Harmonic Synthesis — Theoretical Bounds

Gavin Brown
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Gavin Brown: University of New South Wales

A chapter in Fourier Techniques and Applications, 1985, pp 171-181 from Springer

Abstract: Abstract At its simplest harmonic synthesis amounts to combining a finite number of pure vibrations — in other words constructing a trigonometric polynomial. Simple though it sounds, there is active research in the area and I will discuss four examples: I. I. There exists εn → 0 and phases φk = φk (n) such that $$\left({1 - {\varepsilon _n}} \right)\sqrt n \le \left| {\sum\limits_{k = 1}^n {{e^{ik\theta}}}} \right.{e^{i\phi}}\left. k \right| \le (1 + {\varepsilon _n})\sqrt n,$$ for all real θ (Byrnes, Körner, Kahane; 1977, 1980). II. Given decreasing positive Ak with Ak -1 concave there are phases φk = φk (m,n) such that $$\mathop {\sup}\limits_\theta \left| {\sum\limits_n^m {{A_k}{e^{ik\theta}}\left. {{e^{i\phi}}_k} \right|}} \right. \le C{\left({\sum\limits_n^m {A_k^2}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}},$$ where C is an absolute constant and the last sum is taken to be less than one (Brown, Hewitt; 1978)

Keywords: London Math; Trigonometric Polynomial; Absolute Constant; Theoretical Bound; Sine Series (search for similar items in EconPapers)
Date: 1985
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DOI: 10.1007/978-1-4613-2525-3_9

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