A continuous Euler transformation and its application to the Fourier transform of a slowly decaying function

Ooura, Takuya

doi:10.1016/s0377-0427(99)00378-7

Cited by 16 publications

(15 citation statements)

References 2 publications

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“…Step 1 and 3 are respectively based on accurate numerical formulas (3.3) and (3.56) for the Fourier transform proposed by Ooura (2001Ooura ( , 2005, which are respectively combined with the nonuniform FFT and the fractional FFT to speed up the computations.…”

Section: Discussionmentioning

confidence: 99%

“…Step 1 and an indefinite integration by the sinc-Gauss sampling formula proposed in Section 3.2 to obtain approximate values of (Ooura, 2001), and the fractional FFT (Bailey & Swarztrauber, 1991;Chourdakis, 2005;Tanaka, 2014a) to obtain the approximate values of the solution p(x,t) for x = nĥ (n = −N + 1, . .…”

Section: Outline Of the Proposed Methodsmentioning

confidence: 99%

“…Therefore, we can compute approximate solutions of p(x,t) for any t with the same computational cost, and errors of numerical time integration do not occur. The second objective of the proposed method is to present new applications of Ooura's methods (Ooura, 2001(Ooura, , 2005 for the Fourier transforms to obtain the solution of PIDEs. The high accuracy of the proposed method is due to the very fast convergence of Ooura's methods, and speeding up of computations by Ooura's methods is realized by combining them with the nonuniform FFT (Dutt & Rokhlin, 1993, 1995Greengard & Lee, 2004;Potts et al, 2001;Steidl, 1998) or the fractional FFT (Bailey & Swarztrauber, 1991;Chourdakis, 2005;Tanaka, 2014a).…”

Section: R\{0}mentioning

confidence: 99%

“…, N) denote the approximations of (2.5) or (2.7) computed in Step 2. In order to approximate the inverse Fourier transform (2.8), we use the formula for the Fourier transform with a continuous Euler transform introduced by Ooura (2001). Define w(y; p, q) by…”

Section: Step 3: Computation Of the Inverse Fourier Transformmentioning

confidence: 99%

“…To propose the method, we use Ooura's accurate numerical formulas for the Fourier transform (Ooura, 2001(Ooura, , 2005, and propose a numerical indefinite integration formula based on the sinc-Gauss sampling formula (Tanaka et al, 2008) to compute the integrals with respect to the Lévy measures in the equation. Furthermore, we combine the fast Fourier transform (FFT) with these formulas to speed up the numerical computations.…”

Section: Introductionmentioning

confidence: 99%

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A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

Tanaka¹

2015

IMA J Numer Anal

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In this paper, we propose a fast and accurate numerical method based on Fourier transform to solve Kolmogorov forward equations of symmetric scalar Lévy processes. The method is based on the accurate numerical formulas for Fourier transform proposed by Ooura. These formulas are combined with nonuniform fast Fourier transform (FFT) and fractional FFT to speed up the numerical computations. Moreover, we propose a formula for numerical indefinite integration on equispaced grids as a component of the method. The proposed integration formula is based on the sinc-Gauss sampling formula, which is a function approximation formula. This integration formula is also combined with the FFT. Therefore, all steps of the proposed method are executed using the FFT and its variants. The proposed method allows us to be free from some special treatments for a non-smooth initial condition and numerical time integration. The numerical solutions obtained by the proposed method appeared to be exponentially convergent on the interval if the corresponding exact solutions do not have sharp cusps. Furthermore, the real computational times are approximately consistent with the theoretical estimates.

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Section: Discussionmentioning

confidence: 99%

Section: Outline Of the Proposed Methodsmentioning

confidence: 99%

Section: R\{0}mentioning

confidence: 99%

Section: Step 3: Computation Of the Inverse Fourier Transformmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

Tanaka¹

2015

IMA J Numer Anal

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Appendix 1: Performance of Grounding Systems in Transient Conditions

2021

Electrical Safety Engineering of Renewable Energy Systems

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Drell-Yan qT resummation of fiducial power corrections at N3LL

Ebert

Michel

Stewart

et al. 2021

J. High Energ. Phys.

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We consider Drell-Yan production pp → V*X → LX at small qT ≪ Q, where qT and Q are the total transverse momentum and invariant mass of the leptonic final state L. Experimental measurements require fiducial cuts on L, which in general introduce enhanced, linear power corrections in qT/Q. We show that they can be unambiguously predicted from factorization, and resummed to the same order as the leading-power contribution. For the fiducial qT spectrum, they constitute the complete linear power corrections. We thus obtain predictions for the fiducial qT spectrum to N3LL and next-to-leading-power in qT/Q. Matching to full NNLO ($$ {\alpha}_s^2 $$ α s 2 ), we find that the linear power corrections are indeed the dominant ones, and once included by factorization, the remaining fixed-order corrections become almost negligible below qT ≲ 40 GeV. We also discuss the implications for more complicated observables, and provide predictions for the fiducial ϕ* spectrum at N3LL+NNLO. We find excellent agreement with ATLAS and CMS measurements of qT and ϕ*. We also consider the $$ {p}_T^{\mathrm{\ell}} $$ p T ℓ spectrum. We show that it develops leptonic power corrections in qT/(Q − 2$$ {p}_T^{\mathrm{\ell}} $$ p T ℓ ), which diverge near the Jacobian peak $$ {p}_T^{\mathrm{\ell}} $$ p T ℓ ∼ Q/2 and must be kept to all powers to obtain a meaningful result there. Doing so, we obtain for the first time an analytically resummed result for the $$ {p}_T^{\mathrm{\ell}} $$ p T ℓ spectrum around the Jacobian peak at N3LL+NNLO. Our method is based on performing a complete tensor decomposition for hadronic and leptonic tensors. We show that in practice this is equivalent to often-used recoil prescriptions, for which our results now provide rigorous, formal justification. Our tensor decomposition yields nine Lorentz-scalar hadronic structure functions, which for Z/γ* → ℓℓ or W → ℓν directly map onto the commonly used angular coefficients, but also holds for arbitrary leptonic final states. In particular, for suitably defined Born-projected leptons it still yields a LO-like angular decomposition even when including QED final-state radiation. Finally, we also discuss the application to qT subtractions. Including the unambiguously predicted fiducial power corrections significantly improves their performance, and in particular makes them applicable near kinematic edges where they otherwise break down due to large leptonic power corrections.

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A continuous Euler transformation and its application to the Fourier transform of a slowly decaying function

Cited by 16 publications

References 2 publications

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

Appendix 1: Performance of Grounding Systems in Transient Conditions

Drell-Yan qT resummation of fiducial power corrections at N3LL

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