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A sparse approximation for fractional Fourier transform

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Abstract

The paper promotes a new sparse approximation for fractional Fourier transform, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the upper half-plane. Under this methodology, the local polynomial Fourier transform characterization of Hardy space is established, which is an analog of the Paley-Wiener theorem. Meanwhile, a sparse fractional Fourier series for chirp \( L^2 \) function is proposed, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the unit disk. Besides the establishment of the theoretical foundation, the proposed approximation provides a sparse solution for a forced Schr\(\ddot{\textrm{o}}\)dinger equations with a harmonic oscillator.

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References

  1. Almeida, L.: The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process 42(11), 3084–3091 (1994)

    Article  Google Scholar 

  2. Cartan, H.: Calcul differentiel, Hermann (1967)

  3. Cordero, E., Nicola, F.: Metaplectic representation on Wiener amalgam spaces and applications to the Schr\(\ddot{\rm o}\)dinger equation. J. Funct. Anal. 254(2), 506–534 (2008)

    Article  MathSciNet  Google Scholar 

  4. Cordero, E., Tabacco, A., Wahlberg, P.: Schr\(\ddot{\rm o}\)dinger-type propagators, pseudodifferential operators and modulation spaces. J. Lond. Math. Soc. 88(2), 375–395 (2013)

    Article  MathSciNet  Google Scholar 

  5. Coifman, R., Steinerberger, S.: Nonlinear phase unwinding of functions. J. Fourier Anal. Appl. 23, 778–809 (2017)

    Article  MathSciNet  Google Scholar 

  6. Coifman, R., Peyriere, J.: Phase unwinding, or invariant subspace decompositions of Hardy spaces. J. Fourier Anal. Appl. 25, 684–695 (2019)

    Article  MathSciNet  Google Scholar 

  7. Duren, P.L.: Theory of \(H^p\) spaces. Pure Appl. Math. 38, (2013)

  8. Feynman, R.P., Hibbs, A.R.: Quantum mechanics and path integrals. McGraw-Hill, New York (1965)

    Google Scholar 

  9. Fu, Z.W., Hou, X.M., Wu, Q. Y.: Convergence of fractional Fourier series on the torus and applications (2022)

  10. Holstein, B.R.: The harmonic oscillator propagator. Am. J. Phys. 66(7), 583–583 (1998)

    Article  MathSciNet  Google Scholar 

  11. Kerr, F.H.: Namias’ fractional Fourier transforms on \( L^2 \) and applications to differential equations. J. Math. Anal. Appl. 136, 404–418 (1988)

    Article  MathSciNet  Google Scholar 

  12. Kou, K.I., Morais, J.: Asymptotic behaviour of the quaternion linear canonical transform and the Bochner-Minlos theorem. Appl. Math. Comput. 247(15), 675–688 (2014)

    Article  MathSciNet  Google Scholar 

  13. Lopez, R.M., Suslov, S.K.: The Cauchy problem for a forced harmonic oscillator. Revista Mexicana De Fisica E 55(2), 196–215 (2009)

    MathSciNet  Google Scholar 

  14. Li, X., Bi, G., Stankovic, S., et al.: Local polynomial Fourier transform: a review on recent developments and applications. Signal Process. 91(6), 1370–1393 (2011)

    Article  Google Scholar 

  15. McBride, A.C., Kerr, F.H.: On Namias’s fractional Fourier transforms. IMA J. Appl. Math. 39, 159–175 (1987)

    Article  MathSciNet  Google Scholar 

  16. Mi, W., Qian, T.: Frequency-domain identification: an algorithm based on an adaptive rational orthogonal system. Automatica 48(6), 1154–1162 (2012)

    Article  MathSciNet  Google Scholar 

  17. Mi, W., Qian, T., Wan, F.: A fast adaptive model reduction method based on Takenaka-Malmquist systems. Syst. Control Lett. 61(1), 223–230 (2012)

    Article  MathSciNet  Google Scholar 

  18. Mo, Y., Qian, T., Mi, W.: Sparse Representation in Szego kernels through reproducing kernel Hilbert space theory with applications. Int. J. Wavelet Multiresolut. Inf. Process. 13(4), 1550030 (2015)

    Article  MathSciNet  Google Scholar 

  19. Mai, W., Shao, G.: On the Bergman kernel in weighted monogenic Bargmann-Fock spaces. Adv. Math. 415, (2023)

  20. Nikolski, N.: Hardy spaces, Cambridge University Press, (2019)

  21. Namias, V.: The fractional order Fourier transform and its application to quantum mechanics. IMA J. Appl. Math. 25, 241–265 (1980)

    Article  MathSciNet  Google Scholar 

  22. Qian, T.: Intrinsic mono-component decomposition of functions: an advance of Fourier theory. Math. Methods Appl. Sci. 33, 880–891 (2010)

