New developments in convergence of wavelet expansion of functions Lᴾ (R²), Sobolev space Hˢ (R²) and Lᴾ (S²)

In this work, we highlight to some methods that can develop the convergence of wavelet expansions under some new forms of partial sums operators. We improve some requirements on classical wavelet expansions on R² domain. In addition, it is interesting to consider the spherical wavelets which are defined by polar coordinates on R3 domain, and establish a convergence of unique expansions called spherical wavelet expansions. We introduce a generalization of wavelet expansions principle in two dimensions with new conditions under some associated operators which are Wavelet Projection Operator, Hard Sampling Operator and Soft Sampling Operator. The expansions can be generalized to expand functions for different types of functional spaces such as Lᴾ (R²), Sobolev space Hˢ (R²) and Lᴾ (S²). The wavelet expansions are analyzed by two methods of analysis which are classical multi-resolution analysis and spherical multi-resolution analysis. We investigate the sufficient conditions for a wavelet function and its expansions to achieve the convergence of wavelet expansions of the function under its related operators. For instance, after imposing a minimal regularity on the wavelet functions we can establish the rapidly decreasing property in two and four dimensions, that is, the expansion of any wavelet function is dependent on four integer parameters ( j₁, j₂,k₁,k₂) in analyzing the wavelet. It is important as well to take the boundedness property of wavelet expansions of functions into consideration. A special technique is established to achieve the convergence of wavelet expansions ofLᴾ (R²) and Lᴾ (S²) functions by limiting the operator’s magnitude with another bounded operator such as Hardy-Littlewood maximal operator and spherical Hardy-Littlewood maximal operator. While other techniques like use the boundedness condition of Zak transform and the structure of Meyer wavelet, are used to prove the convergence of wavelet expansions of Sobolev spaces functions with using high-regularity wavelet function. Some basic properties of wavelet functions as well as sharp examples are also given. The performance of some partial sums operators developed by improving the conditions of their wavelet expansion. The two dimensional wavelet expansions of functions for some functional spaces converged in the two cases of classical wavelet and spherical wavelet expansions. Depending on some required properties for wavelet and its expansions, the convergence appeared almost everywhere along the Lebesgue set points of Lᴾ (R²) and Lᴾ (S²) functions. On the other hand, new type of convergence produced by making equivalent between the wavelet expansion and Fourier expansion of Sobolev space functions Hˢ (R²). By this, the partial sums operator behaved likes a truncated parts of inverse Fourier transformation, such that the convergence appeared uniformly at the singularity points of the partial sums operator kernel.

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