On the convergence of the quasi-periodic approximations on a finite interval

Authors

  • Arnak Poghosyan Institute of Mathematics, NAS RA
  • Lusine Poghosyan Institute of Mathematics, NAS RA
  • Rafayel Barkhudaryan Institute of Mathematics, NAS RA; Yerevan State University

DOI:

https://doi.org/10.52737/18291163-2021.13.10-1-44

Keywords:

Truncated Fourier series, convergence acceleration, quasi-periodic interpolation, quasi-periodic approximation

Abstract

We investigate the convergence of the quasi-periodic approximations in different frameworks and reveal exact asymptotic estimates of the corresponding errors. The estimates facilitate a fair comparison of the quasi-periodic approximations to other classical well-known approaches. We consider a special realization of the approximations by the inverse of the Vandermonde matrix, which makes it possible to prove the existence of the corresponding implementations, derive explicit formulas and explore convergence properties. We also show the application of polynomial corrections for the convergence acceleration of the quasi-periodic approximations. Numerical experiments reveal the auto-correction phenomenon related to the polynomial corrections so that utilization of approximate derivatives surprisingly results in better convergence compared to the expansions with the exact ones.

References

J. W. Gibbs, Fourier's series, Letter in Nature, 59 (1899), pp. 200-200.

M. Bôcher, Introduction to the theory of Fourier's series, Ann. of Math. (2), 7 (1906), no. 3, pp. 81-152.

E. Hewitt and R. E. Hewitt, The Gibbs-Wilbraham phenomenon: an episode in Fourier analysis, Arch. Hist. Exact Sci., 21 (1979/80), no. 2, pp. 129-160. https://doi.org/10.1007/bf00330404

A. J. Jerri, The Gibbs phenomenon in Fourier analysis, splines and wavelet approximations, vol. 446 of Mathematics and its Applications, Dordrecht: Kluwer Academic Publishers, 1998. https://doi.org/10.1007/978-1-4757-2847-7_4

A. Zygmund, Trigonometric Series. Vol. 1,2, Cambridge Univ. Press, 1959.

L. Fejer, Untersuchungen über Fouriersche reihen, Math. Ann., 58 (1904), pp. 51-69.

K. Knopp, Infinite sequences and series, New York: Dover Publications Inc., 1956.

P. Wynn, On a device for computing the $e_ m(S_n)$ tranformation, Math. Tables Aids Comput., 10 (1956), pp. 91-96.

A. Haug, Theoretical solid state physics, New York: Pergamon Press Oxford, 1972.

D. Levin, Development of non-linear transformations of improving convergence of sequences, Internat. J. Comput. Math., 3 (1973), pp. 371-388.

A. Majda, J. McDonough, and S. Osher, The Fourier method for nonsmooth initial data, Math. Comp., 32 (1978), no. 144, pp. 1041-1081. https://doi.org/10.1090/s0025-5718-1978-0501995-4

J. Wimp, Sequence transformations and their applications, vol. 154 of Mathematics in Science and Engineering, New York: Academic Press, 1981.

D. A. Smith and W. F. Ford, Numerical comparisons of nonlinear convergence accelerators, Math. Comp., 38 (1982), no. 158, pp. 481-499. https://doi.org/10.1090/s0025-5718-1982-0645665-1

S. Biringen and K. H. Kao, On the application of pseudospectral FFT techniques to nonperiodic problems, Internat. J. Numer. Methods Fluids, 9 (1989), no. 10, pp. 1235-1267. https://doi.org/10.1002/fld.1650091006

J. P. Boyd, Sum-accelerated pseudospectral methods: the Euler-accelerated sinc algorithm, Appl. Numer. Math., 7 (1991), no. 4, pp. 287-296. https://doi.org/10.1016/0168-9274(91)90065-8

H. Vandeven, Family of spectral filters for discontinuous problems, J. Sci. Comput., 6 (1991), no. 2, pp. 159-192. https://doi.org/10.1007/bf01062118

C. Brezinski and M. Redivo Z., Extrapolation methods, vol. 2 of Studies in Computational Mathematics, Amsterdam: North-Holland Publishing Co., 1991.