    Article  MathSciNet  Google Scholar 

  23. Qian, T.: Sparse representations of random signals. Math. Methods Appl. Sci. 45(8), 4210–4230 (2022)

    Article  MathSciNet  Google Scholar 

  24. Qian, T., Wang, Y.B.: Remarks on adaptive Fourier decomposition. Int. J. Wavelets Multiresolution Inf. Process. 11(1), 1–14 (2013)

    Article  MathSciNet  Google Scholar 

  25. Qian, T., Chen, Q., Tan, L.: Rational orthogonal systems are Schauder bases. Complex Var. Elliptic Equ. 59(6), 841–846 (2014)

    Article  MathSciNet  Google Scholar 

  26. Qian, T.: Two-dimensional adaptive Fourier decomposition. Math. Methods Appl. Sci. 39(10), 2431–2448 (2016)

    Article  MathSciNet  Google Scholar 

  27. Qian, T.: A novel Fourier theory on non-linear phase and applications. Adv. Math. (China) 47(3), 321–347 (2018)

    MathSciNet  Google Scholar 

  28. Qian, T., Sproessig, W., Wang, J.X.: Adaptive Fourier decomposition of functions in quaternionic Hardy spaces. Math. Methods Appl. Sci. 35(1), 43–64 (2012)

    Article  MathSciNet  Google Scholar 

  29. Qian, T., Wang, Y.B.: Adaptive Fourier series-a variation of greedy algorithm. Adv. Comput. Math. 34(3), 279–293 (2011)

    Article  MathSciNet  Google Scholar 

  30. Qian, T.: Reproducing kernel sparse representations in relation to operator equations. Complex Anal. Oper. Theorem 14(2), 1–15 (2020)

    MathSciNet  Google Scholar 

  31. Qu, W., Chui, C.K., Deng, G.T., Qian, T.: Sparse representation of approximation to identity. Anal. Appl. 20(4), 815–837 (2021)

    Article  MathSciNet  Google Scholar 

  32. Qu, W., Dang, P.: Rational approximation in a class of weighted Hardy spaces. Complex Anal. Oper. Theory 13(4), 1827–1852 (2019)

    Article  MathSciNet  Google Scholar 

  33. Qu, W., Dang, P.: Reproducing kernel approximation in weighted Bergman spaces: algorithm and applications. Math. Methods Appl. Sci. 42(12), 4292–4304 (2019)

    Article  MathSciNet  Google Scholar 

  34. Rudin, W.: Real and complex analysis 2nd. McGraw-Hill, (1974)

  35. Rudin, W.: Principles of mathematical analysis. McGraw-hill, New York (1976)

    Google Scholar 

  36. Saitoh, S., Sawano, Y.: Theory of reproducing kernels and applications. Developments in Mathematics 44, (2016)

  37. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean space, Princeton University Press, (1971)

  38. Toft, J., Bhimani, D., Manna, R.: Fractional Fourier transforms, powers of harmonic oscillators and their propagators on Pilipovic and modulation spaces (2021)

  39. Katkovnik, V.: A new form of the Fourier transform for time-varying frequency estimation. Signal Process 47(2), 187–200 (1995)

    Article  MathSciNet  Google Scholar 

  40. Wiener, N.: Hermitian polynomials and Fourier analysis. J. Math. Phys. 8, 70–73 (1929)

    Article  Google Scholar 

  41. Yosida, B.: Functional analysis, Springer-Verlag, (1978)

  42. Yang, F., Chen, M., Li, P.T., Qu, W., Chen, J.C., Qian, T.: Sparse series solutions of random boundary and initial value problems, International Journal of Wavelets, Multiresolution and Information Processing (2023)

  43. Zhu, K.: Analysis on Fock spaces, Springer Science and Business Media (2012)

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Acknowledgements

The authors would like to thank Dr. Wei Qu and Prof. Bingzhao Li for their suggestions.

Funding

This work was supported by Research grant of Macau University of Science and Technology (grant number FRG-22-075-MCMS), Macao Government Research Funding (grant number FDCT0128/2022/A, 0020/2023/RIB1, 0111/2023/AFJ, 005/2022/ALC), and the National Natural Science Foundation of China (grant number 12271042, 12071437).

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, China

    Fang Yang & Jiecheng Chen

  2. School of General Education, Wenzhou Polytechnic, Wenzhou, China

    Fang Yang

  3. Macau Center for Mathematical Sciences, Macau University of Science and Technology, Macau, China

    Tao Qian

  4. School of Mathematical Sciences, Institution of Mathematics and Mathematical Education, Key Laboratory of Mathematics and Complex Systems Ministry of Education, Beijing Normal University, Beijing, China

    Jiman Zhao

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  1. Fang Yang
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  2. Jiecheng Chen
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Correspondence to Tao Qian.

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Communicated by: Zydrunas Gimbutas

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Yang, F., Chen, J., Qian, T. et al. A sparse approximation for fractional Fourier transform. Adv Comput Math 50, 50 (2024). https://doi.org/10.1007/s10444-024-10127-6

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