Methods for fast computation of integral transforms

This paper is concerned with two aspects of the numerical calculation of integral transforms. The first is finding a necessary and sufficient condition that enables converting an integral transform into a correlation (convolution) form. The condition and the transformation that implements it are generalizations of the Gardner transformation and derived in the paper. This technique can be applied to a wide class of integral transforms and is shown to reduce the computational complexity and storage requirements of the resulting algorithm. The second issue addressed in the paper is the accuracy of the calculation of the correlation integral, obtained by the above transformation, for a given number of samples. It is shown how the standard FFT method can be applied in combination with various numerical integration rules. This proves to be an important factor in expediting the computations, reducing the storage requirements, and improving the accuracy.