pade table. Journ. Comp. Appl. Math., 6 (4), 1980,259-266.

[BU7] A.BULTHEEL: Recursive algorithms for the matrix Pade problem. Math. of Comp., 35 (151), 1980,875-892.

[BU8] A.BULTHEEL, M. VAN BAREL: Pade techniques for model reduction in linear system theory: A survey., J. Comp. Appl. Math., 14 , 1985, 401-438.

[BU9] A.BULTHEEL : Triangular decomposition of Toeplitz and related matrices: a guided tour in the algorithmic aspects. Tijdschrift Belgisch Wiskundig Genootschap, Series A, 36 (3), 1986.

[BUlO] A.BULTHEEL: Error analysis of incoming and outgoing schemes for the trigonometric momemt problem. in Pade approximaJ:ion and its applicat;ions, Amsterdam 1980, M.G. de Bruin, H. van Rossum (eds.), Lecture Notes in Math. 888, Springer Verlag, Berlin, 1981, 100-109.

[CYB] G.CYBENKO: The numerical stability of the Levinson-Durbin algorithm for Toeplitz systems of equations. SIAM J. Scient. Stat. Comp. 1, 1980, 303-319.

[DA y] K.M.DA Y : Toeplitz matrices generated by Laurent series expansion of an arbitrary rational function, Trans. Amer. Math. Soc., 206 , 1975, 224-245.

[DED] P.DEWILDE, H.DYM : Schur recursions, error formulas and convergence of rational estimators for stationary stochastic sequences, IEEE trans. on Information Theory, IT-27 , 1981,446-461.

[DRA] A.DRAUX : PolynOmes ortlwgonaux jormels - applicat;ions., Springer Verlag, Berlin, 1983.

[GED] K.O.GEDDES: Block structure in the Chebyshev-Pade table, SIAM J. Numer. Anal., 18, 1981,844-861.

[GER] L.Y.GERONlMUS: Ortlwgonal polynomials, Consultants Bureau, New york, 1961.

REFERENCES 259

[GOF] LZ.GOHBERG, LA. FELDMAN : Convolution equations and projection methods for their solution. Translations of mathematical monographs 41 ,AMS, Rhode Island, 1974.

[GR1] W.B.GRAGG: Laurent, Fourier and Chebyshev-Pade tables, in Pade and rational approximation, Theory and Applications, E.B.Saff, R.S.Varga (eds.), Acad. Press, New York, 1977,61-72.

[GR2] W.B.GRAGG: The Pade table and its relation to certain algorithms in numerical analysis, SIAM Review, 14, 1972, 1-69.

[GR3] W.B.GRAGG: Matrix interpretation and applications of the continued fraction algorithm. Rocky Mountain J. Math., 4, 1974,491-500.

[GRL] W.B.GRAGG, A.lINDQUIST : On the partial realization problem. Lin. Alg. Appl., 50 , 1983,277-319.

[ORJ] W.B.GRAGG, G.D.JOHNSON : The Laurent-Pade table, In Information Processing 74, North Holland, Amsterdam, 1974,632-637.

[GRE] U.GRENANDER, G.SZEGO : Toeplitz forms and their applications ,John Wiley & Sons, 1968.

[HAM] G.HAMEL : Bine characteristische Eigenschaft beschranter analytische Funktionen, Math. Annal., 78 , 1918,257-269.

[HER] G. HEINIG, K.ROST : Algebraic methods for Toeplitz-like matrices and operators, Akademie-Verlag, Berlin, and Birkhauser Verlag, Basel, 1984.

[HEN] P.HENRICI: Applied and Computational complex analysis, John Wiley & sons, 1977.

[JOM] W.B.JONES, A.MAGNUS : Computation of poles of two-point Pade approximants and their limits, J.Comput. App1. Math., 6 , 1980, 105-119.

[JOT] W.B.JONES, W.J.THRON : Contirwed fractions: Analytic theory and applications, Addison Wesley Publ. Comp., Reading, Massachusetts, 1980.

[KAI] T.KAILATH, S.-Y.KUNG, M.MORF: Displacement ranks of matrices and linear equations, J. Math. Anal. App1., 68 ,1979, 395-407.

[KVM] T.KAILATH, A.VIERA, M.MORF : Inverses of Toeplitz operators, innovations and orthogonal polynomials, SIAM review, 20 ,1978, 106-119.

