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The first sign change of a cosine polynomial


Authors: James D. Nulton and Kenneth B. Stolarsky
Journal: Proc. Amer. Math. Soc. 84 (1982), 55-59
MSC: Primary 42A05; Secondary 33A10
DOI: https://doi.org/10.1090/S0002-9939-1982-0633277-7
MathSciNet review: 633277
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Abstract: It is reasonable to expect the first sign change of a real cosine polynomial to decrease when its smallest frequency is increased. Many cases in which this is true are exhibited, but it is shown that there exist (presumably unusual) cosine polynomials for which the first sign change may increase by an arbitrarily large amount.


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  • [1] P. J. Cohen, On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960), 191-212. MR 0133397 (24:A3231)
  • [2] H. Davenport, On a theorem of P. J. Cohen, Mathematika 7 (1960), 93-97. MR 0124681 (23:A1992)
  • [3] D. G. Dickson, The asymptotic distribution of zeros of exponential sums, Publ. Math. Debrecen 11 (1964), 295-300. MR 0173770 (30:3979)
  • [4] -, Zeros of exponential sums, Proc. Amer. Math. Soc. 78 (1965), 84-89. MR 0171903 (30:2129)
  • [5] M. Kac, On the distribution of values of trigonometric sums with linearly independent frequencies, Amer. J. Math. 65 (1943), 609-615. MR 0009061 (5:96a)
  • [6] J. E. Littlewood, Some problems in real and complex analysis, Heath Math. Mono., Heath, Lexington, Mass., 1968. MR 0244463 (39:5777)
  • [7] S. K. Pichorides, A lower bound for the $ {L^1}$ norm of exponential sums, Mathematika 21 (1974), 155-159. MR 0371831 (51:8048)
  • [8] -, Norms of exponential sums, Publ. Math. D'Orsay, nos. 77-73, Université de Paris-Sud, Orsay, 1976.
  • [9] G. Polya, Geometrisches über die Verteilung der Nullstellen gewisser ganzer transzendenter Funktionen, Münchener Sitzungsberichte 50 (1920), 285-290.
  • [10] -, On polar singularities of power series, and of Dirichlet series, Proc. London Math. Soc. (2) 33 (1932), 85-101 (esp. p. 89).
  • [11] A. J. van der Poorten, On the number of zeros of functions, Enseign. Math. 23 (1977), 19-38.
  • [12] A. J. van der Poorten and R. Tijdeman, On common zeros of exponential polynomials, Enseign. Math. 21 (1975), 57-67. MR 0379387 (52:292)
  • [13] J. J. Price, Topics in orthogonal functions, Amer. Math. Monthly 82 (1975), 594-609. MR 0370048 (51:6277)
  • [14] K. F. Roth, On cosine polynomials corresponding to sets of integers, Acta Arith. 24 (1973), 87-98. MR 0342475 (49:7221)
  • [15] E. Schwengeler, Geometrisches über die Verteilung der Nullstellen spezieller ganzer funktionen, Thesis, Zürich, 1925.
  • [16] P. Stein, On the real zeros of a certain trigonometric function, Proc. Cambridge Philos. Soc. 31 (1935), 455-457.
  • [17] A. A. Talalyan, The representation of measurable functions by series, Russian Math. Surveys 15 (5) (1960), 77-136. MR 0125401 (23:A2704)
  • [18] M. Uchiyama (née Katayama) and S. Uchiyama, On the cosine problem, Proc. Japan Acad. 36 (1960), 475-479. MR 0123545 (23:A870)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0633277-7
Keywords: Cosine polynomial, exponential polynomial, frequencies, sign changes, zeros
Article copyright: © Copyright 1982 American Mathematical Society

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