Best Meromorphic Approximation of Markov Functions on the Unit Circle

Foundations of Computational Mathematics

Volumn 1, Issue 4, 2001, Pages 385-416

  Baratchart, L ^a   Prokhorov, V A ^b   Saff, E B ^c  

^a INRIA   (France)

^b UNIVERSITY OF SOUTH ALABAMA   (United States)

^c UNIVERSITY OF SOUTH FLORIDA   (United States)

Author keywords

Best approximation; Markov functions; Meromorphic approximation

Indexed keywords

FUNCTIONS;

ANALYTIC FUNCTIONS; BEST APPROXIMATIONS; BOREL MEASURES; COMPACT SETS; LIMITING DISTRIBUTIONS; MEROMORPHIC APPROXIMATIONS; MEROMORPHIC FUNCTION; UNIT CIRCLES;

POLYNOMIAL APPROXIMATION;

EID: 0009573926     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s002080010015     Document Type: Article

References (58)

1. N. I. Achiezer, Elements of the Theory of Elliptic Functions, 2nd rev. ed., Nauka, Moscow, 1970; English transl., American Mathematical Society, Providence, RI, 1990.
2. V. M. Adamyan, D. Z. Arov, and M. G. Kreǐn, Infinite Hankel matrices and generalized Carathéodory-Fejér and Riesz problem, Functional Anal. Appl. 2 (1968), 1-18.
3. V. M. Adamyan, D. Z. Arov, and M. G. Kreǐn, Analytic properties of Schmidt pairs, Hankel operators, and the generalized Schur-Takagi problem, Mat. Sb. 86 (128) (1971), 34-75; English transl., Math. USSR-Sb. 15(1971).
4. V. M. Adamyan, D. Z. Arov, and M. G. Kreǐn, Analytic properties of Schmidt pairs, Hankel operators, and the generalized Schur-Takagi problem, Mat. Sb. 86 (128) (1971), 34-75; English transl., Math. USSR-Sb. 15(1971).
5. J.-E. Andersson, Best rational approximation to Markov functions, J. Approx. Theory 76 (1994), 219-232.
6. G. A. Baker and P. R. Graves-Morris, Padé Approximants, 2nd ed., Encyclopedia of Mathematics, Vol. 59, Cambridge University Press, Cambridge, 1996.
7. p of the circle: System-theoretic motivations, critical points and error rates, in Computational Methods in Function Theory (N. Papamichael, St. Ruscheweyh, and E. B. Saff, eds.), World Scientific, Singapore, 1999, pp. 45-78.
8. L. Baratchart, J. Leblond, F. Mandrea, and E. B. Saff, How can meromorphic approximation help to solve 2D inverse problems for the Laplacian?, Inverse Problems 15 (1999), 79-90.
9. L. Baratchart, V. A. Prokhorov, and E. B. Saff, On Hankel operators associated with Markov functions, Proceedings IWOTA Conference, Bordeaux, France, 2000.
10. 2 rational approximation, J. Analyse Math. 70 (1996), 225-266.
11. p analog to the A.A.K. theory for p ≥ 2 (submitted).
12. 2 rational best aproximants to Markov functions, J. of Approx. Theory (submitted).
13. 2 of the circle, Constr. Approx. 17 (2001), 103-138.
14. 2 and Stieltjes integrals: a uniqueness theorem, Constr. Approx. 9 (1993), 1-21.
15. W. Barrett, On the convergence of rational approximations to analytic functions of a certain class, J. Inst. Math. Appl. 7 (1971), 308-323.
16. D. Braess, Rational approximation of Stieltjes functions by the Carathéodory-Fejér method, Constr. Approx. 3 (1987), 43-50.
17. R. W. Brockett and P. A. Fuhrmann, Normal symmetric dynamical systems, SIAM J. Control Optim. 14 (1976), 107-119.
18. J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory, Macmillan, New York, 1992.
19. S. D. Fisher and C. A. Micchelli, The n-widths of sets of analytic functions, Duke Math. J. 47 (1980), 789-801.
20. S. D. Fisher, Function Theory on Planar Domains, Wiley, New York, 1983.
21. q. J. Approx. Theory 67 (1991), 347-356.
22. ∞ Control Theory, Lecture Notes in Control and Information Sciences, Vol. 88, Springer-Verlag, New York, 1987.
23. J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
24. ∞-error bounds, Internat. J. Control 39 (6) (1984), 1115-1193.
25. A. A. Gonchar, On the degree of rational approximation of certain analytic functions, Mat. Sb. 105 (147) (1978), 147-163; English transl., Math. USSR-Sb. 34 (1978).
26. A. A. Gonchar, On the degree of rational approximation of certain analytic functions, Mat. Sb. 105 (147) (1978), 147-163; English transl., Math. USSR-Sb. 34 (1978).