[LAN] LP.LANDAU: Otails et modeZes matlu!matiques pour l'autama:tique,l'analyse de systemes et le traitement du signal, Ed. du centre de 1a recherche scientifique, Paris, 1982.

[LEV] N.LEVINSON: The Wiener RMS (root mean square) error criterion in filter design and prediction, J.Math. Phys;, 25, 1947,261-278.

260 REFERENCES

[MAR] J.D.MARKEL, A.H.GRAY Jr. : Linear prediction of speech, Springer Verlag, Berlin, 1976.

[MCM] J.MCCABE, J.A.MURPHY : Continued fractions which correspond to power series expansions in two points, J. Inst. Math. Applies., 17 , 1976, 233-247.

[MC1] J.MCCABE: A formal extension of the Pade table to include two-point Pade quotients, J. Inst. Math. Applies., 15, 1975, 363-372.

[MC2] J.MCCABE: The quotient-difference algorithm and the Pade table: an alternative form and a general continued fraction, Math. Comp. 41 , 1983, 183-197.

[MES] H.MESCHKOWSKl : Hilbertsche Rliume mit Kernfunktion, Springer Verlag, Berlin, 1962.

[MUS] B.R.MUSICUS: Levinson and fast Cho1esky algorithms for Toeplitz and almost Toeplitz matrices. Manuscript MIT, 1984.

[NEV] R.NEVANLINNA: Ueber beschrankte Funktionen die in gegeben Punkten vorgeschriebene Werte annehmen, Ann. Acad. Sci. Fenn. Ser A, 13 , 1919.

[OPS] A.V.OPPENHEIM, R.W.SCHAFER : Digital signal processing, Prentice­Hall, Englewood Cliffs, 1975.

[PRAB] C.V.K.PRABHAKARA RAO : A system approach to the 77U.dtiport inverse scattering problem, Ph. D. Thesis, T.H. Delft, The Netherlands, 1985.

[PIC] G.PICK Ueber beschrankte Funktionen mit vorgegebenen Wertzuordnung, Ann. Acad. Sci. Fenn. Ser A, 15 , 1920.

[RED] R.REDHEFFER: On the relation of transmission-line theory to scattering and transfer. J. Math. Phys., 41 , 1962, 1-41.

[ROB] E.A.ROBINSON: Waves propagationg in random media as statistical time series. in Applied time series analysis, D.F.Findley (ed.), Ac.Press, New York, 1978,287-323.

[RUD] W.RUDIN: Real and Complex analysis, 2nd edition, McGraw Hill, New York, 1974.

[RUT] H.RUTISHAUSER: Der Quotienten-diJferenzen Algorithnu.ts, Mittlg. Inst. f. Angew. Math., ETH nr 7, Birkbauser Verlag, 1957.

[SCH] J.SCHUR : Ueber Potenzreihen die im Innern des Einheitskreises beschrankt sind, Z. Reine Angew. Math., 147 , 1917,205-232, 148, 1918, 122-145.

[SEE] W.SEEW ALD : Quotienten- Differenzen- Algorithmus: Beweis der Regeln von RUtiShaUseT, Numer. Math., 40 , 1982,93-98.

REFERENCES 261

[SWE] D.R.SWEET : Numerical metlwds lor Toeplitz rrud;rices. Ph.D. Thesis, Dept. of computer Science, Univ. of Adelaide, 1982.

[SZE] G.SZEOO: Ort/wgonal polynomials, AMS Colloquium Publ., XXIII, AMS, Providence, 1939.

[TRE1] W.F.TRENCH: An algorithm for the inversion of finite Toeplitz matrices, SIAM J. Appl. Math., 12, 1964,515-522.

[TRE2] W.F.TRENCH: An algorithm for the inversion of finite Hankel matrices, SIAM J. Appl. Math., 13, 1965, 1102-1107.

[ZOH1] S.ZOHAR : Toeplitz matrix inversion, the algorithm of W.F.Trench, J. Assoc. Comput. Mach., 16, 1969,592-601.

[ZOH2] S.ZOHAR : The solution of a Toeplitz set of linei"-r equations, J. Assoc. Comp. Mach., 21 , 1975, 272-276.