D. Gottlieb, C.-W. Shu, A. Solomonoff, and H. Vandeven, On the Gibbs phenomenon. I. Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. Comput. Appl. Math., 43 (1992), no. 1-2, pp. 81-98. https://doi.org/10.1016/0377-0427(92)90260-5

D. Gottlieb, Issues in the application of high order schemes, in Algorithmic trends in computational fluid dynamics (1991), ICASE/NASA LaRC Ser., pp. 195-218, New York: Springer, 1993. https://doi.org/10.1007/978-1-4612-2708-3_12

H. H. H. Homeier, Some applications of nonlinear convergence accelerators, IJQC, 45 (1993), no. 6, pp. 545-562.

J. P. Boyd, A lag-averaged generalization of Euler's method for accelerating series, Appl. Math. Comput., 72 (1995), no. 2-3, pp. 143-166. https://doi.org/10.1016/0096-3003(94)00180-c

A. Dillmann and G. Grabitz, On a method to evaluate Fourier-Bessel series with poor convergence properties and its application to linearized supersonic free jet flow, Quart. Appl. Math., 53 (1995), no. 2, pp. 335-352. https://doi.org/10.1090/qam/1330656

D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon. IV. Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comp., 64 (1995), no. 211, pp. 1081-1095. https://doi.org/10.1090/s0025-5718-1995-1284667-0

D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon. V. Recovering exponential accuracy from collocation point values of a piecewise analytic function, Numer. Math., 71 (1995), no. 4, pp. 511-526. https://doi.org/10.1007/s002110050155

D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon. III. Recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function, SIAM J. Numer. Anal., 33 (1996), no. 1, pp. 280-290. https://doi.org/10.1137/0733015

L. Vozovoi, M. Israeli, and A. Averbuch, Analysis and application of Fourier-Gegenbauer method to stiff differential equations, SIAM J. Numer. Anal., 33 (1996), no. 5, pp. 1844-1863. https://doi.org/10.1137/s0036142994263591

C. Oleksy, A convergence acceleration method of Fourier series, Comput. Phys. Comm., 96 (1996), no. 1, pp. 17-26.

J. Geer and N. S. Banerjee, Exponentially accurate approximations to piece-wise smooth periodic functions, J. Sci. Comput., 12 (1997), no. 3, pp. 253-287.

A. Gelb and D. Gottlieb, The resolution of the Gibbs phenomenon for 'spliced' functions in one and two dimensions, Computers Math. Applic., 33 (1997), no. 11, pp. 35-58.

D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon and its resolution, SIAM Rev., 39 (1997), no. 4, pp. 644-668. https://doi.org/10.1137/s0036144596301390

L. Vozovoi, A. Weill, and M. Israeli, Spectrally accurate solution of nonperiodic differential equations by the Fourier-Gegenbauer method, SIAM J. Numer. Anal., 34 (1997), no. 4, pp. 1451-1471. https://doi.org/10.1137/s0036142994278814

J. P. Boyd, Two comments on filtering (artificial viscosity) for Chebyshev and Legendre spectral and spectral element methods: preserving boundary conditions and interpretation of the filter as a diffusion, J. Comput. Phys., 143 (1998), no. 1, pp. 283-288. https://doi.org/10.1006/jcph.1998.5961

G. Kvernadze, Determination of the jumps of a bounded function by its Fourier series, J. Approx. Theory, 92 (1998), no. 2, pp. 167-190. https://doi.org/10.1006/jath.1997.3125

S. L. Marshall, Convergence acceleration of Fourier series by analytical and numerical application of Poisson's formula, J. Phys. A, 31 (1998), no. 11, pp. 2691-2704. https://doi.org/10.1088/0305-4470/31/11/016

A. Gelb and E. Tadmor, Detection of edges in spectral data, Appl. Comput. Harmon. Anal., 7 (1999), no. 1, pp. 101-135.

A. Gelb and E. Tadmor, Detection of edges in spectral data. II. Nonlinear enhancement, SIAM J. Numer. Anal., 38 (2000), no. 4, pp. 1389-1408. https://doi.org/10.1137/s0036142999359153

G. Kvernadze, T. Hagstrom, and H. Shapiro, Detecting the singularities of a function of $V_ p$ class by its integrated Fourier series, Comput. Math. Appl., 39 (2000), no. 9-10, pp. 25-43. https://doi.org/10.1016/s0898-1221(00)00084-5

A. Gelb, A hybrid approach to spectral reconstruction of piecewise smooth functions, J. Sci. Comput., 15 (2000), no. 3, pp. 293-322.

H. N. Mhaskar and J. Prestin, On the detection of singularities of a periodic function, Adv. Comput. Math., 12 (2000), no. 2-3, pp. 95-131.

