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On the Autoconvolution Equation and Total Variation Constraints
Author(s) -
Fleischer G.,
Gorenflo R.,
Hofmann B.
Publication year - 1999
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/(sici)1521-4001(199903)79:3<149::aid-zamm149>3.0.co;2-n
Subject(s) - mathematics , interval (graph theory) , mathematical analysis , combinatorics
This paper is concerned with the numerical analysis of the autoconvolution equation x × x = y restricted to the interval [0,1]. We present a discrete constrained least squares approach and prove its convergence in L p (0,1), 1≤ p ≤ ∞, where the regularization is based on a prescribed bound for the total variation of admissible solutions. This approach includes the case of non‐smooth solutions possessing jumps. Moreover, an adaptation to the Sobolev space H 1 (0,1) is added. A numerical case study concerning the determination of non‐monotone smooth and non‐smooth functions x from the autoconvolution equation with noisy data y completes the paper.