LIST OF SYMBOLS

The most important symbols used are given. The number following the short description of the symbol indicates the page on which the symbol is first introduced.

o

o <.,.>(m) [.,. ](m)

{.,.}(m)

(./.)

11·11

II· lip 1I·lIp,p

end of a proof, 12

continued fraction, 18

bar means complex conjugation, 83

composition of transformations, 13

bilinear form, 84

bilinear form, 98

bilinear form, 101

couple of elements, 11

operator norm, 196

LP or HP norm, 207

l~ norm, 196

a-parameter of Moebius transform tb 12

a-parameter for a Laurent series, 67

b-parameter of Moebius transform tb 12

b-parameter for a Laurent series, 67

a-parameter for a Laurent series, 78

b-parameter for a Laurent series, 78

square block grid, 148

c-parameter of Moebius transform tb 12

d-parameter of Moebius transform tb 12

264

det(M)

A

ekln,ekln

!k f:n fis

fps

ind(o)

k~m)(z)

l~m)(z)

l~m)(z)

l~,l~+,l~ m(m) n(m) n , n

0(0)

ord+,ord­

p*(z)

pX(z)

t (o),tn (o),t ~m)( 0)

t(o),tn(o)

t,.(o)

tn(o)

u(m) u(m) n , n

Ak

A~m)(z),A~m)(z)

LIST OF SYMBOLS

determinant of M, 104

forward and backward prediction error, 220

forward and backward innovations, 220

coefficients of fis, 22

1h.fm, 38

formal Laurent series, 22

formal power series, 23

winding number or index, 156

reproducing kernel for <o,o>(m), 94

reproducing kernel for [0,0 ](m), 100

reproducing kernel for {o, 0 } (m), 100

Banach spaces, 196

coefficients of a CF, 48

Landau symbol small oh, 160

+ and - order, 23

parahermitean conjugate of a polynomial, 83

X operation on a polynomial, 98

Moebius transforms, 11,12,56

dual Moebius transforms, 13

composite Moebius transform, 12

dual composite Moebius transform, 13

u and u parameters of a fis, 30

square norm of prediction error, 220

v and v parameters of a fis, 30

stochastic proces, 219

forward and back ward predictor, 220

A-parameter of composite Moebius transform tb 12 A

A and A polynomials of the second kind, 38

An(z) = A~O)(x)

A (m) A (m) n , n

BTc

B (p )~m)(z ),B(p )~m)(z)

B~m)(z),B~m)(z)

Bn(z) = B~O)(z) B(m) B(m)

n , n

rn (m) IBA (m) n , n

CTc

CF

C

DTc n(m)

n

ID(m) n

ID

E

IE

IBn F(z)

F +(z),F _(z)

F(z) F(m)

n

IF G~m)

c(m) n

GpCo)

LIST OF SYMBOLS

A polynomial, 214

bi-diagonal matrices, 127

265

B-parameter of composite Moebius transform tb 12

B and B Laurent polynomials, 40

Band B Laurent polynomials for p = 0,142

B polynomial, 214

bi-diagona1 matrices, 128

square block grids, 148,151

C-parameter of composite Moebius transform tb 12

continued fraction, 18

complex plane, 181

D-parameter of composite Moebius transform tb 12

diagonal matrix, 114,122

intersection of square block grids, 151

complex plane without 0 and 00, 155

regions in the complex plane, 169

pointed discs, 203

Expectation operator, 219

annular region, 156

annular region, 170

formal Laurent series, 22

analytic and coana1ytic factors, 156

the formal Laurent series F(1 / z), 131

F-parameter for a Laurent series, 72

F-parameter for a Laurent series, 79

function class, 209

G-parameter for a Laurent series, 72

G-parameter for a Laurent series, 79

geometric mean,coefficient in factorization, 156

266

H(z) H(m)

n

HP H(m)

n

In J (m) JA(m)

n , n

KP

K(p )~m)(z),K(p )~m)(z)

K~m)(z),K~m)(z)

Kn(z) = K~O)(z)

Kn

IK.