R. K. Wright, A robust method for accurately representing non-periodic functions given Fourier coefficient information, J. Comput. and Appl. Math., 140 (2002), no. 1, pp. 837-848. https://doi.org/10.1016/s0377-0427(01)00518-0

E. Tadmor and J. Tanner, Adaptive mollifiers for high resolution recovery of piecewise smooth data from its spectral information, Found. Comput. Math., 2 (2002), no. 2, pp. 155-189. https://doi.org/10.1007/s102080010019

A. Gelb and E. Tadmor, Spectral reconstruction of piecewise smooth functions from their discrete data, Math. Model. Numer. Anal., 36 (2002), no. 2, pp. 155-175. https://doi.org/10.1051/m2an:2002008

J.-H. Jung and B. D. Shizgal, Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon, J. Comput. Appl. Math., 172 (2004), no. 1, pp. 131-151. https://doi.org/10.1016/j.cam.2004.02.003

R. K. Wright, Local spline approximation of discontinuous functions and location of discontinuities, given low-order Fourier coefficient information, J. Comput. and Appl. Math., 164-165 (2004), pp. 783-795. https://doi.org/10.1016/s0377-0427(03)00647-2

R. Archibald, A. Gelb, and J. Yoon, Polynomial fitting for edge detection in irregularly sampled signals and images, SIAM J. Numer. Anal., 43 (2005), no. 1, pp. 259-279. https://doi.org/10.1137/s0036142903435259

J.-H. Jung and B. D. Shizgal, Inverse polynomial reconstruction of two dimensional Fourier images, J. Sci. Comput., 25 (2005), no. 3, pp. 367-399. https://doi.org/10.1007/s10915-004-4795-3

E. Tadmor and J. Tanner, Adaptive filters for piecewise smooth spectral data, IMA J. Numer. Anal., 25 (2005), no. 4, pp. 635-647. https://doi.org/10.1093/imanum/dri026

A. Nersessian and A. Poghosyan, Accelerating the convergence of trigonometric series, Cent. Eur. J. Math., 4 (2006), no. 3, pp. 435-448. https://doi.org/10.2478/s11533-006-0016-7

J.-H. Jung and B. D. Shizgal, On the numerical convergence with the inverse polynomial reconstruction method for the resolution of the Gibbs phenomenon, J. Comput. Phys., 224 (2007), no. 2, pp. 477-488. https://doi.org/10.1016/j.jcp.2007.01.018

S. Paszkowski, Convergence acceleration of orthogonal series, Numer. Algorithms, 47 (2008), no. 1, pp. 35-62. https://doi.org/10.1007/s11075-007-9146-7

J. P. Boyd, Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and H ermite coefficients and Fourier transforms, J. Engrg. Math., 63 (2009), no. 2-4, pp. 355-399. https://doi.org/10.1007/s10665-008-9241-3

B. Adcock, Gibbs phenomenon and its removal for a class of orthogonal expansions, BIT, 51 (2011), no. 1, pp. 7-41. https://doi.org/10.1007/s10543-010-0301-5

A. Poghosyan, On a convergence of the Fourier-Pade approximation, Armen. J. Math., 4 (2012), no. 2, pp. 49-79.

A. Poghosyan, On the convergence of rational-trigonometric-polynomial approximations realized by roots of Laguerre polynomials, Izv. Nats. Akad. Nauk Armenii Mat., 48 (2013), no. 6, pp. 82-91. https://doi.org/10.3103/s1068362313060101

A. Poghosyan, On a convergence of the Fourier-Pade interpolation, Armen. J. Math., 5 (2013), no. 1, pp. 1-25.

A. Nersessian, On some fast implementations of Fourier interpolation, in Operator Theory and Harmonic Analysis (A. N. Karapetyants, V. V. Kravchenko, E. Liflyand, and H. R. Malonek, eds.), pp. 463-477, Springer International Publishing, 2021. https://doi.org/10.1007/978-3-030-77493-6_27

A. Krylov, Lectures on approximate calculations (lectures delivered in 1906 in Russian). St. Petersburg: Tipolitography of Birkenfeld, 1907.

G. P. Tolstov, Fourier series. Translated from the Russian by Richard A. Silverman, Englewood Cliffs, N.J.: Prentice-Hall Inc., 1962.

