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Linear transforms and convolution. (English) Zbl 0622.65143
In modern computation the circular convolution of two sequences U and V is often computed by means of the discrete Fourier transform: $$U*V=D^{- 1}(D(U)\otimes D(V)).$$ This paper deals with generalizations of this result by enquiring when three operators $$L_ 1$$, $$L_ 2$$, $$L_ 3$$ support circular convolution. The paper is concerned with the following:
Let $$L_ 1$$, $$L_ 2$$, $$L_ 3$$ be linear transforms of the sequences of length N of elements from the ring R. The 3-tuple $$(L_ 1,L_ 2,L_ 3)$$ is said to support circular convolution or briefly it is SCC if for each sequence $$x_ 1$$ and each sequence $$x_ 2$$ of length N of elements from the ring R the equality $$x_ 1*x_ 2=L_ 3(L_ 1(x_ 1)\otimes L_ 2(x_ 2))$$ is satisfied.