LPA

LPF LP

L+,L_

L(p )~m)(z),i(p )~m)(z)

IL(m) it (m) IL (m) n , n , n

lL(Y)

IN

0(·)

O+(zm),O_(zn)

PA PF P+,P-

p~m)(z),p~m)(z)

Pn(z) = Pn°)(z)

Pnm,m +h)(z ),p~m,m +h)(z)

p(m,m+h)(z),p(m,m+h)(z)

p(m) p(m) n , n

LIST OF SYMBOLS

system function, 222

Hankel determinant, 104

Hardy space of the unit disc, 207

Hankel matrix, 104

unit matrix, 110

Jacobi matrices, 110

Hardy space for the complement of the unit disc, 208

Laurent-Pade approximants, 41

Laurent-Pade approximants for p = 0, 142

Laurent-Pade approximant, 230

anti-unit matrix, 104

compact subset of C, 181

Laurent-Pade approximant, 26

Laurent-Pade form, 27

Hilbert space, 208

+ and - expansion operators, 24

numerator of Laurent-Pade approximants, 41

triangular grids, 148,150,153

lower triangular Toeplitz matrix, 120

the set of nonnegative integers, 39

Landau symbol big oh, 157

set of semi infinite Laurent series, 23

Pade approximant, 24

Pade form, 26

index boundaries for poles, 155

p and P (numerator) series for a f1.s, 32

P series, 217

dual Rutishauser polynomials, 188

polynomials, 187,188

triangular matrices, 113,116

IP Q~m)(z),Q~m)

Qn(z) = Q~O)(z)

Q(m) () QA (m) () n,n+k Z, n,n+k Z

Qk,n (Z) = Qi?~ (z)

Qn,n+h(Z),Qn,n +h(Z)

Q(m) QA (m) n , n

R~m)(z ),R ~m)(z)

Rn(z) = R~O)(z)

R(m) R(m) n , n

SS

s*(z)

$ T(m)

n

T = T(O) n n T(m)

n

T = T(O) n n

'II' , 'II'+ , rr-u~m)(z),i!~m)(z)

Un(z) = u~m)(z)

u(m)(z) i!(m)(z) k,n , Ic,n

U (z) = U(O) (z) k,n k,n

U (m) U(m) n , n

V~m)(z), v~m)(z)

Vn(z) = v~m)(z)

vi,:,j(z), vi,:,j(z)

Vk,n(z) = vi?~(z) V (m) v(m)

n , n

W (m) w(m) n , n

LIST OF SYMBOLS 267

CarathOOdory class or positive real functions, 210

Q polynomials of the first kind and the coef. vector, 30

Q polynomial, 214

Rutishauser polynomials, 173,174,179

Rutishauser polynomials, 214

polynomials, 177,181

triangular matrices, 113,122

Rand R (residual) series for a fis, 32

R series, 217

triangular matrices, 116,114,122

Schur-Szego, 32

lower star operation on fis, 83

Schur class of contractions, 210

Toeplitz determinant, 104

Toeplitz determinant, 214

Toeplitz matrix, 24

Toeplitz matrix, 218

operator spaces, 196

U and U polynomials for a fis, 44

U polunomial, 216 A

partial U and U polynomials for a fis, 62

partial U polynomial, 214

triangular matrices, 116

V and V polynomials for a fis, 44

V polunomial, 216

partial V and V polynomials for a fis, 62

partial V polynomial, 214

lower triangular Toeplitz matrices, 120,124

lower triangular Toeplitz matrices, 120,125

268

X (m) x(m) n , n

y(m) y(m) n , n

Z+,Z-

z(m)(z ),z(m)(z)

Z(z) = Z(O)(z)

7Z

7Tk

p~m)

A (m) Pn a(m)

n

11m :n

II~m)(z)

rrn(z) = rr~O)(z)

II(m)(Z) rr(m)(z) k,n , k,n II (z)=II(O)(z) k,n k,n

r~m)(z), r~m)(z)

r n(Z) = r~O)(z)

LIST OF SYMBOLS

lower triangular Toeplitz matrices, 120,124

lower triangular Toeplitz matrices, 120,125

index boundaries for zeros, 155

halves of a Laurent series, 38

CarathOOdory function, 210

the set of integers, 39

SS parameters, 30

reflection coefficient, 214

winding number, 156

linear functional, 83

linear functional, 98

poles of meromorphic function, 156

p-parameter for a Laurent series, 77

p-parameter for a Laurent series, 79

a-parameter for a Laurent series, 77

a-parameter for a Laurent series, 79

~oebiustransforms, 60

transmission coefficient, 226

e matrix of the recurrence, 55

zeros of meromorphic function, 156

projection operator, 22

II series for a fls, 57

II series, 216

partial II series for a fls, 62

II series, 216

rand r series for a fls, 57

r series, 214

r(z) A A(-l) ,

{}~1n)(z), O~1n)(z)

