ISSN:
1432-0940
Keywords:
42A05
;
42A10
;
26C05
;
Trigonometric polynomials
;
Inequalities
;
Zeros
;
Geometric convergence
;
Bounded analytic functions
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract This is a contribution to the theory of “incomplete trigonometric polynomials”T n , but mainly for the case when their zeros are not concentrated at just one point, but are distributed in some intervalI whose length is not too large. We begin with the simple theorem that if ∥T n ∥ ≤ 1 and ifT n has ≥θn, 0〈θ〈 2, zeros at 0, thenT n (t) must be small on the interval |t|〈2 arcsin (θ/2). There are similar but more complicated and more difficult to prove results whenT n has ≥θn zeros onI. These results have the following application: IfT n →f a.e., and if ∥T n 〉∥∞〈-1, thenf vanishes on a set of the circleT whose measure is controlled by lim sup (N n /n), whereN n is the number of zeros ofT n onT. In turn, this has further applications to series of polynomials, to norms of Lagrange operators, and to Hardy classes.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01890020
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