Notice of Retraction: The Inverse Problem of Anti-circulant Matrices in Signal Processing

Zhao, Wenling

doi:10.1109/kese.2009.21

Cited by 4 publications

(6 citation statements)

References 2 publications

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“…Circulant matrices have a wide range of applications, for examples in signal processing, coding theory, image processing, digital image disposal, self-regress design and so on. Numerical solutions of the certain types of elliptic and parabolic partial differential equations with periodic boundary conditions often involve linear systems associated with circulant matrices [9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal–Lucas Numbers

Bozkurt

Tam

2012

Applied Mathematics and Computation

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Let n ≥ 3 and Jn := circ(J1, J2, . . . , Jn) and ‫ג‬n := circ(j0, j1, . . . , jn−1) be the n × n circulant matrices, associated with the nth Jacobsthal number Jn and the nth Jacobsthal-Lucas number jn, respectively. The determinants of Jn and ‫ג‬n are obtained in terms of the Jacobsthal and Jacobsthal-Lucas numbers. These imply that Jn and ‫ג‬n are invertible. We also derive the inverses of Jn and ‫ג‬n), i.e., where J k and j k are the kth Jacobsthal and Jacobsthal-Lucas numbers, respectively, with the recurrence relations J k = J k−1 + 2J k−2 , j k = j k−1 + 2j k−2 , and the initial conditions J 0 = 0, J 1 = 1, j 0 = 2, and j 1 = 1 (k ≥ 2). Let α and β be

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Section: Introductionmentioning

confidence: 99%

Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal–Lucas Numbers

Bozkurt

Tam

2012

Applied Mathematics and Computation

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“…The remaining conclusions now follow from Proposition 1. ✷ In the proof of Proposition 2, we use direct calculation to show (15), (18), (21) and (22). Are (15) and (18) still true for m = 12l + 6 with l ≥ 4?…”

Section: 2mentioning

confidence: 99%

Positive semi-definiteness of generalized anti-circulant tensors

Wang

2016

Communications in Mathematical Sciences

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Anti-circulant tensors have applications in exponential data fitting. They are special Hankel tensors. In this paper, we extend the definition of anti-circulant tensors to generalized anti-circulant tensors by introducing a circulant index r such that the entries of the generating vector of a Hankel tensor are circulant with module r. In the special case when r = n, where n is the dimension of the Hankel tensor, the generalized anticirculant tensor reduces to the anti-circulant tensor. Hence, generalized anti-circulant tensors are still special Hankel tensors. For the cases that GCD(m, r) = 1, GCD(m, r) = 2 and some other cases, including the matrix case that m = 2, we give necessary and sufficient conditions for positive semi-definiteness of even order generalized anti-circulant tensors, and show that in these cases, they are sum of squares tensors. This shows that, in these cases, there are no PNS (positive semidefinite tensors which are not sum of squares) Hankel tensors.

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“…Convolution operations, and so circulant matrices, arise in number of applications: digital signal processing, image compression, physics/engineering simulations, number theory, coding theory, cryptography, etc. Numerical solutions of certain types of elliptic and parabolic partial differential equations with periodic boundary conditions often involve linear systems associated with circulant matrices [1]- [3]. A certain type of transformation of a set of numbers can be represented as the multiplication of a vector by a square matrix.…”

Section: Introductionmentioning

confidence: 99%

“…Let A 6 = circ 6 (5, 4, 9, 0, 8, −2) be a circulant pentadiagonal matrix. we find the eigenvalues of A 6 by using (5000 + 6, 0621i,λ 5000 + 4, 3301i, λ 4 = 20, λ 5 = λ 3 = −4, 5000 − 4, 3301i, λ 6 = λ 2 = −2, 5000 − 6, 0621iand from Theorem 4.1, the entries of A3 6 circ 6 (3778, 1008, 3483, 938, 3651, 966) . Let A 9 = circ 9 (−2, 3, −4, 9, 0, 0, 6, 5, −1) be a circulant heptadiagonal matrix.…”

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confidence: 99%

Circulant $m$-Diagonal Matrices Associated with Chebyshev Polynomials

Öteleş

2021

Fundamental Journal of Mathematics and Applications

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In this study, we deal with an m banded circulant matrix, generally called circulant mdiagonal matrix. This special family of circulant matrices arise in many applications such as prediction, time series analysis, spline approximation, difference solution of partial differential equations, and so on. We firstly obtain the statements of eigenvalues and eigenvectors of circulant m-diagonal matrix based on the Chebyshev polynomials of the first and second kind. Then we present an efficient formula for the integer powers of this matrix family depending on the polynomials mentioned above. Finally, some illustrative examples are given by using maple software, one of computer algebra systems (CAS).

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Notice of Retraction: The Inverse Problem of Anti-circulant Matrices in Signal Processing

Cited by 4 publications

References 2 publications

Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal–Lucas Numbers

Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal–Lucas Numbers

Positive semi-definiteness of generalized anti-circulant tensors

Circulant $m$-Diagonal Matrices Associated with Chebyshev Polynomials

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