C. Lanczos, Discourse on Fourier series. Edinburgh: Oliver and Boyd, 1966.

W. B. Jones and G. Hardy, Accelerating convergence of trigonometric approximations, Math. Comp., 24 (1970), pp. 547-560. https://doi.org/10.1090/s0025-5718-1970-0277086-x

J. N. Lyness, Computational techniques based on the Lanczos representation, Math. Comp., 28 (1974), pp. 81-123. https://doi.org/10.1090/s0025-5718-1974-0334458-6

A. Poghosyan, Asymptotic behavior of the Krylov-Lanczos interpolation, Anal. Appl. (Singap.), 7 (2009), no. 2, pp. 199-211. https://doi.org/10.1142/s0219530509001359

A. Poghosyan, On a pointwise convergence of trigonometric interpolations with shifted nodes, Armen. J. Math., 5 (2013), no. 2, pp. 105-122.

D. Gottlieb and S. A. Orszag, Numerical analysis of spectral methods: theory and applications. Philadelphia, Pa.: Society for Industrial and Applied Mathematics, 1977.

G. Baszenski and F.-J. Delvos, Error estimates for sine series expansions, Math. Nachr., 139 (1988), pp. 155-166. https://doi.org/10.1002/mana.19881390114

P. J. Roache, A pseudo-spectral FFT technique for non-periodic problems, J. Comput. Phys., 27 (1978), no. 2, pp. 204-220. https://doi.org/10.1016/0021-9991(78)90005-0

D. Gottlieb, L. Lustman, and S. A. Orszag, Spectral calculations of one-dimensional inviscid compressible flows, SIAM J. Sci. Statist. Comput., 2 (1981), no. 3, pp. 296-310. https://doi.org/10.1137/0902024

S. Abarbanel and D. Gottlieb, Information content in spectral calculations, in Progress and supercomputing in computational fluid dynamics (Jerusalem, 1984), 6 of Progr. Sci. Comput., pp. 345-356, Boston, MA: Birkhäuser Boston, 1985. https://doi.org/10.1007/978-1-4612-5162-0_18

S. Abarbanel, D. Gottlieb, and E. Tadmor, Spectral methods for discontinuous problems, in Numerical methods for fluid dynamics, II (Reading, 1985), 7 of Inst. Math. Appl. Conf. Ser. New Ser., pp. 129-153, New York: Oxford Univ. Press, 1986.

W. Cai, D. Gottlieb, and C. W. Shu, Essentially non oscillatory spectral Fourier methods for shock wave calculations, Math. Comp., 52 (1989), pp. 389-410. https://doi.org/10.1090/s0025-5718-1989-0955749-2

D. Gottlieb, Spectral methods for compressible flow problems, in Ninth international conference on numerical methods in fluid dynamics (Saclay, 1984), 218 of Lecture Notes in Phys., pp. 48-61, Berlin: Springer, 1985. https://doi.org/10.1007/3-540-13917-6_109

S. A. Orszag, Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representations, Studies in Appl. Math., 50 (1971), pp. 293-327. https://doi.org/10.1002/sapm1971504293

H. Wengle and J. H. Seinfeld, Pseudospectral solution of atmospheric diffusion problems, J. Computational Phys., 26 (1978), no. 1, pp. 87-106. https://doi.org/10.1016/0021-9991(78)90101-8

K. S. Eckhoff, Accurate and efficient reconstruction of discontinuous functions from truncated series expansions, Math. Comp., 61 (1993), pp. 745-763. https://doi.org/10.1090/s0025-5718-1993-1195430-1

K. S. Eckhoff, Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions, Math. Comp., 64, pp. 671-690. https://doi.org/10.1090/s0025-5718-1995-1265014-7

K. S. Eckhoff, On a high order numerical method for functions with singularities, Math. Comp., 67 (1998), pp. 1063-1087. https://doi.org/10.1090/s0025-5718-98-00949-1

A. Sidi, Practical extrapolation methods, 10 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge: Cambridge University Press, 2003.

J. K. Shaw, L. W. Johnson, and R. D. Riess, Accelerating convergence of eigenfunction expansions, Math. Comp., 30 (1976), no. 135, pp. 469-477. https://doi.org/10.1090/s0025-5718-1976-0418404-4

G. Baszenski, F.-J. Delvos, and M. Tasche, A united approach to accelerating trigonometric expansions, Computers Math. Applic., 30 (1995), no. 3-6, pp. 33-49. https://doi.org/10.1016/0898-1221(95)00084-4

M. Tasche, Zur Konvergenzbeschleunigung von Fourier-Reihen, Math. Nachr., 90 (1979), pp. 123-134. https://doi.org/10.1002/mana.19790900110

A. Barkhudaryan, R. Barkhudaryan, and A. Poghosyan, Asymptotic behavior of Eckhoff's method for Fourier series convergence acceleration, Anal. Theory Appl., 23 (2007), no. 3, pp. 228-242. https://doi.org/10.1007/s10496-007-0228-0

A. Nersessian and A. Poghosyan, Convergence acceleration of two-dimensional Fourier interpolation, Armen. J. Math., 1 (2008), no. 1, pp. 50-63.