{}n(z) = n~O)(z)

};~)(z),i~1n)(z)

};n(z) = };~O)(z) 6~)

LIST OF SYMBOLS

Schur function, 210

shift operator and its left inverse, 196

{} and.o series for a fis, 61

.0 series, 217

}; and }; series for a fis, 60

}; series, 217

9 matrix of composite recurrence, 55

269

SUBJECT INDEX

Algorithms Algorithm 1, 31 Algorithm 2,34 Algorithm 2a, 58 Algorithm 3a, 69 Algorithm 3b, 80 Algorithm 4, 76 Algorithm 5, 76 Algorithm inverse la, 59 Algorithm inverse 1 b, 62 Algorithm of S~hur, 215 Algorithm of Schur-Cohn, 216 Algorithm of Trench-Zohar, Algorithm of Levinson, 221 Algorithm of Pick-Nevanlinna, 229 qd algorithm, 81 1TZ; algorithm, 81 Gram-Schmidt algorithm, 118 Choleskya1gorithm, 118

Bilinear forms < . . >"(m) 84218 [.,.j(m),98 ' {.,.}(m),101

Blaschke product, 208,215 Block structure

of Pade table, 145 of Laurent-Pade table, 149,219 of two-point Pade table, 149 of Chebyshev-Pade table, 219 of Fourier-Pade table, 219 of T-tab1e, 163

Cauchy product, 23 Christoffel-Darboux formula

for <.,.>(m), 95 for <.,.>(m), 100 for {.,}(m), 102

Continued fraction, 18,47 definition, 18 convergent of, 18 contraction of, 20 expansions, 49,51 forward evaluation scheme, 50 determinant formula, 13,38,89

Contraction of a CF even contraction, 20 odd contraction, 20

Coefficients parial correlation, 221 reflection, 224 transmission, 227 Fourier, 195 McLaurin, 197

Determinants Toeplitz determinants, 104,142,212 Hankel determinants, 104 asymptotics for Toeplitz det, 157,159 Jacobi's identity for Toeplitz

determinants, 105 determinant formula, 13,38,89

Equations telegraphers equations, 224 wave equations, 225 Toeplitz operator equation, 195

272 SUBJECT INDEX

Wiener-Hopf equation, 198 Yule-Walker equations, 220

Equivalence of pairs, 20 Extension

pseudomeromorphic, 208,219

Factorization inner-outer factorization, 208 Beurling factorization, 208 of Toeplitz matrix, 115 of Hankel matrix, 123 of meromorphic fct, 156 of chain scattering matrix, 229 of Toeplitz operators, 197

Filter ARMA filtering, 230 lattice filter, 223 modeling filter, 222 whitening filter, 222 Wiener filter, 222 back ward error filter, 222 forward error filter, 222 minimal phase filter, 222

Flow graph see also Port definition, 14 node of a flow graph, 14 branch of a flow graph, 14 transmittance of a flow graph, 14 lattice flow graph, 15,21,223 ladder flow graph, 15,22,223

Formal Laurent series Cauchy product of, 23 definition, 22 projection operator for, 22 inverse of, 23 quotient of, 23 normal fls, 24

Functionals A. (Tn) 83 , )..(Tn) 98 ,

Functions inner function, 208

meromorphic function, 155,198 outer function, 208 positive real function, 210 Caratheodory function, 210 Schur function, 210,229 contraction, 210 scattering function, 229 transmission function, 229 system function, 222

Function classes Caratheooory class, 210 Schur class, 210 IF class, 209

Gohberg-Semencu1 formula, 121 Gram-Schmidt, 118

Index of a function see winding number

Jacobi matrices, 110,128,179 identity for Toeplitz det., 105

Laurent polynomial definition, 23

B and B Laurent pols., 40

Matrices Toeplitz matrix, 104,112 Hankel matrix, 104,122 Jacobi matrix, 110,128,179 inversion of Toeplitz matrix, 120,127 inversion of Hankel matrix, 125 factorization of Toeplitz matrix, 115 factorization of Hankel matrix, 123 anti-unit matrix, 104 Gram matrix, 98,118,218 9 matrix, 55,213 scattering matrix, 228 chain scattering matrix, 227,229

