Chebyshev polynomials and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"><mml:mi>r</mml:mi></mml:math>-circulant matrices

Pucanović, Zoran S.; Pešović, Marko

doi:10.1016/j.amc.2022.127521

Cited by 4 publications

(3 citation statements)

References 6 publications

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“…We require that exactly q of the n Chebyshev roots in [−a, a] are included in [−1, 1]. It is worth mentioning that the roots of T n (x), defined in (5), and of T n (x), defined in Equation ( 13), are not equally spaced in the standard sense; however, they are equally spaced along the unit semicircle and a semicircle with radius a, respectively, (see Figure 1). In the following proposition, we provide a criterion for the choice of a, so that the q Chebyshev roots included in [−1, 1] are almost equally spaced.…”

Section: Chebyshev Interpolation With Almost Equally Spaced Pointsmentioning

confidence: 99%

“…Since their introduction by the renowned Russian mathematician Pafnuty Chebyshev [1], these orthogonal polynomials have played a pivotal role in approximation theory [2], providing an invaluable tool in numerical analysis [3]. Furthermore, Chebyshev polynomials are widely used in several fields including scientific computing [4], matrix theory [5], integral transforms [6], and image processing [7], as well as engineering [8,9], machine learning [10], quantum computing [11], and medical imaging [12,13].…”

Section: Introductionmentioning

confidence: 99%

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Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography

Marinakis,

Fokas,

Kastis

et al. 2023

Mathematics

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Since their introduction, Chebyshev polynomials of the first kind have been extensively investigated, especially in the context of approximation and interpolation. Although standard interpolation methods usually employ equally spaced points, this is not the case in Chebyshev interpolation. Instead of equally spaced points along a line, Chebyshev interpolation involves the roots of Chebyshev polynomials, known as Chebyshev nodes, corresponding to equally spaced points along the unit semicircle. By reviewing prior research on the applications of Chebyshev interpolation, it becomes apparent that this interpolation is rather impractical for medical imaging. Especially in clinical positron emission tomography (PET) and in single-photon emission computerized tomography (SPECT), the so-called sinogram is always calculated at equally spaced points, since the detectors are almost always uniformly distributed. We have been able to overcome this difficulty as follows. Suppose that the function to be interpolated has compact support and is known at q equally spaced points in −1,1. We extend the domain to −a,a, a>1, and select a sufficiently large value of a, such that exactlyq Chebyshev nodes are included in −1,1, which are almost equally spaced. This construction provides a generalization of the concept of standard Chebyshev interpolation to almost equally spaced points. Our preliminary results indicate that our modification of the Chebyshev method provides comparable, or, in several cases including Runge’s phenomenon, superior interpolation over the standard Chebyshev interpolation. In terms of the L∞ norm of the interpolation error, a decrease of up to 75% was observed. Furthermore, our approach opens the way for using Chebyshev polynomials in the solution of the inverse problems arising in PET and SPECT image reconstruction.

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Section: Chebyshev Interpolation With Almost Equally Spaced Pointsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography

Marinakis,

Fokas,

Kastis

et al. 2023

Mathematics

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“…Marin has also discussed examples of concrete usage areas of studies similar to those in [14,15]. Pucanovic et al by taking the 𝑟 −circulant matrix and Chebyshev polynomials as the subject, handled the inputs of the r-circulant matrices by constructing them from Chebyshev polynomials [16]. In this paper, we study 𝑟 −circulant matrices with generalized Fermat numbers.…”

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confidence: 99%

On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers

KULOǦLU,

ESER,

ÖZKAN

2023

Sakarya University Journal of Science

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-circulant matrices have applied in numerical computation, signal processing, coding theory, etc. In this study, our main goal is to investigate the r-circulant matrices of generalized Fermat numbers which are shown by We obtain the eigenvalues, determinants, sum identity of matrices. Also we find upper and lower bounds for the spectral norms of generalized Fermat r-circulant matrices. Beside these, we present 〖GR〗_(a,b,r)^* matrix in the form of the Hadamard product of two matrices as 〖GR〗_(a,b,r)^*=A.B. In addition, we get the right and skew-right circulant matrices for . Finally, we examine their different norms (Spectral and Euclidean) and limits for matrix norms.

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On K-Chebsyhev Sequence

Özkan

Akkuş

2023

Wseas Transactions on Mathematics

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In this study, firstly the k-Chebsyhev sequence is defined, and some terms of this sequence are given. Then, the relations between the terms of the k-Chebsyhev sequence are presented and the generating function of this sequence is obtained. In addition, the Catalan transformation of the sequence is given and the generating function of the Catalan k-Chebsyhev sequence is obtained. Finally, the Hankel transform is applied to the Catalan k-Chebsyhev transform.

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Chebyshev polynomials and r-circulant matrices

Cited by 4 publications

References 6 publications

Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography

Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography

On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers

On K-Chebsyhev Sequence

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