THE POINT WISE BEHAVIOR OF 2-DIMENSIONAL WAVELET EXPANSIONS IN $L^p(R^2)$

Abstract

We show that the two dimensional wavelet expansion of L p (R 2 ) function for 1 < p < ∞ converges pointwise almost everywhere under wavelet projection operator. This convergence can be established by assuming some minimal regularity to get the rapidly decreasing for two dimensional wavelet ψ j 1 ,j 2 ,k 1 ,k 2 . The Kernel function of the wavelet projection operator in two dimension converges absolutely, distributionally and is bounded. Also the wavelet expansions in two dimension are controlled in a magnitude by the maximal function operator. All these conditions can be utilized to achieve the convergence almost everywhere.

The Convergence of Operator With Rapidly Decreasing Wavelet Functions

“…In a different work, [13] used a prolate spheroidal wavelet to examine the pointwise convergence of wavelet expansions of L 2 (R) functions. In addition, [12]and [11] looked at how L P functions defined on the S 2 and R 2 domains converge. By utilizing a spherical multi-resolution analysis on S 2 surface functions, [11] elucidated the pointwise behavior of spherical wavelet expansion with respect to the spherical wavelet projection operator.…”

“…By utilizing a spherical multi-resolution analysis on S 2 surface functions, [11] elucidated the pointwise behavior of spherical wavelet expansion with respect to the spherical wavelet projection operator. Hence, redefining the projection operator into 2D wavelet projections operators and expanding the 0-regular wavelet function with two scaling and shifting parameters, [12] enhanced Tao's work. To reach convergence, the effort entailed verifying the wavelet function's 2D fast decreasing property.…”

“…Proof: Assuming the convergent Wavelet Projection Operator in [12], we can consider that the Wavelet Projection Operator P J is written in the following equation…”

A Bounded Maximal Function Operator and Its Acting on <img src=image/13425593_01.gif> Functions

“…Moreover, Tao [11] highlighted that if one regards certain minimal regularity with respect to ψ, the point wise classical wavelet operators convergence can be extended to the whole Lebesgue set with respect to f . Furthermore, Raghad et al [18] expanded the dimension of a wavelet function for four-dimensional wavelet function to improve the results of Tao, in which the results were examined on four-dimensional wavelet projections operators. Also, the convergence of non-convolution integral operators in Lebesque space is studied in [19].…”