Approximation of Periodic Functions by Wavelet Fourier Series

Main Article Content

Varsha Karanjgaokar, Snehal Rahatgaonkar, Laxmi Rathour, Lakshmi Narayan Mishra, Vishnu Narayan Mishra


This paper aims to examine the expansion of periodic functions using wavelet bases. M. Skopina [8] obtained a Wavelet analog of the classical Jackson’s theorem for trigonometric approximation. Our result generalizes the result of M. Skopina [8] and V. Karanjgaokar et al. [15].

Article Details


  1. A. Grossmann, J. Morlet, Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape, SIAM J. Math. Anal. 15 (1984), 723–736.
  2. L. Dengfeng, P. Silong, Characterization of Periodic Multiresolution Analysis and an Application, Acta Math. Sinica. 14 (1998), 547–554.
  3. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conferences Series in Applied Mathematics, SIAM, Philadelphia, PA, (1992).
  4. J. Morlet, G. Arens, E. Fourgeau, D. Glard, Wave Propagation and Sampling Theory–Part I: Complex Signal and Scattering in Multilayered Media, GEOPHYSICS. 47 (1982), 203–221.
  5. J. Prestin, K.K. Selig, On a Constructive Representation of an Orthogonal Trigonometric Schauder Basis for C2π, in: J. Elschner, I. Gohberg, B. Silbermann (Eds.), Problems and Methods in Mathematical Physics, Birkhäuser, Basel, 2001: pp. 402–425.
  6. L. Debnath, Wavelet Transforms and Their Applications, Birkhäuser, Boston, 2002.
  7. L. N. Mishra, V. N. Mishra, K. Khatri, Deepmala, On the Trigonometric Approximation of Signals Belonging to Generalized Weighted Lipschitz W(L r , ξ(t))(r ≥ 1)− Class by Matrix (C 1 .Np) Operator of Conjugate Series of Its Fourier Series, Appl. Math. Comput. 237 (2014), 252–263.
  8. M. Skopina, Wavelet Approximation of Periodic Functions, J. Approx. Theory. 104 (2000), 302–329.
  9. M. Skopina, Localisation Principle for Wavelet Expansion, Self Seminar System, In: Proceedings of the international Workshop, Dubna, (1999), 125–133.
  10. P. Chandra, Taylor means and functions from H(α, p) space, Jñanabha, 47 (2017), 17–30.
  11. R.A. DeVore, G.G.Lorentz, Constructive Approximation, Springer, New York, (1993).
  12. S. Lal, S. Kumar, Best Wavelet Approximation of Functions Belonging to Generalized Lipschitz Class Using Haar Scaling Function, Thai J. Math. 15 (2017), 409–419.
  13. S.E. Kelly, M.A. Kon, L.A. Raphael, Local Convergence for Wavelet Expansions, J. Funct. Anal. 126 (1994), 102–138.
  14. V. Karanjgaokar, N. Shrivastav, A Note on the Convergence of Wavelet Fourier Series, Trans. A. Razm. Math. Inst. 177 (2023), 77–83.
  15. V. Karanjgaokar, N. Shrivastav and S. Rahatgaonkar, On the rate of Convergence of Wavelet Fourier Series, Jñanabha, 51 (2021), 12–18.
  16. V. Karanjgaokar, On the Rate of Convergence of Wavelet Expansion, J. Indian Math. Soc. 85 (2018), 100–110.
  17. V.N. Mishra, K. Khatri, L.N. Mishra, Deepmala, Trigonometric Approximation of Periodic Signals Belonging to Generalized Weighted Lipschitz W’(Lr, ξ(t)),(r ≥ 1)− Class by Nörlund-Euler (N, pn)(E, q) Operator of Conjugate Series of Its Fourier Series, J. Class. Anal. 5 (2014), 91–105.
  18. Y. Meyer, Wavelets: Their Past and Their Future. In: Y. Meyer, S. Roques, (eds.) Progress in Wavelet Analysis and Applications, Toulouse, 1992, pp. 9–18. Frontieres, Gif-sur-Yvette, (1993).