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Fourier transform inversion: Bounded variation, polynomial growth, Henstock–Stieltjes integration

  • Erik Talvila EMAIL logo
Published/Copyright: February 17, 2023
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 73 Issue 1

Abstract

In this paper we prove pointwise and distributional Fourier transform inversion theorems for functions on the real line that are locally of bounded variation, while in a neighbourhood of infinity are Lebesgue integrable or have polynomial growth. We also allow the Fourier transform to exist in the principal value sense. A function is called regulated if it has a left limit and a right limit at each point. The main inversion theorem is obtained by solving the differential equation df(t)−iωf(t)=g(t) for a regulated function f, where ω is a complex number with positive imaginary part. This is done using the Henstock–Stieltjes integral. This is an integral defined with Riemann sums and a gauge. Some variants of the integration by parts formula are also proved for this integral. When the function is of polynomial growth its Fourier transform exists in a distributional sense, although the inversion formula only involves integration of functions and returns pointwise values.

Keywords: Fourier transform; inversion; Henstock–Stieltjes integral; contour integral; principal value; Heaviside step function; integration by parts; bounded variation; distribution
MSC 2010: Primary 42A38; Secondary 26A39; 46F12

  1. (Communicated by Marcus Waurick)

Acknowledgement

An anonymous referee provided helpful comments and several important references.

References

[1] Appell, J.—Banas, J.—Merentes Díaz, N. J.: Bounded Variation and Around, De Gruyter, Berlin, 2014.Search in Google Scholar

[2] Arredondo, J. H.—Mendoza, F. J.—Reyes, A.: On the norm continuity of the HK-Fourier transform, Electron. Res. Announc. Math. Sci. 25 (2018), 36–47.Search in Google Scholar

[3] Bachman, G.—Narici, L.—Beckenstein, E.: Fourier and Wavelet Analysis, Springer, New York, 2000.Search in Google Scholar

[4] Benedetto, J. J.: Harmonic Analysis and Applications, CRC Press, Boca Raton, 1997.Search in Google Scholar

[5] Broucke, F.—Debruyne, G.—Vindas, J.: An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions, J. Math. Anal. Appl. 494 (2021), Art. ID 124450, 15 pp.Search in Google Scholar

[6] Estrada, R.—Vindas, J.: Distributional point values and convergence of Fourier series and integrals, J. Fourier Anal. Appl. 13 (2007), 551–576.Search in Google Scholar

[7] Estrada, R.—Vindas, J.: Distributional versions of Littlewood's Tauberian theorem, Czechoslovak Math. J. 63(138) (2013), 403–420.Search in Google Scholar

[8] Gel'fand, I. M.—Shilov, G. E.: Generalized Functions, vol. I (trans. E. Saletan), Academic Press, New York, 1964.Search in Google Scholar

[9] Koekoek, J.: On the Fourier integral theorem, Nieuw Arch. Wisk. 5(4) (1987), 83–85.Search in Google Scholar

[10] Liflyand, E.: The Fourier transform of a function of bounded variation: symmetry and asymmetry, J. Fourier Anal. Appl. 24 (2018), 525–544.Search in Google Scholar

[11] Liflyand, E.: Functions of Bounded Variation and their Fourier Transforms, Birkhäuser, Cham, Switzerland, 2019.Search in Google Scholar

[12] Mcleod, R. M.: The Generalized Riemann Integral, The Mathematical Association of America, Washington, 1980.Search in Google Scholar

[13] Mikolás, M.: Recent contributions to the theory of non-absolutely convergent Fourier transforms, Integral Transform. Spec. Funct. 1 (1993), 33–40.Search in Google Scholar

[14] Monteiro, G. A.—SlavíK, A.—Tvrdý, M.: Kurzweil–Stieltjes Integral, World Scientific, Singapore, 2019.Search in Google Scholar

[15] Móricz, F.: Pointwise behavior of Fourier integrals of functions of bounded variation over ℝ. Special issue dedicated to John Horváth, J. Math. Anal. Appl. 297 (2004), 527–539.Search in Google Scholar

[16] Ortner, N.: Fourier transforms in the "classical sense", Schur spaces and a new formula for the Fourier transforms of slowly increasing, O(p, q)-invariant functions, Indag. Math. (N.S.) 24 (2013), 142–160.Search in Google Scholar

[17] Pilipovic, S.—Stankovic, B.—Vindas, J.: Asymptotic Behavior of Generalized Functions, World Scientific, Singapore, 2011.Search in Google Scholar

[18] Riesz, M.—Livingston, A. E.: A short proof of a classical theorem in the theory of Fourier integrals, Amer. Math. Monthly 62 (1955), 434–437.Search in Google Scholar

[19] Talvila, E.: Fourier transform inversion using an elementary differential equation and a contour integral, Amer. Math. Monthly 126 (2019), 717–727.Search in Google Scholar

[20] Titchmarsh, E. C.: Introduction to the Theory of Fourier Integrals, Chelsea, New York, 1986.Search in Google Scholar

[21] Tvrdý, M.: Differential and integral equations in the space of regulated functions, Mem. Differential Equations Math. Phys. 25 (2002), 1–104.Search in Google Scholar

[22] Whittaker, E. T.—Watson, G. N.: A course in Modern Analysis, Cambridge University Press, Cambridge, 1927.Search in Google Scholar
Received: 2021-03-24
Accepted: 2022-03-25
Published Online: 2023-02-17
Published in Print: 2023-02-23

© 2023 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2023-0013
Keywords for this article
Fourier transform; inversion; Henstock–Stieltjes integral; contour integral; principal value; Heaviside step function; integration by parts; bounded variation; distribution

Articles in the same Issue

  1. RNDr. Stanislav Jakubec, DrSc. passed away
  2. Schur m-power convexity for general geometric Bonferroni mean of multiple parameters and comparison inequalities between several means
  3. Conditions forcing the existence of relative complements in lattices and posets
  4. Radically principal MV-algebras
  5. A topological duality for dcpos
  6. Padovan or Perrin numbers that are concatenations of two distinct base b repdigits
  7. Addendum to “A generalization of a result on the sum of element orders of a finite group”
  8. Remarks on w-distances and metric-preserving functions
  9. Solution of logarithmic coefficients conjectures for some classes of convex functions
  10. Multiple periodic solutions of nonautonomous second-order differential systems with (q, p)-Laplacian and partially periodic potentials
  11. Asymptotic stability of nonlinear neutral delay integro-differential equations
  12. On Catalan ideal convergent sequence spaces via fuzzy norm
  13. Fourier transform inversion: Bounded variation, polynomial growth, Henstock–Stieltjes integration
  14. (ε, A)-approximate numerical radius orthogonality and numerical radius derivative
  15. Wg-Drazin-star operator and its dual
  16. On the Lupaş q-transform of unbounded functions
  17. K-contact and (k, μ)-contact metric as a generalized η-Ricci soliton
  18. Compactness with ideals
  19. Number of cells containing a given number of particles in a generalized allocation scheme
  20. A new generalization of Nadarajah-Haghighi distribution with application to cancer and COVID-19 deaths data
  21. Compressive sensing using extropy measures of ranked set sampling
  22. A note on star partial order preservers on the set of all variance-covariance matrices
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