Abstract
The use of Padé-approximants for the solution of mathematical problems in science has great development. Padé-approximants have proved to be very useful in numerical analysis too: the solution of a nonlinear equation, acceleration of convergence, numerical integration by using nonlinear techniques, the solution of ordinary and partial differential equations. Especially in the presence of singularities the use of Padé-approximants has been very interesting.
Yet we have tried to generalize the concept of Padé-approximant to operator theory, departing from "power-series-expansions" as is done in the classical theory*. A lot of interesting properties of classical Padé-approximants remain valid and the classical Padé-approximant is now a special case of the theory. The notion of abstract Padé-table is introduced; it also consists of squares of equal elements as in the classical theory.
This work is supported by I.W.O.N.L. (Belgium)
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Abbreviations
- R +o :
-
{positive real numbers}
- X,Y:
-
always normed vectorspaces or Banach-spaces or Banach-algebras with unit
- L(X,Y):
-
{linear bounded operators L : X → Y}
- L(Xk,Y):
-
{k-linear bounded operators L : X → L(Xk−1,Y)}
- Λ:
-
field R or C
- λ,μ,...:
-
elements of Λ
- O:
-
unit for addition in a Banach-space, or multilinear operator L ≠ L(Xk,Y) such that Lx1 ... xk = 0 ∨(x1,...,xk) ≠ Xk
- I:
-
unit for multiplication in a Banach-algebra
- 1:
-
unit for multiplication in Λ
- F,G,...:
-
non-linear operators : X → Y
- B(xo,r):
-
open ball with centre xo ≠ X and radius r > o
- \(\bar B(x_0 ,r)\) :
-
closed ball with centre xo ≠ X and radius r > o
- P,Q,R,S,T,...:
-
non-linear operators : X → Y, usually abstract polynomials
- ϖP,ϖQ,...:
-
exact degree of the abstract polynomial P,Q,...
- F(k)(xo):
-
kth Fréchet-derivative of the operator F : X → Y in xo
- D(G):
-
{x ≠ X|G(x) is regular in Y} for the operator G : X → Y (=Banach-algebra)
- Ai,Bj,Ck,Ds :
-
i-linear, j-linear, k-linear, s-linear operators
References
de BRUIN, M.G. and van ROSSUM, H. Formal Padé-approximation. Nieuw Archief voor Wiskunde (3), 23, 1975, pp. 115–130.
GODEMENT, R. Algebra. Kershaw Publ. Co. Ltd., London, 1969.
KARLSSON, J. and WALLIN, H. Rational approximation by an interpolation procedure in several variables. In: Staff, E.B. and Varga, R.S. Padé and rational approximation: theory and appl. Academic Press, London, 1977, pp. 83–100.
LARSEN, R. Banach-Algebras, an introduction. Marcel Dekker, Inc. New York, 1973.
RALL, L.B. Computational Solution of Nonlinear Operator Equations. J. Wiley and Sons. New York, 1969.
BESSIS, J.D. and TALMAN, J.D. Variational approach to the theory of operator Padé approximants. Rocky Mountain Journ. Math. 4(2), 1974, pp. 151–158.
CHENEY, E.W. Introduction to Approximation Theory chapter 5 section 6, McGraw-Hill, New York, 1966.
CHISHOLM, J.S.R. N-variable rational approximants. In: Saff, E.B. and Varga, R.S. Padé and rational approximation: theory and appl. Academic Press, London, 1977, pp. 23–42.
COMMON, A.K. and GRAVES-MORRIS, P.R. Some properties of Chisholm Approximants. Journ. Inst. Math. Applics 13, 1974, pp. 229–232.
GAMMEL, J.L. Review of two recent generalizations of the Padé approximant. In: Graves-Morris, P.R. Padé-approximations and their appl. Academic Press, London, 1973, pp. 3–9.
GRAVES-MORRIS, P.R. and HUGHES JONES, R. and MAKINSON, G.J. The calculation of some rational approximants in two variables. Journ. Inst. Math. Applics 13, 1974, pp. 311–320.
HUGHES JONES, R. General Rational Approximants in N-Variables. Journal of approximation Theory 16, 1976, pp. 201–233.
KARLSSON, J. and WALLIN, H. Rational approximation by an interpolation procedure in several variables. In: Saff, E.B. and Varga, R.S. Padé and rational approximation: theory and appl. Academic Press, London, 1977, pp. 83–100.
LUTTERODT, C.H. Rational approximants to holomorphic functions in n-dimensions. Journ. Math. Anal. Applic. 53, 1976, pp. 89–98.
SHAFER, R.E. On quadratic approximation. SIAM Journ. Num. Anal. 11(2), 1974, pp.447–460.
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Cuyt, A.A.M. (1979). Abstract Padé-approximants in operator theory. In: Wuytack, L. (eds) Padé Approximation and its Applications. Lecture Notes in Mathematics, vol 765. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085575
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DOI: https://doi.org/10.1007/BFb0085575
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