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Linear operators generated by Sonnenschein matrices

Published online by Cambridge University Press:  09 April 2009

B. Wood
B. Wood
Affiliation:
Department of Mathematics, The University of Arizona, Tucson, Arizona 85721, U.S.A.
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Abstract

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An approximation method based on a certain Sonnenschein matrix is studied. Results are obtained for approximation in an interval and in the complex plane. A connection between convergence of the approximation process and regularity of the matrix is also discussed.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 3 , May 1978 , pp. 375 - 384
Copyright
Copyright © Australian Mathematical Society 1978

References

Agnew, R. P. (1944), ‘Euler Transformations’, American Journal of Mathematics 66, 313–338.CrossRefGoogle Scholar
Bjasanski, B. (1956), ‘Sur une classe générale de procédés de sommations du type d'Euler-Borel’ Publ. Inst. Math. Beograd 10, 131–152.Google Scholar
Clunie, J. and Vermes, P. (1959), ‘Regular Sonnenschein Type Summability Methods’ Acad Roy. Belg. Bull. Cl. Sci. 45, 930–945.Google Scholar
Cooke, R. G. (1950), Infinite Matrices and Sequence Spaces (Macmillan, N.Y.).Google Scholar
Hardy, G. (1949), Divergent Series (Oxford, England).Google Scholar
King, J. (1968), ‘A Class of Positive Linear Operators’, Canad. Math. Bull. 11, 51–59.CrossRefGoogle Scholar
King, J. and Swetits, J. J. (1970), ‘Positive Linear Operators and Summability’, J. Australian Math. Soc. 11, 3, 281–290.CrossRefGoogle Scholar
Knopp, K. (1926), ‘Uber Polynomentwicklungen im Mittag-Letflerschem Stern’, Acta Mathematica 47, 313–335.CrossRefGoogle Scholar
Korovkin, P. P. (1960), Linear Operators and Approximation Theory (Gordon and Breach, N.Y.).Google Scholar
Lorentz, G. G. (1948), ‘A contribution to the Theory of Divergent Sequences’, Acta Math. 80, 167–190.CrossRefGoogle Scholar
Lorentz, G. G. (1953), Bernstein Polynomials (Math. Expositions, No. 8, University of Toronto Press, Toronto).Google Scholar
Perron, O. (1923), ‘Uber eine Verallgemeinerung der Eulerschen Reihentransformation’, Math. Zeitschrift 18, 157–172.CrossRefGoogle Scholar
Ramanujan, M. S. (1963), ‘On the Sonnenschein Methods of Summability’, Proc. Japan Acad. 39, 432–434.Google Scholar
Sledd, W. T. (1962), ‘The Gibbs Phenomenon and Lebesgue Constants for Regular Sonnenschein Matrices’, Canadian Journal of Math. 14, 723–728.CrossRefGoogle Scholar
Sledd, W. T. (1963), ‘Summability of Ordinary Dirichler Series by Perron-Type Matrices’, Michigan Math. Journal 10, 33–41.CrossRefGoogle Scholar
Sledd, W. T. (1963), ‘Regularity Conditions for Karamata Matrices,’ Journal London Math. Soc. 38, 105–107.CrossRefGoogle Scholar
Sonnenschein, J. (1949), ‘Sur les series divergentes’, Bull. Acad. Royale de Beigique 35, 594–601.Google Scholar
Vermes, P. (1949), ‘Series to Series Transformations and Analytic Continuation by Matrix Methods’, American Journal of Math. 71, 54–562.CrossRefGoogle Scholar
Vermes, P. (1957), ‘The Transpose of a Summability Matrix’, Colloque sur la Theorie des Suites, Centre Belge de Recherches Mathematiques, Bruxelles, 60–86.Google Scholar