J. P. Boyd, Acceleration of algebraically-converging Fourier series when the coefficients have series in powers in $1/n$, J. Comput. Phys., 228 (2009), no. 5, pp. 1404-1411, 2009. https://doi.org/10.1016/j.jcp.2008.10.039

A. Poghosyan, Asymptotic behavior of the Eckhoff method for convergence acceleration of trigonometric interpolation, Anal. Theory Appl., 26 (2010), no. 3, pp. 236-260. https://doi.org/10.1007/s10496-010-0236-3

A. Nersessian, Convergence acceleration of Fourier series revisited, Armen. J. Math., 3 (2010), no. 4, pp. 152-161.

A. Poghosyan, On an auto-correction phenomenon of the Krylov-Gottlieb-Eckhoff method, IMA J. Numer. Anal., 31 (2011), no. 2, pp. 512-527. https://doi.org/10.1093/imanum/drp043

A. Poghosyan, On an auto-correction phenomenon of the Eckhoff interpolation, Aust. J. Math. Anal. Appl., 9 (2012), no. 1, pp. 1-31.

A. Poghosyan, Asymptotic behavior of the Eckhoff approximation in bivariate case, Anal. Theory Appl., 28 (2012), no. 4, pp. 329-362.

D. Batenkov, N. Sarig, and Y. Yomdin, An ‘algebraic’ reconstruction of piecewise-smooth functions from integral measurements, Functional Differential Equations, 19 (2012), no. 1-2, pp. 13-30.

A. Nersessian and N. Oganesyan, Quasiperiodic interpolation, Reports of NAS RA, 101 (2001), no. 2, pp. 115-121.

L. Poghosyan and A. Poghosyan, On the pointwise convergence of a quasiperiodic trigonometric interpolation, Izv. Nats. Akad. Nauk Armenii Mat., 49 (2014), no. 3, pp. 68-80.

L. Poghosyan and A. Poghosyan, Asymptotic estimates for the quasi-periodic interpolations, Armen. J. Math., 5 (2013), no. 1, pp. 34-57.

L. Poghosyan, On $L_2$-convergence of the quasi-periodic interpolation, Dokl. Nats. Akad. Nauk Armen., 113 (2013), no. 3, pp. 240-247.

A. Poghosyan and L. Poghosyan, On a pointwise convergence of quasi-periodic-rational trigonometric interpolation, Int. J. Anal., 2014 (2014), no. Art. ID 249513, pp. 1-10. https://doi.org/10.1155/2014/249513

L. Poghosyan, Convergence acceleration of quasi-periodic and quasi-periodic-rational interpolations by polynomial corrections, Armen. J. Math., 5 (2013), no. 2, pp. 123-138.

A. Poghosyan, L. Poghosyan, and R. Barkhudaryan, On some quasi-periodic approximations, Armen. J. Math., 12 (2020), no. 10, pp. 1-27.

A. Nersessian, On an over-convergence phenomenon for Fourier series. Basic approach, Armen. J. Math., 10 (2018), no. 9, pp. 1-22.

A. Nersessian, A correction to the article ‘On an over-convergence phenomenon for Fourier series. Basic approach’, Armen. J. Math., 11 (2019), no. 1, pp. 1-2.

A. Nersessian, A fast method for numerical realization of Fourier tools, IntechOpen, 2020.

A. Iserles and S. P. Nørsett, From high oscillation to rapid approximation. I. Modified Fourier expansions, IMA J. Numer. Anal., 28 (2008), no. 4, pp. 862-887. https://doi.org/10.1093/imanum/drn006

A. Iserles and S. P. Nørsett, From high oscillation to rapid approximation III: Multivariate expansions, IMA J. Numer. Anal., 29 (2009), no. 4, pp. 882-916. https://doi.org/10.1093/imanum/drn020

M. G. Krein, On a special class of differential operators, Doklady AN USSR, 2 (1935), pp. 345-349.

S. Olver, On the convergence rate of a modified Fourier series, Math. Comp., 78 (2009), no. 267, pp. 1629-1645. https://doi.org/10.1090/s0025-5718-09-02204-2

B. Adcock, Univariate modified Fourier methods for second order boundary value problems, BIT, 49 (2009), no. 2, pp. 249-280. https://doi.org/10.1007/s10543-009-0224-1

B. Adcock, Modified Fourier expansions: theory, construction and applications, PhD Thesis, University of Cambridge, 2010.

