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Solving singular convolution equations using the inverse fast Fourier transform. (English) Zbl 1265.42020
The paper is concerned with the application of the convolution theorem of the Fourier transform theory to a singular convolution equation. The use of this theorem via the inverse Fourier transform is well-known for a nonsingular convolution. Apparently, no statement of this kind has been published for the singular equation. The authors present the problem and prove that if the transfer function is a trigonometric polynomial with simple zeros on the unit circle, then they can find the (non-unique) unknown solution by a generalization of the standard process.
Numerical importance of the paper consists in the fact that the authors’ generalized solution procedure can be carried out by the discrete Fourier transform via the fast Fourier transform. The authors also present two simple numerical examples of such a process. The paper is interesting for both theoretical and numerical analyst.

MSC:
42A85 Convolution, factorization for one variable harmonic analysis
65R10 Numerical methods for integral transforms
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References:
[1] I. Babuška: The Fourier transform in the theory of difference equations and its applications. Arch. Mech. 11 (1959), 349–381. · Zbl 0092.12201
[2] R. Beals: Advanced Mathematical Analysis. GTM 12. Springer, New York-Heidelberg-Berlin, 1973. · Zbl 0294.46029
[3] B. Fisher: The product of distributions. Q. J. Math. 22 (1971), 291–298. · Zbl 0213.13104
[4] H. Jarchow: Locally Convex Spaces. B.G.Teubner, Stuttgart, 1981. · Zbl 0466.46001
[5] W. Rudin: Functional Analysis. McGraw-Hill, New York, 1973.
[6] E. Vitásek: Periodic distributions and discrete Fourier transforms. Pokroky mat., fyz. astronom. 54 (2009), 137–144. (In Czech.)
[7] G.G. Walter: Wavelets and Other Orthogonal Systems with Applications. CRC Press, Boca Raton, 1994. · Zbl 0866.42022
[8] J. L. Walsh, W.E. Sewell: Note on degree of approximation to an integral by Riemann sums. Am. Math. Monthly 44 (1937), 155–160. · Zbl 0016.29901
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