Rate of convergence of wavelet series by Cesàro means. (Rate of convergence of wavelet series by Cesàro means.) (English) Zbl 1449.42051

Summary: Wavelet frames have become a useful tool in time frequency analysis and image processing. Many authors worked in the field of wavelet frames and obtained various necessary and sufficient conditions. A. Ron and Z. Shen [J. Funct. Anal. 148, No. 2, 408–447 (1997; Zbl 0891.42018)] gave a charactarization of wavelet frames. J. J. Benedetto and O. M. Treiber [in: Wavelet transforms and time-frequency signal analysis. Boston, MA: Birkhäuser. 3–36 (2001; Zbl 1036.42032)], presented different works on the wavelet frames. Any function \(f\in L^2(R)\) can be expanded as an orthonormal wavelet series and pointwise convergence and uniform convergence of series have been discussed extensively by various authors [S. E. Kelly et al., Bull. Am. Math. Soc., New Ser. 30, No. 1, 87–94 (1994; Zbl 0788.42014)]. In this paper we investigate the pointwise convergence of orthogonal wavelet series in Pringsheim’s sense. Furthermore, we investigate Cesàro \(\vert C,1,1\vert\) summability and the strong Cesàro \(\vert C,1,1\vert\) summability of wavelet series.


42C15 General harmonic expansions, frames
40A30 Convergence and divergence of series and sequences of functions
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
68U10 Computing methodologies for image processing
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