Differentiation by integration using orthogonal polynomials, a survey

Diekema, Enno; Koornwinder, Tom H.

doi:10.1016/j.jat.2012.01.003

Cited by 26 publications

(31 citation statements)

References 48 publications

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“…In [1], a comprehensive survey is given about the orthogonal derivative. This derivative has a long, but almost forgotten history.…”

Section: Introductionmentioning

confidence: 99%

The Fractional Orthogonal Difference with Applications

Diekema¹

2015

Mathematics

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This paper is a follow-up of a previous paper of the author published in Mathematics journal in 2015, which treats the so-called continuous fractional orthogonal derivative. In this paper, we treat the discrete case using the fractional orthogonal difference. The theory is illustrated with an application of a fractional differentiating filter. In particular, graphs are presented of the absolutel value of the modulus of the frequency response. These make clear that for a good insight into the behavior of a fractional differentiating filter, one has to look for the modulus of its frequency response in a log-log plot, rather than for plots in the time domain.

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“…In [1], a comprehensive survey is given about the orthogonal derivative. This derivative has a long, but almost forgotten history.…”

Section: Introductionmentioning

confidence: 99%

The Fractional Orthogonal Difference with Applications

Diekema¹

2015

Mathematics

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“…In [8] (Theorem 3.1), the following theorem is proved (see [8], Section 2.1, for generalities about orthogonal polynomials):…”

Section: The Fractional Weyl Transform For the Orthogonal Derivativementioning

confidence: 99%

“…For example, the ordinary derivative does not exist for the function f (x) = |x| for x = 0, but the orthogonal derivative for this function does exist. For less trivial examples, see [8] (Section 3.8).…”

Section: The Fractional Weyl Transform For the Orthogonal Derivativementioning

confidence: 99%

“…The above formulas remain valid for ν = n, where we get the transfer functions as in [8] (Section 5). …”

Section: (1022) After Substitution There Remainsmentioning

confidence: 99%

“…In [8], we reviewed a formula for the so-called orthogonal derivative, which has a long history. This derivative (of order n) can be computed as a limit of a certain integral.…”

Section: Introductionmentioning

confidence: 99%

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The Fractional Orthogonal Derivative

Diekema¹

2015

Mathematics

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This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann-Liouville fractional derivative is considered by replacing the n-th derivative by its approximation in the formula for the fractional derivative. In the case of, for instance, Jacobi polynomials, an explicit formula for the kernel of this approximate fractional derivative can be given. Next, we consider the fractional derivative as a filter and compute the frequency response in the continuous case for the Jacobi polynomials and in the discrete case for the Hahn polynomials. The frequency response in this case is a confluent hypergeometric function. A different approach is discussed, which starts with this explicit frequency response and then obtains the approximate fractional derivative by taking the inverse Fourier transform.

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Direct and inverse computation of Jacobi matrices of infinite iterated function systems

Mantica

2013

Numer. Math.

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We introduce a new set of algorithms to compute the Jacobi matrices associated with invariant measures of infinite iterated function systems, composed of one–dimensional, homogeneous affine maps. We demonstrate their utility in the study of theoretical problems, like the conjectured almost periodicity of such Jacobi matrices, the singularity of the measures, and the logarithmic capacity of their support. Since our technique is based on a reversible transformation between pairs of Jacobi matrices, it can also be applied to solve an inverse/approximation problem. The proposed algorithms are tested in significant, highly sensitive cases: they perform in a stable fashion, and can reliably compute Jacobi matrices of large order

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Differentiation by integration using orthogonal polynomials, a survey

Cited by 26 publications

References 48 publications

The Fractional Orthogonal Difference with Applications

The Fractional Orthogonal Difference with Applications

The Fractional Orthogonal Derivative

Direct and inverse computation of Jacobi matrices of infinite iterated function systems

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