The Function sin x x

Gearhart, W. B.; Shultz, Harris S.

doi:10.2307/2686748

Cited by 29 publications

(21 citation statements)

References 3 publications

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“…Vieta discovered that the sinc function can be represented as infinite product of the cosine functions [1,5]:…”

Section: Incomplete Cosine Expansionmentioning

confidence: 99%

“…ð Þ . Then the expansion of the function g t ð Þ in terms of sinc function series (1) that may not be necessarily integrable or expressed in terms of the rational functions for efficient computation since the sinc function is not an elementary function. In this work we propose a methodology of sampling based on the incomplete cosine expansion series of the sinc function that effectively resolves this problem and, as an example, we demonstrate its effectiveness in derivation of a new approximation of the Voigt/complex error function where only 16 summation terms are sufficient to obtain accuracy better than 10 À14 over almost all the complex domain of practical importance.…”

Section: Introductionmentioning

confidence: 99%

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Sampling by incomplete cosine expansion of the sinc function: Application to the Voigt/complex error function

Abrarov

Quine

2015

Applied Mathematics and Computation

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a b s t r a c tA new sampling methodology based on an incomplete cosine expansion series is presented as an alternative to the traditional sinc function approach. Numerical integration shows that this methodology is efficient and practical. Applying the incomplete cosine expansion we obtain a rational approximation of the complex error function that with the same number of the summation terms provides an accuracy exceeding the Weideman's approximation accuracy by several orders of the magnitude. Application of the expansion results in an integration consisting of elementary function terms only. Consequently, this approach can be advantageous for accurate and rapid computation.

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“…Vieta discovered that the sinc function can be represented as infinite product of the cosine functions [1,5]:…”

Section: Incomplete Cosine Expansionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Sampling by incomplete cosine expansion of the sinc function: Application to the Voigt/complex error function

Abrarov

Quine

2015

Applied Mathematics and Computation

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“…We start by considering [Gea90], the cardinal sinus can be expressed as an infinite product, i.e., If we truncate the infinite product to a finite product with J factors, then, thanks to the cosine product-to-sum identity [Qui13], we have…”

Section: Coefficients Computationmentioning

confidence: 99%

“…We present two different alternatives to compute the coefficients. While the first one is based on Vieta's formula [Gea90], which allows us to expand the cardinal sinus function into a combination of cosines, the second one relies on the application of Parseval's identity. In the second method, we benefit from compact support features of the Fourier transform of the Shannon basis functions, avoiding this way the truncation of the infinite domain of integration.…”

Section: Swiftmentioning

confidence: 99%

A Highly Efficient Shannon Wavelet Inverse Fourier Technique for Pricing European Options

Ortiz-Gracia

Oosterlee

2015

SSRN Journal

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Abstract. In the search for robust, accurate, and highly efficient financial option valuation techniques, we here present the SWIFT method (Shannon wavelets inverse Fourier technique), based on Shannon wavelets. SWIFT comes with control over approximation errors made by means of sharp quantitative error bounds. The nature of the local Shannon wavelets basis enables us to adaptively determine the proper size of the computational interval. Numerical experiments on European-style options show exponential convergence and confirm the bounds, robustness, and efficiency.

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“…Using the classical Vieta formula [5], the cardinal sinus can be expressed as an infinite product, i.e.,…”

Section: Density Coefficientsmentioning

confidence: 99%

A Highly Efficient Pricing Method for European-Style Options Based on Shannon Wavelets

Ortiz-Gracia

Oosterlee

2017

Trends in Mathematics

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In the search for robust, accurate and highly efficient financial option valuation techniques, we present here the SWIFT method (Shannon Wavelets Inverse Fourier Technique), based on Shannon wavelets. SWIFT comes with control over approximation errors made by means of sharp quantitative error bounds. The nature of the local Shannon wavelets basis enables us to adaptively determine the proper size of the computational interval. Numerical experiments on European-style options confirm the bounds, robustness and efficiency.

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The Function sin x x

Cited by 29 publications

References 3 publications

Sampling by incomplete cosine expansion of the sinc function: Application to the Voigt/complex error function

Sampling by incomplete cosine expansion of the sinc function: Application to the Voigt/complex error function

A Highly Efficient Shannon Wavelet Inverse Fourier Technique for Pricing European Options

A Highly Efficient Pricing Method for European-Style Options Based on Shannon Wavelets

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