Abstract
In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval \(x \in [-\chi, \chi]\), to a function \(\tilde{f}\) which is periodic on the larger interval \(x \in [-\Theta, \Theta]\). We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval \(x \in [-\chi, \chi]\), identically zero for \(|x| < \Theta\), and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) \(\mathcal{T}(x; L, \chi, \Theta)=(1+\mbox{erf}(z))/2\) where \(z=L \xi/\sqrt{1-\xi^{2}}\) where \(\xi \equiv -1 + 2 (\Theta-x)/(\Theta - \chi)\). By applying steepest descents to approximate the coefficient integrals in the limit of large degree j, we show that when the width L is fixed, the Fourier cosine coefficients a j of \(\mathcal{T}\) on \(x \in [-\Theta, \Theta]\) are proportional to \(a_{j} \sim (1/j) \exp(- L \pi^{1/2} 2^{-1/2} (1-\chi/\Theta)^{1/2} j^{1/2}) \Lambda(j)\) where Λ(j) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the Nth term, the width should be chosen to increase with N as \(L=0.91 \sqrt{1 - \chi/\Theta} N^{1/2}\). We derive similar asymptotics for the function f(x)=x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergence
Similar content being viewed by others
References
Averbuch A., Braverman E., Coifman R., Israeli M., Sidi A. (2000). Efficient computation of oscillatory integrals via adaptive multiscale local Fourier bases. Appl. Comput. Harmonic Anal 9:19–53
Averbuch A., Ioffe L., Israeli M., Vozovoi L. (1997). Highly scalable two- and three-dimensional Navier-Stokes parallel solvers on MIMD multiprocessors. J. Supercomput 11(1):7–39
Averbuch A., Ioffe L., Israeli M., Vozovoi L. (1998). Two-dimensional parallel solver for the solution of Navier–Stokes equations with constant and variable coefficients using ADI on cells. Parallel Comput. 24(5–6):673–699
Averbuch A., Israeli M., Vozovoi L., (1995). Parallel implementation of nonlinear evolution problems using parabolic domain decomposition. Parallel Comput 21(7):1151–1183
Averbuch A., Vozovoi L., Israeli M., (1997). On a fast direct elliptic solver by a modified Fourier method. Numer. Algorithms 15(3–4):287–313
Bittner K., Chui C.K. (1999). From local cosine bases to global harmonics. Appl. Comput. Harmonic Anal 6:382–399
Boyd J.P. (1982). The optimization of convergence for Chebyshev polynomial methods in an unbounded domain. J. Comput. Phys 45: 43–79
Boyd J.P. (1984). The asymptotic coefficients of Hermite series. J. Comput. Phys 54:382–410
Boyd J.P. (1987). Orthogonal rational functions on a semi-infinite interval. J. Comput. Phys 70:63–88
Boyd J.P. (1987). Spectral methods using rational basis functions on an infinite interval. J. Comput. Phys 69:112–42
Boyd J.P. (1989). New directions in solitons and nonlinear periodic waves: Polycnoidal waves, imbricated solitons, weakly non-local solitary waves and numerical boundary value algorithms. In Wu T.-Y., Hutchinson J.W (eds). Advances in Applied Mechanics. Academic Press, New York, pp. 1–82
Boyd J. P. (1994). The rate of convergence of Fourier coefficients for entire functions of infinite order with application to the Weideman-Cloot sinh-mapping for pseudospectral computations on an infinite interval. J. Comput. Phys 110:360–72
Boyd J.P. (1996). Asymptotic Chebyshev coefficients for two functions with very rapidly or very slowly divergent power series about one endpoint. Appl. Math. Lett 9(2):11–15
Boyd J.P. (1997). Construction of Lighthill’s unitary functions: The imbricate series of unity. Applied Mathematics Comput. 86:1–10
Boyd J.P. (2001). Additive blending of local approximations into a globally- valid approximation with application to new analytic approximations to the dilogarithm. Appl. Math. Lett 14:477–481
Boyd J.P. (2001). Chebyshev and Fourier Spectral Methods. Dover, Mineola, New York, 2nd edition, 665 pp.
