Abstract
This work is intended to directly supplement the previous work by Coutsias and Kazarinoff on the foundational understanding of lacunary trigonometric systems and their relation to the Fresnel integrals, specifically the Cornu spirals [Physica 26D (1987) 295]. These systems are intimately related to incomplete Gaussian summations. The current work provides a focused look at the specific system built off of the triangular numbers. The special cyclic character of the triangular numbers modulo m carries through to triangular lacunary trigonometric systems. Specifically, this work characterizes the families of Cornu spirals arising from triangular lacunary trigonometric systems. Special features such as self-similarity, isometry, and symmetry are presented and discussed.
Highlights
With the current work, the authors seek to connect with and augment a very interesting study by Coutsias and Kazarinoff, which was published in 1987 [1]
This work added to the foundational understanding of lacunary trigonometric systems and their relation to the Fresnel integrals and the Cornu spirals
Such systems are intimately related to incomplete Gaussian summations and a thorough analysis was previously provided by Coutsias and Kazarinoff
Summary
The authors seek to connect with and augment a very interesting study by Coutsias and Kazarinoff, which was published in 1987 [1]. Those authors looked at the lacunary trigonometric system characterized by the sequence, S, of partial summations. This sequence of partial summations gives rise to a beautiful family of Cornu spirals [1]. The triangular lacunary system is less general the one studied by Coutsias and Kazarinoff, but on the other hand, its periodic nature offers an interesting generalization to that work. All of the numerical calculations for this work were a direct summation of the lacunary sequence members by MATHEMATICA, some recent work by Yamada and Ikeda has investigated the use of Padé approximate methods for speeding up the convergence of summations for lacunary functions [11,12]
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