Chebyshevian splines

Wronicz, Zygmunt

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Chebyshevian splines

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Zygmunt Wronicz

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Rozprawy Matematyczne tom/nr w serii: 305 wydano: 1990

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CONTENTS
Introduction...........................................................................................................5
I.   Canonical complete Chebyshev systems
   1. Canonical complete Chebyshev systems.......................................................7
   2. Interpolation by generalized polynomials and divided differences................12
   3. The Markov inequality for generalized polynomials......................................16
II.   Chebyshevian splines
   1. Basic properties...........................................................................................18
   2. B-splines......................................................................................................21
   3. The Marsden identity...................................................................................28
   4. De Boor's inequalities..................................................................................32
   5. A recurrence relation for B-splines...............................................................37
   6. Bounds on zeros..........................................................................................41
III.   Spline operators
   1. Orthogonal spline projections .....................................................................46
   2. Biorthogonal systems..................................................................................49
   3. Equivalence of spline bases .......................................................................57
   4. Positive spline operators and orthogonal splines .......................................60
IV.    Generalized moduli of smoothness and approximation by splines
   1. Generalized moduli of smoothness .............................................................64
   2. Generalization of the Whitney Theorem.......................................................70
   3. Best approximation by splines......................................................................72
   4. The Bernstein type inequality for splines ....................................................77
V.   Applications to approximation of analytic functions
   1. Approximation by analytic splines................................................................78
   2. Biorthogonal systems in the complex space A(D)........................................83
   3. Systems conjugate to biorthogonal spline systems......................................86
References.........................................................................................................94
List of symbols....................................................................................................98

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Kategoryzacja MSC:

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 305

Liczba stron

98

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CCCV

Daty

wydano

1990

otrzymano

1989-12-06

Twórcy

autor

Zygmunt Wronicz

Institute of Mathematics, Academy of Mining and Metallurgy, Al. Mickiewicza 30, 30-059 Kraków, Poland

