Malaysian Journal of Mathematical Sciences, December 2022, Vol. 16, No. 4
The Convergence of Operator With Rapidly Decreasing Wavelet Functions
Shamsah, R. S., Ahmedov, A., Kilicman, A., and Zainuddin, H.
Corresponding Email: raghad.sahib@uokerbala.edu.iq
Received date: 12 March 2020
Accepted date: 5 October 2022
Abstract:
The expansion of (2D) wavelet functions with respect to $L^p(R^2)$ space converging almost everywhere for $1 < p < \infty$ throughout the length of the Lebesgue set points of space functions is investigated in this research. The convergence is established by assuming some wavelet function minimal regularity $\psi_{j_{1},j_{2},k_{1},k_{2}}$ under the current description of the wavelet projection operator known as 2D Hard Sampling Operator. Note that the feature of fast decline in 2D is derived here. Another condition is used, for instance, the wavelet expansion's boundedness under the Hard Sampling Operator. The bound (limit) is governed in magnitude with respect to the maximal equality of the Hardy-Littlewood maximal operator. Some ideas presented in this work are to find a new method to prove the convergence theory for a new type of conditional wavelet operator. Propose some conditions for wavelets functions and their expansion can support the operator to be convergence. It also performs a comparison with the identity convergent operator is our method for achieving this convergence.
Keywords: boundedness; convergence; maximal function; rapidly decreasing; wavelet expansion
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