The point wise behavior of 2-dimensional wavelet expansions in Lᴾ (R²)

We show that the two dimensional wavelet expansion of Lᴾ (R²) 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 ψj1,j2,k1,k2. 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