Reverse Bernstein Inequality on the Circle
Abstract
The more then hundred years old Bernstein inequality states that the supremum norm of the derivative of a trigonometric polynomial of fixed degree can be bounded from above by supremum norm of the polynomial itself. The reversed Bernstein inequality, that we prove in this note, says that the reverse inequality holds for functions in the orthogonal complement of the space of polynomials of fixed degree. In fact, we derived a more general result for the lower bounds on higher derivatives. These bounds are better then those obtained by applying bound for the first derivative successively several times.
 Publication:

arXiv eprints
 Pub Date:
 February 2023
 DOI:
 10.48550/arXiv.2302.10122
 arXiv:
 arXiv:2302.10122
 Bibcode:
 2023arXiv230210122J
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 42A05
 EPrint:
 8 pages, additional references in v2