About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Kofroň, Josef
Interpolation operators on the space of holomorphic functions on the unit circle. (English). Applications of Mathematics, vol. 46 (2001), issue 3, pp. 161-189
MSC: 30E05, 41A05, 41A50, 41A80, 46N40, 65D05, 65D30 | MR 1828304 | Zbl 1065.30035 | DOI: 10.1023/A:1013787823049
Keywords:
numerical interpolation; optimal interpolatory rule with prescribed nodes; optimal interpolatory rule with free nodes; remainder estimation
Summary:
The aim of the paper is to get an estimation of the error of the general interpolation rule for functions which are real valued on the interval $[-a,a]$, $a\in (0,1)$, have a holomorphic extension on the unit circle and are quadratic integrable on the boundary of it. The obtained estimate does not depend on the derivatives of the function to be interpolated. The optimal interpolation formula with mutually different nodes is constructed and an error estimate as well as the rate of convergence are obtained. The general extremal problem with free weights and knots is solved.
References:
[1] K. Yosida: Functional Analysis. Springer-Verlag, Berlin, 1965. Zbl 0126.11504
[2] N. I. Achiezer: Lectures on the Theory of Approximation. Nauka, Moscow, 1965. (Russian)
[3] H. Engels: Ein numerisches Differentiationsverfahren mit ableitungsfreien Fehlerschranken. Z. Angew. Math. Mech. 52 (1972), 145–154. DOI 10.1002/zamm.19720520303 | MR 0298938 | Zbl 0251.65012
[4] A. Paulik: Zur Existenz optimaler Quadraturformeln mit freien Knoten bei Integration analytischer Funktionen. Numer. Math. 27 (1977), 395–405. MR 0436555 | Zbl 0363.65020
[5] G. Meinardus: Approximation von Funktionen und ihre numerische Behandlung. Springer-Verlag, Berlin, 1964. MR 0176272 | Zbl 0124.33103
[6] I. P.  Natanson: Konstruktive Funktionentheorie. Akademie-Verlag, Berlin, 1955. MR 0069915 | Zbl 0065.29502
[7] A. Ralston: A First Course in Numerical Analysis. Mc Graw-Hill, New York, 1965. MR 0191070 | Zbl 0139.31603
[8] W. Knauff, A. Paulik: A note on Davis type error bounds. BIT 18 (1978), 175–183. DOI 10.1007/BF01931694 | MR 0499233
[9] J. Kofroň: Die ableitungsfreien Fehlerabschätzungen von Interpolationsformeln. Apl. Mat. 17 (1972), 137–152. MR 0295527
[10] W. Rudin: Real and Complex Analysis. Mc Graw-Hill, New York, 1966, 1974. MR 0344043
Partner of
EuDML logo