TY - JOUR
T1 - Bounding multivariate trigonometric polynomials
AU - Pfister, Luke
AU - Bresler, Yoram
N1 - Funding Information:
Manuscript received March 2, 2018; revised August 2, 2018, October 25, 2018, and November 21, 2018; accepted November 21, 2018. Date of publication November 29, 2018; date of current version December 19, 2018. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Eleftherios Kofidis. This work was supported by the National Science Foundation (NSF) under Grant CCF-1320953. (Corresponding author: Luke Pfister.) L. Pfister is with the Electrical and Computer Engineering, University of Illinois at Urbana Champaign, Champaign, IL 61801 USA (e-mail:, lpfiste2@ illinois.edu).
Publisher Copyright:
© 1991-2012 IEEE.
PY - 2019/2/1
Y1 - 2019/2/1
N2 - The extremal values of multivariate trigonometric polynomials are of interest in fields ranging from control theory to filter design, but finding the extremal values of such a polynomial is generally NP-Hard. In this paper, we develop simple and efficiently computable estimates of the extremal values of a multivariate trigonometric polynomial directly from its samples. We provide an upper bound on the modulus of a complex trigonometric polynomial, and develop upper and lower bounds for real trigonometric polynomials. For a univariate polynomial, these bounds are tighter than existing bounds, and the extension to multivariate polynomials is new. As an application, the lower bound provides a sufficient condition to certify global positivity of a real trigonometric polynomial.
AB - The extremal values of multivariate trigonometric polynomials are of interest in fields ranging from control theory to filter design, but finding the extremal values of such a polynomial is generally NP-Hard. In this paper, we develop simple and efficiently computable estimates of the extremal values of a multivariate trigonometric polynomial directly from its samples. We provide an upper bound on the modulus of a complex trigonometric polynomial, and develop upper and lower bounds for real trigonometric polynomials. For a univariate polynomial, these bounds are tighter than existing bounds, and the extension to multivariate polynomials is new. As an application, the lower bound provides a sufficient condition to certify global positivity of a real trigonometric polynomial.
KW - Multivariate polynomials
KW - Oversampling
KW - Positive trigonometric polynomials
KW - Trigonometric interpolation
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U2 - 10.1109/TSP.2018.2883925
DO - 10.1109/TSP.2018.2883925
M3 - Article
AN - SCOPUS:85057818542
SN - 1053-587X
VL - 67
SP - 700
EP - 707
JO - IRE Transactions on Audio
JF - IRE Transactions on Audio
IS - 3
M1 - 8552451
ER -