Boyd J.P. (2002). A comparison of numerical algorithms for Fourier Extension of the First, Second and Third Kinds. J. Comput. Phys 178:118–60
Boyd, J. P. (2005). Fourier embedded domain methods: Extending a function defined on an irregular region to a rectangle so that the extension is spatially periodic and c ∞. Appl. Math. Comput 161(2):591–97
Boyd J.P. (2005). Limited-area Fourier spectral models and data analysis schemes: Windows, Fourier extension, Davies relaxation and all that. Month. Weather Rev 133:2030–2043
Cloot A., Weideman J.A.C. (1992). An adaptive algorithm for spectral computations on unbounded domains. J. Computational Phys. 102:398–406
Coifman R., Meyer Y. (1991). Remarques sur l’analyse de Fourier á fenêtre, série. C. R. Acad. Sci. Paris Sér I. Math 312:259–61
Elghaoui M., Pasquetti R. (1996). Mixed spectral-boundary element embedding algorithms for the Navier-Stokes equations in the vorticity-stream function formulation. J. Comput. Phys 153:82–100
Elghaoui M., Pasquetti R. (1996). A spectral embedding method applied to the advection-diffusion equation. J. Comput. Phys 125:464–76
Elliott D. (1964). The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function. Math. Comp 18:274–84
Elliott, D. (1965). Truncation errors in two Chebyshev series approximations. Math. Comp 19:234–48
Elliott D., Szekeres G. (1965). Some estimates of the coefficients in the Chebyshev expansion of a function. Math. Comp 19:25–32
Elliott D., Tuan P.D. (1974). Asymptotic coefficients of Fourier coefficients. SIAM J. Math. Anal 5:1–10
Garbey M. (2000). On some applications of the superposition principle with Fourier basis. SIAM J. Sci. Comput 22(3): 1087–116
Garbey M., Tromeur-Dervout D. (1998). A new parallel solver for the nonperiodic incompressible Navier-Stokes equations with a Fourier method: Application to frontal polymerization. J. Comput. Phys 145(1):316–31
Haugen J. E., Machenhauer B. (1993). A spectral limited-area model formulation with time-dependent boundary conditions applied to the shallow-water equations. Monthly Weather Rev 121:2618–30
Högberg M., Henningson D.S. (1998). Secondary instability of cross-flow vortices in falkner-skan-cooke boundary layers. J. Fluid Mech 368:339–57
Israeli M., Vozovoi L., Averbuch A. (1993). Spectral multidomain technique with local Fourier basis. J. Sci. Comput 8(2):135–49
Israeli M., Vozovoi L., Averbuch A. (1994). Domain decomposition methods with local fourier basis for parabolic problems. Contemp. Math 157:223–30
Jawerth B., Sweldens W. (1995). Biorthogonal smooth local trigonometric bases. J. Fourier Anal. Appl 2(2):109–33
Matviyenko G. (1996). Optimized local trigonometric bases. Appl. Comput. Harmonic Anal 3:301–23
Miller G.F. (1966). On the convergence of Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal 3:390–409
Németh G. (1992). Mathematical Approximation of Special Functions: Ten Papers on Chebyshev Expansions. Nova Science Publishers, New York, 200 pp
Nordström J., Nordin N., Henningson D. (1999). The fringe region technique and the Fourier method used in the direct numerical simulation of spatially evolving viscous flows. SIAM J. Sci. Comput 20(4):1365–93
Schumer J.W., Holloway J.P. (1998). Vlasov simulations using velocity-scaled Hermite representations. J. Comput. Phys 144(2):626–61
Shen J. (2000). Stable and efficient spectral methods in unbounded domains using Laguerre functions. SIAM J. Numer. Anal 38:1113–33
Tang T. (1993). The Hermite spectral method for Gaussian-type functions. SIAM J. Scientific Comput. 14:594–606
Vozovoi L., Israeli M., Averbuch A. (1996). Multi-domain Local Fourier method for PDEs in complex geometries. J. Comput. Appl. Math 66(1-2):543–55
Vozovoi L., Weill A., Israeli M. (1997). Spectrally accurate solution of non-periodic differential equations by the Fourier-Gegenbauer method. SIAM J. Numer. Anal 34(4):1451–71
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boyd, J.P. Asymptotic Fourier Coefficients for a C∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem. J Sci Comput 29, 1–24 (2006). https://doi.org/10.1007/s10915-005-9010-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-005-9010-7