| This work deals with the fundamental, mathematical properties and the applications of the Distributed Approximating Functionals. In the first part, different DAFs are discussed, the general formalism and the basic properties of the well-tempered DAFs, the interpolating DAFs and the Sine-DAFs are provided. The DAFs have the ability to approximate a function and its derivatives with a controllable accuracy is the principal reason for their successful applications in solving PDEs, in wavelet analysis and signal processing. The second part concerns the basic properties of Hermite DAFs. As an approximating low pass filter, its properties on the high pass band, low pass band, and transition band are presented. With properly chosen parameters, the HDAFs can make an infinite approximation to the ideal low pass filter, but with more good quality in the time and frequency domain. We also proved the HDAFs are infinitely approximate to 1/2 at the inflection point. In the third part, the low pass filter obtained from the periodization of the HDAF and normalization at the frequency zero and pi is proved to be an interpolating filter, and this is a good start to construct the interpolating wavelets. Then we presented the decay of the refinement function. In the last part of this dissertation, the theory is implemented in the field of image processing. Based on the well tempered property of HDAF and the iterative method, a new algorithm is used to restore the impulse noise corrupted images. |