B. Adcock, Multivariate modified Fourier series and application to boundary value problems, Numer. Math., 115 (2010), no. 4, pp. 511-552. https://doi.org/10.1007/s00211-010-0287-6

B. Adcock, Convergence acceleration of modified Fourier series in one or more dimensions, Math. Comp., 80 (2011), no. 273, pp. 225-261. https://doi.org/10.1090/s0025-5718-2010-02393-2

D. Huybrechs, A. Iserles, and S. P. Nørsett, From high oscillation to rapid approximation IV: Accelerating convergence, IMA J. Numer. Anal., 31 (2011), no. 2, pp. 442-468. https://doi.org/10.1093/imanum/drp046

T. Bakaryan, On a convergence of the modified Fourier-Pade approximations, Armen. J. Math., 8 (2016), no. 2, pp. 120-144.

T. Bakaryan, On a convergence of rational approximations by the modified Fourier basis, Armen. J. Math., 9 (2017), no. 2, pp. 68-83.

A. Poghosyan and T. Bakaryan, Optimal rational approximations by the modified Fourier basis, Abstr. Appl. Anal., 2018 (2018), pp. 1-21. https://doi.org/10.1155/2018/1705409

A. Poghosyan and T. Bakaryan, On interpolation with respect to a modified trigonometric system, Izv. Nats. Akad. Nauk Armenii Mat., 53 (2018), no. 3, pp. 72-83.

J. P. Boyd, A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds, J. Comput. Phys., 178 (2002), no. 1, pp. 118-160. https://doi.org/10.1006/jcph.2002.7023

O. P. Bruno, Y. Han, and M. M. Pohlman, Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis, J. Comput. Phys., 227 (2007), no. 2, p. 1094–1125. https://doi.org/10.1016/j.jcp.2007.08.029

D. Huybrechs, On the Fourier extension of nonperiodic functions, SIAM J. Numer. Anal., 47 (2010), no. 6, pp. 4326-4355. https://doi.org/10.1137/090752456

B. Adcock and D. Huybrechs, On the resolution power of Fourier extensions for oscillatory functions, J. Comput. Appl. Math., 260 (2014), pp. 312-336. https://doi.org/10.1016/j.cam.2013.09.069

B. Adcock, D. Huybrechs, and J. Martín-Vaquero, On the numerical stability of Fourier extensions, Found. Comput. Math., 14 (2014), pp. 635-687. https://doi.org/10.1007/s10208-013-9158-8

D. Batenkov and Y. Yomdin, Algebraic Fourier reconstruction of piecewise smooth functions, Math. Comp., 81 (2012), no. 277, pp. 277-318. https://doi.org/10.1090/s0025-5718-2011-02539-1

D. Batenkov, Complete algebraic reconstruction of piecewise-smooth functions from Fourier data, Math. Comp., 84 (2015), no. 295, pp. 2329-2350. https://doi.org/10.1090/s0025-5718-2015-02948-2

A. Björk and V. Pereyra, Solution of Vandermonde systems of equations, Math. Comp., 24 (1970), pp. 893-904.

N. J. Higham, Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems, Numer. Math., 50 (1987), no. 5, pp. 613-632. https://doi.org/10.1007/bf01408579

J. L. López and N. M. Temme, Two-point Taylor expansions of analytic functions, Stud. Appl. Math., 109 (2002), no. 4, pp. 297-311. https://doi.org/10.1111/1467-9590.00225

J. Riordan, Combinatorial identities. New York: John Wiley & Sons Inc., 1968.

A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), no. 2057, pp. 1383-1399. https://doi.org/10.1098/rspa.2004.1401

A. Deaño, D. Huybrechs, and A. Iserles, Computing highly oscillatory integrals. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2018. https://doi.org/10.1137/1.9781611975123

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2021-12-10

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On the convergence of the quasi-periodic approximations on a finite interval. (2021). Armenian Journal of Mathematics, 13(10), 1-44. https://doi.org/10.52737/18291163-2021.13.10-1-44