Bibliografia

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[8] S. V. Bochkarev, Conjugate Franklin system—a basis in the space of continuous functions, Dokl. Akad. Nauk SSSR 285 (1985), 521-526 (in Russian).
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[26] G. Freud and V. Popov, On approximation by spline functions, in: Proc. Conf. on Constructive Theory of Functions, G. Alexits and S. B. Stechkin (eds.), Akadémiai Kiadó, Budapest 1969, 163-172 (in Russian).
[27] G. Freud and V. Popov, Some questions related to approximation by spline functions and polynomials, Studia Sei. Math. Hungar. 5 (1970), 161-171 (in Russian).
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[32] H. Johnen, Inequalities connected with the moduli of smoothness, Math. Ves. 9 (1972), 289-303.
[33] H. Johnen and K. Scherer, Direct and inverse theorems for best approximation by A-splines, in: Spline Functions, K. Böhmer, G. Meinardus and W. Schemp (eds.), Lecture Notes in Math. 501, Springer, New York 1975, 116-131.
[34] S. Karlin, Total Positivity, Vol. 1, Stanford Univ. Press, 1968.
[35] S. Karlin and W. J. Studden, Tchebycheff Systems: with Applications in Analysis and Statistics, Interscience, New York 1966.
[36] B. S. Kashin and A. A. Saakjan, Orthogonal Series, Nauka, Moscow 1984 (in Russian).
[37] M. T. Krein and A. A. Nudel'man, The Markov Moment Problem and Extremal Problems, Nauka, Moscow 1973 (in Russian).
[38] F. Lej a, Theory of Analytic Functions, PWN, Warszawa 1957 (in Polish).
[39] T. Lyche, A recurrence relation for Chebyshevian B-splines, Cons.tr. Approx. 1 (1985), 155-173.
[40] M. J. Marsden, An identity for spline functions with applications to variation-diminishing spline approximation, J. Approx. Theory 3 (1970), 7-49.
[41] G. Meinardus, Approximation of Functions, Theory and Numerical Methods, Springer, Heidelberg 1967.
[42] G. Mühlbach, A recurrence formula for generalized divided differences and some applications, J. Approx. Theory 9 (1973), 165-172.
[43] G. Mühlbach, Čebyšev-Systeme und Lipschitzklassen, ibid., 192-203.
[44] G. Mühlbach, The general Neville-Aitken-algorithm and some applications, Numer. Math. 31
(1978), 97-110.
[45] G. Mühlbach, The general recurrence relation for divided differences and the general Newton-interpolation-algorithm with applications to trigonometric interpolation, ibid. 32
(1979), 393-408.
[46] T. Popoviciu, Sur quelques propriétés des fonctions d'une ou de deux variables réelles, Mathematica (Cluj) 8 (1934), 1-85.
[47] G. Mühlbach, Sur la reste dans certaines formules linéaires d'approximation de l'analyse, ibid. 1 (24) (1959), 95-142.
[48] I. I. Privalov, Boundary Properties of Analytic Functions, Gostekhizdat, Moscow 1950 (in Russian).
[49] L. A. Rubel, A. L. Shields and B. A. Taylor, Mergelian sets and the modulus of continuity of analytic functions, J. Approx. Theory 15 (1975), 23-40.
[50] M. A. Rutman, Integral representation of functions forming a Markov series, Dokl. Akad. Nauk SSSR 164 (1965), 989-992 (in Russian).
[51] P. Sablonnière, Positive spline operators and orthogonal splines, J. Approx. Theory 52 (1988), 28-42.
[52] K. Scherer and L. L. Schumaker, A dual basis for L-splines and applications, ibid. 29 (1980), 151-169.
[53] I. J. Schoenberg, On trigonometric spline interpolation, J. Math. Mech. 13 (1964), 795-825.
[54] S. Schonefeld, Schauder bases in the Banach space C²(T²), Trans. Amer. Math. Soc. 165 (1972), 300-318.
[55] L. L. Schumaker, On Tchebycheffian spline functions, J. Approx. Theory 18 (1976), 278-303.
[56] L. L. Schumaker, Towards a constructive theory of generalized spline functions, in: Spline Functions, K. Böhmer, G. Meinardus and W. Schemp (eds.), Lecture Notes in Math. 501, Springer, New York 1976, 265-331.
[57] L. L. Schumaker, Spline Functions: Basic Theory, Wiley, New York 1981.
[58] B. Sendov, A new proof of the H. Whitney's Theorem, C. R. Acad. Bulg. Sci. 35 (1982), 609-611.
[59] L. L. Schumaker, The constants of H. Whitney are bounded, ibid. 38 (1985), 1299-1302.
[60] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.
[61] Yu. N. Subbotin, Approximation by splines and smooth bases in C(0,2π), Mat. Zametki 12 (1972), 43-52 (in Russian).
[62] P. M. Tamrazov, Smoothness and Polynomial Approximation, Naukova Dumka, Kiev 1975 (in Russian).
[63] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Fizmatgiz, Moscow 1960 (in Russian).
[64] H. Whitney, On functions with bounded nth differences, J. Math. Pures Appl. 36 (1957), 67-95.
[65] Z. Wronicz, On the approximation and interpolation by multiple splines, Zeszyty Nauk. Uniw. Jagielloń. Prace Mat. 17 (1975), 147-157.
[66] Z. Wronicz, On the application of the orthonormal Franklin system to the approximation of analytic functions, in: Approximation Theory, Z. Ciesielski (ed.), Banach Center Publ. 4, PWN, Warszawa 1979, 305-316.
[67] Z. Wronicz, Approximation by complex splines, Zeszyty Nauk. Uniw. Jagielloń. Prace Mat. 20 (1979), 67-88.
[68] Z. Wronicz, Construction of an orthonormal basis in the space of functions analytic in a disc and of class C^n in its closure, ibid. 21 (1979), 91-96.
[69] Z. Wronicz, Interpolation by complex cubic splines, in: Constructive Function Theory '77, Bl. Sendov and D. Vačov (eds.), Publ. House of the Bulgar. Acad. Sci., Sofia 1980, 549-558.
[70] Z. Wronicz, On approximation by analytic splines, in: Approximation and Function Spaces, Z. Ciesielski (ed.), PWN and North-Holland, Warszawa-Amsterdam 1981, 867-879.
[71] Z. Wronicz, The Bernstein type inequality for splines, Bull. Acad. Polon. Sci. 30 (1982), 235-237.
[72] Z. Wronicz, On approximation by complex splines, in: Constructive Function Theory '81, B. Sendov, B. Boyanov, D. Vačov, R. Maleev, S. Markov and T. Boyanov (eds.), Publ. House of the Bułgar. Acad. Sci., Sofia 1983, 577-583.
[73] Z. Wronicz, Moduli of smoothness associated with Chebyshev systems and approximation by L-splines, in: Constructive Theory of Functions '84, Bl. Sendov, P. Petrushev, R. Maleev and S. Tashev (eds.), Publ. House of the Bułgar. Acad. Sci., Sofia 1984, 906-916.
[74] Z. Wronicz, On some properties of LB-splines, Ann. Polon. Math. 46 (1985), 379-388.
[75] Z. Wronicz, The Marchaud inequality for generalized moduli of smoothness, in: Rational Approximation and its Applications in Mathematics and Physics, J. Gilewicz, M. Pindor, W. Siemaszko (eds.), Lecture Notes in Math. 1237, Springer, 1987, 134-144.
[76] Z. Wronicz, On equivalence of spline bases in L_p spaces, Bull. Polish Acad. Sci. Math. 36 (1988), 273-278.
[77] Z. Wronicz, Systems conjugate to biorthogonal spline systems, ibid., 279-288.
[78] A. Zygmund, Trigonometric Series, Vol. I, Cambridge Univ. Press, 1959.

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1985 Mathematics Subject Classification: 41A15, 46E15, 46B15

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Książka - szczegóły

Chebyshevian splines

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