Interpolation properties of generalized perfect splines and the solutions of certain extremal problems. I.(English)Zbl 0303.41011

MSC:

 41A15 Spline approximation 41A05 Interpolation in approximation theory 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

 [1] S. D. Fisher and J. W. Jerome, Perfect spline solutions to $${L^\infty }$$ extremal problems (preprint). · Zbl 0292.49021 [2] G. Glaeser, Prolongement extrémal de fonctions différentiables d’une variable, J. Approximation Theory 8 (1973), 249 – 261 (French). Collection of articles dedicated to Isaac Jacob Schoenberg on his 70th birthday, III. · Zbl 0259.41011 [3] Samuel Karlin, Total positivity. Vol. I, Stanford University Press, Stanford, Calif, 1968. [4] Samuel Karlin, Total positivity, interpolation by splines, and Green’s functions of differential operators, J. Approximation Theory 4 (1971), 91 – 112. · Zbl 0228.41002 [5] Samuel Karlin, Some variational problems on certain Sobolev spaces and perfect splines, Bull. Amer. Math. Soc. 79 (1973), 124 – 128. · Zbl 0253.41010 [6] Samuel Karlin and John M. Karon, Poised and non-poised Hermite-Birkhoff interpolation, Indiana Univ. Math. J. 21 (1971/72), 1131 – 1170. · Zbl 0262.41002 [7] Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. · Zbl 0153.38902 [8] M. G. Kreĭn, The ideas of P. L. Čebyšev and A. A. Markov in the theory of limiting values of integrals and their further development, Amer. Math. Soc. Transl. (2) 12 (1959), 1 – 121. M. G. Kreĭn and P. G. Rehtman, Development in a new direction of the Čebyšev-Markov theory of limiting values of integrals, Amer. Math. Soc. Transl. (2) 12 (1959), 123 – 135. [9] R. Louboutin, Sur une bonne partition de l’unite. Appeared in Le prolongateur de Whitney. Vol. II; Ed. Glaeser, Univ. of Rennes, 1967. [10] I. J. Schoenberg, The perfect \?-splines and a time-optimal control problem, Israel J. Math. 10 (1971), 261 – 274. · Zbl 0273.41006 [11] I. J. Schoenberg and A. Cavaretta, Solution of Landau’s problem concerning higher derivatives on the half line, Report No. 1050, M.R.C., University of Wisconsin, Madison, Wis., 1970. · Zbl 0229.41004 [12] V. M. Tihomirov, Best methods of approximation and interpolation of differentiable functions in the space \?[-1,1]., Mat. Sb. (N.S.) 80 (122) (1969), 290 – 304 (Russian). · Zbl 0204.13301
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