Historical Backround of wavelets and orthonormal systems: Recent results on positive linear operators reconstructed via wavelets
Recent results on positive linear operators reconstructed via wavelets
Keywords:
wavelets, positive linear operators, approximation, bounded variation, asymptotic propertiesAbstract
This survey paper provides a historical overview of wavelets and orthonormal systems, alongside recent findings related to linear positive operators reconstructed using wavelets. The first section delves into the historical and chronological development of wavelets, highlighting some of their significant properties. The second section examines linear positive operators constructed through wavelets, discussing their structural characteristics and approximation results. While the list included in this paper is comprehensive, it is not exhaustive. We apologize to any authors whose works on wavelets and wavelet-based operators are not cited in this paper.
References
U. Abel, V. Gupta: An estimate on the rate of convergence of Bezier variant of Baskakov Kantorovitch operators, Demonstr. Math., 36 (l) (2003), 123–136.
A. M. Acu, I. C. Buscu and I. Ra¸sa: Generalized Kantorovich modifications of positive linear operators, Math. Found. Comput., 6 (1) (2023), 54–62.
J. A. Adell, J. Bustamante and J. M. Quesada: Estimates for the moments of Berntein polynomials, J. Math. Anal. Appl., 432 (2015), 114–128.
O. Agratini: Construction of Baskakov-type operators by wavelets, Rev. Anal. Numér. Théor. Approx., 26 (1-2) (1997), 3–11.
G. A. Anastassiou: Rate of convergence of some neural network operators to the unit-univariate case, J. Math. Anal. Appl., 212 (1997), 237–262.
F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, De Gruyter Studies in Mathematics, vol. 17, Walter de Gruyter and Co., Berlin (1994).
C. Bardaro, J. Musielak and G. Vinti: Nonlinear integral operators and applications, De Gruyter Series in Nonlinear Analysis and Applications, vol. 9, New York-Berlin (2003).
P. Bézier: Numerical control mathematics and applications, Wiley, London (1972).
C. S. Burrus, R. A. Gopinath and H. Guo: Introduction to wavelets and wavelet transforms: A primer, Prentice Hall (1998).
P. L. Butzer, R. J. Nessel: Fourier Analysis and Approximation, 1, Academic Press, New York-London (1971).
P. Cardaliaguet, G. Euvrard: Approximation of a function and its derivative with a neural network, Neural Netw., 5 (2) (1992), 207–220.
Z. A. Chanturiya: Modulus of variation of functions and its application in the theory of Fourier series, Dokl. Akad. Nauk SSSR, 214 (1974), 63–66 (in Russian).
G. Cybenko: Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems, 2 (1989), 303–314.
I. Daubechies: Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41 (1988), 909–996.
I. Daubechies: Ten lectures on wavelets, CBMS-NSF Series in Appl. Math., vol. 61, SIAM Publ., Philadelphia (1992).
G. Faber: Uber die orthogonalfunktionen des Herrn Haar , Jahresber Deutsch. Math. Verein., 19 (1910), 104–112.
P. Franklin: A set of continuous orthogonal functions, Math. Ann., 100 (1928), 522–529.
H. H. Gonska, D. X. Zhou: Using wavelets for Szász-type operators, Rev. Anal. Numér. Théor. Approx., 24 (1-2) (1995), 131–145.
V. Gupta: The bezier variant of Kantorovitch operators, Compt & Math Appl., 47 (2-3) (2004), 227–232.
V. Gupta: An estimate on the convergence of Baskakov–Bezier operators, J. Math. Anal. Appl., 312 (2005), 280–288.
V. Gupta, A.M. Acu: On difference of operators with different basis functions, Filomat, 33 (10) (2019), 3023–3034.
A. Haar: Dissertation: Zur theorie der orthogonalen funktionen systeme, Georg-August-Universität Göttingen (1909).
H. Karsli: Extension of the generalized Bezier operators by wavelet, General Math., 30 (2) (2022), 3–15.
H. Karsli: On wavelet type Bernstein operators, Carpathian Math. Publ., 15 (1) (2023), 212–221.
H. Karsli: On wavelet type generalized Bézier operators, Math. Found. Comput., 6 (3) (2023), 439–452.
H. Karsli: A mathematical model for the effects of wavelets and the analysis of neural network operators described using wavelets, 2nd Int. Workshop: Const. Math. Anal. Proc. Book, July (2023), 24–38.
H. Karsli: A mathematical model for linear approximation operators constructed using wavelets, J. Math. Control Sci. Appl., 9 (2) (2023), 9–18.
H. Karsli: Asymptotic properties and quantitative results of the wavelet type Bernstein operators, Dolomites Res. Notes Approx., 17 (1) (2024), 50–62.
P. P. Korovkin: Linear operators and approximation theory, Hindustan Publ., New Delhi (1960).
Y. Meyer: Ondelettes: wavelets and operators, Cambridge University Press, New York (1993).
H. Rademacher: Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann., 87 (1) (1922), 112–138.
J. Schauder: Eine Eigenschaft des Haarschen Orthogonalsystem, Math. Z., 28 (1928), 317–320.
J. O. Strömberg: A modified Franklin system and higher order spline systems on as unconditional bases for Hardy spaces, Conference in Harmonic Analysis in Honor of A. Zygmund, vol. II, Wadsworth, Belmont (1983), 475–493.
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