Generalization of Fourier Transformation of Scaling Function using Riesz basis on L2 (K)

Various applications of multi resolution analysis has been proved in analysis of coding, time frequency etc. One of its important role has been pointed out in the area of applied mathematics which is connected to the wavelets.In this work an analysis of scaling function together with its observation in the closed subspaces of
$\mathbf{L}^{2}(\mathbf{K})$
has been done. Futhermore, it consists of the observation for the riesz basis in the multi resolution analysis with its form in terms of scaling function. It is also proved that the bounds can be found for the scaling function with the use of the transformation of function. The intersection property is proved as single element, with use of translates of riesz basis. It is concluded that the riesz basis can play an important role in the formation of bounds for the scaling function and it is used for the Fourier transformation for the same with respect to the intersection property as single element.