Summary
Index transforms ofm-dimensional arrays inton-dimensional arrays play a significant role in many fast algorithms of multivariate discrete Fourier transforms (DFT's) and cyclic convolutions. The computation ofm-dimensional “long” DFT's or convolutions can be transfered to the parallel computation ofn-dimensional “short” DFT's or convolutions (n>m). In this paper, the nature of index transforms is explored using group-theoretical ideas. We solve the open problems concerning index transforms posed recently by Hekrdla [5, 6].
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References
Agarwal, R.C., Cooley, J.W.: New algorithms for digital convolution. IEEE Trans. ASSP25, 392–410 (1977)
Burrus, C.S.: Index mappings for multidimensional formulation of the DFT and convolution. IEEE Trans. ASSP25, 239–242 (1977)
Elliot, D.F., Rao, K.R.: Fast transforms, algorithms, analyses, applications. 1st Ed. New York, London, Paris, San Diego, Sao Paulo, Sydney, Tokyo: Academic Press 1982
Hasse, H.: Vorlesungen über Zahlentheorie. 1st Ed. Berlin Göttingen, Heidelberg: Springer 1950
Hekrdla, J.: Index transforms forN-dimensional DFT's. Numer. Math.51, 469–480 (1987)
Hekrdla, J.: Index transforms for multidimensional cyclic convolutions and discrete Fourier transforms. IEEE Trans. ASSP34, 996–997 (1986)
Hungerford, T.W.: Algebra. 2nd Ed. New York, Heidelberg, Berlin: Springer 1974
Nussbaumer, H.J.: Fast Fourier transform and convolution algorithms. 1st Ed. Berlin, Heidelberg, New York: Springer 1981
Winograd, S.: On computing the discrete Fourier transform. Math. Comput.32, 175–199 (1978)
Winogradow, I.M.: Elemente der Zahlentheorie. 6th Ed. Berlin: Deutscher Verlag der Wissenschaften 1955