27. A. A. Gonchar and G. L. Lopez, Markov's theorem for multipoint Padé approximants (Russian), Mat. Sb. 105 (147) (1978), 512-524.
28. A. A. Gonchar and E. A. Rakhmanov, The equilibrium measure and the distribution of zeros of extremal polynomials, Mat. Sb. 125 (167) (1984), 117-127; English transl., Math. USSR-Sb. 53 (1986).
29. A. A. Gonchar and E. A. Rakhmanov, The equilibrium measure and the distribution of zeros of extremal polynomials, Mat. Sb. 125 (167) (1984), 117-127; English transl., Math. USSR-Sb. 53 (1986).
30. E. J. Hannan and M. Deistler, The Statistical Theory of Linear Systems, Wiley, New York, 1988.
31. p-Spaces, Cambridge University Press, Cambridge, 1980.
32. N. S. Landkof, Foundations of Modern Potential Theory, Nauka, Moscow, 1966; English transl., Springer-Verlag, Berlin, 1972.
33. p-spaces, 2 ≤ p ≤ ∞ and on the disc algebra, Bull. London Math. Soc. 25 (1993), 275-281.
34. C. Le Merdy, The AAK theorem in spaces of type 2, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 513-517 (French).
35. A. L. Levin and E. B. Saff, Szegõ-type asymptotics for minimal Blaschke products, in Progress in Approximation Theory (A. A. Gonchar and E. B. Saff, eds.), Springer-Verlag, New York, 1992, pp. 105-126.
36. L. Ljung, System Identification: Theory for the User, Prentice-Hall, Englewood Cliffs, NJ, 1987.
37. F. Mandrea, Approximation meromorphe et application la detection de fissures, thèse de doctorat, Université de Nice, 2001.
38. A. Markoff [A. A. Markov], Deux démonstrations de la convergence de certaines fractions continues, Acta Math. 19 (1985), 93-104.
39. Z. Nehari, On bounded bilinear forms, Ann. of Math. 65 (1957), 153-162.
40. O. G. Parfenov, Estimates of the singular numbers of a Carleson operator. Mat. Sb. 131 (173) (1986), 501-518; English transl., Math. USSR-Sb. 59 (1988).
41. O. G. Parfenov, Estimates of the singular numbers of a Carleson operator. Mat. Sb. 131 (173) (1986), 501-518; English transl., Math. USSR-Sb. 59 (1988).
42. O. G. Parfenov, Widths of a class of analytic functions, Mat. Sb. 117 (159) (1982); English transl., Math. USSR-Sb. 45 (1983).
43. O. G. Parfenov, Widths of a class of analytic functions, Mat. Sb. 117 (159) (1982); English transl., Math. USSR-Sb. 45 (1983).
44. O. G. Parfenov, The singular numbers of embedding operators for certain classes of analytic and harmonic functions, J. Soviet Math. 35 (1986), 2193-2200.
45. O. G. Parfenov, Asymptotics of the singular numbers of imbedding operators for certain classes of analytic functions, Math. USSR-Sb. 43 (1982).
46. A. Pietsch, Eigenvalues and s-Numbers, Cambridge University Press, Cambridge, 1987.
47. A. Pinkus, N-Widths in Approximation Theory, Springer-Verlag, New York, 1985.
48. p-generalization of a theorem of Adamyan, Arov, and Kreǐn (manuscript).
49. V. A. Prokhorov, On a theorem of Adamyan, Arov, and Kreǐn, Mat. Sb. 184 (1993), 89-104; English transl., Russian Acad. Sd. Sb. Math. 78 (1994).
50. V. A. Prokhorov, On a theorem of Adamyan, Arov, and Kreǐn, Mat. Sb. 184 (1993), 89-104; English transl., Russian Acad. Sd. Sb. Math. 78 (1994).
51. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Heidelberg, 1997.
52. H. Stahl, Strong asymptotics for orthogonal polynomials with varying weights, Acta Sci. Math. (Szeged) 65 (1999), 717-762.
53. H. Stahl and V. Totik, General Orthogonal Polynomials, Encyclopedia of Mathematics, Vol. 43, Cambridge University Press, New York, 1992.
54. V. Totik, Weighted Approximation with Varying Weights, Lecture Notes in Mathematics, Vol. 1300, Springer-Vertag, Berlin, 1994.
55. M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.
56. J. Weidmann, Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, Vol. 68, Springer-Verlag, New York, 1980.
57. J. C. Willems, Dissipative dynamical systems, Part I; general theory, Part II: linear systems with quadratic supply rates, Arch. Rational Mech. Anal. 45 (1972), 321-351, 352-392.
58. G. Zames, Feedback and optimal sensitivity: model reference transformations, multiplicative semi-norms, and appropriate inverses, IEEE Trans. Automat. Control AC28 (1981), 585-601.