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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 expansion of (2D) wavelet functions with respect to Lp(R2) space converging almost everywhere for 1<p<∞ 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 ψj1,j2,k1,k2 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.

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Abstract.A scaling functions system is a series of subspaces that are embedded and spanned by a group of scaling basis functions . To fully grasp how to construct this system using a unique function when is the Dyadic intervals set, its structure is studied. The Dyadic intervals structure show us that is no any intersection appear between their sub intervals at the different scaling and shifting values. This property introduced a new way to prove the orthonormality of this system by using the supported intervals of the step functions. A new scaling relation called the scaling filter is defined on Dyadic intervals, is used to characterize this system. This filter allows for analyzing and other spaces by multi-resolution analysis, as well as it provides some of the requisite conditions. To explain the structure of this system, the clarity examples are given.