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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.
Reviewer: G.M.L.Gladwell

MSC:
65T40 Numerical methods for trigonometric approximation and interpolation
65F30 Other matrix algorithms (MSC2010)
42A15 Trigonometric interpolation
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References:
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