Summary: The basic fuzzy wavelet type operators \(A_k, B_k, C_k, D_k\), \(k\in\mathbb Z\) were studied in [G. A. Anastassiou, Nonlinear Funct. Anal. Appl. 9, No. 2, 251–269 (2004; Zbl 1078.42025); Comput. Math. Appl. 48, No. 9, 1387–1401 (2004; Zbl 1071.41020)], for their pointwise and uniform convergence with rates to the fuzzy unit operator. They also were studied in [G. A. Anastassiou, J. Concr. Appl. Math. 5, No. 1, 25–52 (2007; Zbl 1143.42035)], in terms of estimating their fuzzy differences and given their pointwise convergence with rates to zero.
For prior related and similar study of convergence to the unit of real analogs of these wavelet type operators see [G. A. Anastassiou, Quantitative approximations, Boca Raton, FL: Chapman & Hall/CRC (2001; Zbl 0969.41001), Section II].
Here in Section 1 we present the complete study of finding uniform estimates for the distances between the real Wavelet type operators \(A_k, B_k, C_k, D_k\), \(k\in\mathbb Z\). Their differences converge to zero with rates. This is done via elegant tight Jackson type inequalities involving the modulus of continuity of the higher order derivative of the engaged real function. Based on these real analysis results in Section 2 we establish the corresponding fuzzy results regarding uniform estimates for the fuzzy differences between the fuzzy wavelet type operators. These fuzzy diferences converge to zero with rates give via fuzzy Jackson type tight inequalities. The last inequalities involve the fuzzy modulus of continuity of the higher order fuzzy derivative of the engaged fuzzy function.
The defining all these operators real scaling function is not assumed to be orthogonal and is of compact support.
Another motivation for this work is [H. Gonska, P. Piţul and I. Raşa, Carpathian J. Math. 22, No. 1–2, 65–78 (2006; Zbl 1150.41012)].