Moment problems, 207 Montessus de Ballore theorem, 167

Node node value, 14

SUBJECT INDEX 273

sink. node, 14 source node, 14

Operator delay operator, 222 expectation operator, 219 projection operator, 22 Toeplitz operator, 195 inverse of Toeplitz operator, 197 shift operator, 196

Order of a fis, 23

Orthogonal orthogonality principle, 220 biorthogonal polynomials for

<o,o>(m),85 orthogonal polynomials for [o,o](m),

98 orthogonal polynomials for {o,o}(m),

101 Szego orthogonal polynomials, 218

Pade approximants, 24,37 definition, 24 Chebyshev-Pade approximant, 154 Fourier-Pade approximant, 219 Laurent-Pade approximant, 26,39,

201,213,230 two point Pade approximant, 27,43,

82 Pade forms

Pade form, 26,143 Chebyshev-Pade form, 154,219 Fourier-Pade form, 219 Laurent-Pade form, 27,143,151 two point Pade form, 28,143150

I ' Pade tables Pade table, 26,149 Chebyshev-Pade table, 153,219 Fourier-Pade table, 219 Laurent-Pade table, 153 two point Pade table, 150

Parahermitean conjugate, 83 Parameters

u and it parameters, 30,32,85,104,212

v and v parameters, 30,32,104,212 Schur-Szego parameters, 30,85,104,

184,219 a and b parameters, 65,104,

184,193,212

a and b parameters, 78,184,193,212 F and G parameters, 73,213

A A

F and G parameters, 79,184,193,213 p and 0 parameters, 75,105,213

p and & parameters, 79,101,105,213 convergence of, 183,193

Paths sawtooth, 65 row, 72,195 staircase, 73 diagonal, 75 column, 173

Polynomials see also Laurent polynomials see also,. Orthogonal Q and Q polynomials, 30,213 of the first kind, 33,213 A and A polynomials, 38,87,213 of the ~econd k.ind, 38,87,213 U and U polynomials, 44

A

partial!! and U polynomials, 62,214 V and V polynomials, 44 partial V and V polynomials, 62,214 Hadamard polynomials, 167 Rutishauser polynomials, 173,179,

200 dual Rutishauser polynomials,

188,191 Port

m-port,14 load of a port, 14

Prediction predictor, 220 prediction error, 220 forward prediction, 220

274 SUBJECT INDEX

back ward prediction, 220 Process

autoregressive, 222 regular, 222 stochastic process, 219 innovation process, 220 white noise, 222

Projection projection operator, 22 projection method, 196

Quadrature, 90

Reproducing kernel for <',.>(m), 94,119 for <',.>(m), 100 for {.,.}(m), 102 recursion for, 97 determinant expression for, 118,124

Residual, 25,29,32 Riesz-Herg1otz representation, 210 Rhombus rules

for ab parameters, 68 for ab parameters, 78 for FG parameters, 73

for FG parameters, 81 qd algorithm, 81 7Tt algorithm, 82

Scattering lossless inverse scattering, 224 scattering matrix, 229 scattering function, 228 chain scattering matrix, 227,229

Series see also Formal Laurent series Fourier series, 210,214 Laurent series, 197 McLaurin series, 167,197 formal power series, 23

A

P and P series, 32,213 A

Rand R series, 32,213 A

II and II series, 57,213

A

partial II and II series, 62,214 A

rand r series, 57,213 A

E and E series, 60,213 A

nand n series, 61,213 Stochastic proces

definition, 219 wide sense stationary, 219

Symbol of Toeplitz operator, 197 System

definition, 221 impulse response of, 221 input of, 221 output of, 221 system function, 222 time invariant, 221

Transforms Moebius transforms definition, 11 inverse of Moebius transform, 12 Moebius transforms, 55,213 dual Moebius transform, 13 determinant formula, 13,38,89 discrete Fourier transform, 219

Transmission line, 224

Wave equation, 225 incoming, 225 outgoing, 225

Wiener-Hopf equation, 198 Wiener-Kinchin theorem, 219 Winding number, 156,196,205

definition, 156 Yule-Walker equations, 220