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Journal of the London Mathematical Society 1992 s2-45(2):255-264; doi:10.1112/jlms/s2-45.2.255
© 1992 by London Mathematical Society
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© Oxford University Press

Remez-Type Inequalities on the Size of Generalized Polynomials

Tamás Erdélyi

Department of Mathematics The Ohio State University 231 West Eighteenth Avenue Columbus Ohio 43210 USA

Generalized polynomials are defined as products of polynomials raised to positive real powers. The generalized degree can be defined in a natural way. A number of classical inequalities holding for polynomials can be extended for generalized polynomials utilizing the generalized degree in place of the ordinary one. Remez established a sharp upper bound for the maximum modulus on [– 1,1] of algebraic polynomials of degree at most n if the measure of the subset of [– 1,1], where the modulus of the polynomial is at most 1, is known. In this paper a numerical version of the Remez inequality is extended for generalized complex algebraic polynomials and its trigonometric and pointwise algebraic analogues are discussed. The results are new even for ordinary polynomials.


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