Generalized Zernike polynomials: operational formulae and generating functions

Aharmim, Bouchra; Amal, El Hamyani; Fouzia, El Wassouli; Ghanmi, Allal

doi:10.1080/10652469.2015.1012510

Cited by 13 publications

(12 citation statements)

References 25 publications

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“…We observe that the normalization adopted in Aharmim et al [1] for the disc polynomials is different from the one we use here.…”

Section: Proof Of the Resultsmentioning

confidence: 64%

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Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Massa¹,

Peron²,

Porcu³

2017

SIGMA

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Abstract. We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Montée and , 1965], allow for similar walks through dimensions. We show that the Montée operators also preserve, up to a constant, strict positive definiteness. For the Descente operators, we show that strict positive definiteness is preserved under some additional conditions, but we provide counterexamples showing that this is not true in general. We also provide a list of parametric families of (strictly) positive definite functions over complex spheres, which are important for several applications.

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“…We observe that the normalization adopted in Aharmim et al [1] for the disc polynomials is different from the one we use here.…”

Section: Proof Of the Resultsmentioning

confidence: 64%

“…The first lemma contains recurrence formulas connecting disc polynomials of different indexes and degrees. They are obtained from equation (5.5) in Aharmim et al [1] and the following properties of the disc polynomials…”

Section: Proof Of the Resultsmentioning

confidence: 99%

Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Massa¹,

Peron²,

Porcu³

2017

SIGMA

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“…For the classical orthogonal polynomials -Hermite, Laguerre, Jacobi -this is in Section 3, and we show that the integrated formulas in case of the Hermite and Laguerre polynomials are essentially given as a change of coordinates. As is clear from the above, several of these results for classical orthogonal polynomials can be traced back in the literature, see [1,2,7,8,12,23,25] and references given there. We show how to generalize these operational formulas and expansion identities to all of the families of orthogonal polynomials in the Askey scheme and its q-analogue.…”

Section: Introductionmentioning

confidence: 86%

“…This part is characterized by having the derivative as the lowering operator. The extension to Zernike polynomials is given in [1], where more references to the literature are given.…”

Section: Introductionmentioning

confidence: 99%

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Ismail¹,

Koelink²,

Román³

2018

SIGMA

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Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its q-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big q-Jacobi polynomials and big q-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials.

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“…Zernike's polynomials functions are a set of two‐dimensional orthogonal polynomials defined on a circular domain of unitary radius 15,16

normalC = [0, 1] \times [0, 2 π] \subset R^{2}

As a natural consequence of their definition domain, Zernike's polynomials are generally expressed in polar coordinates by means of two variables: a polar phase

ϑ

and a radial module

ϱ

.…”

Section: Zernike's Polynomialsmentioning

confidence: 99%

Organic solar cells defects classification by using a new feature extraction algorithm and an EBNN with an innovative pruning algorithm

et al. 2021

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Physical defects reduce the organic solar cells (OSC) functioning. Throughout the OSC fabrication process, the defects can occur, for instance, by scratches or uneven morphologies. In general, bulk defects, interface defects, and interconnect defects can promote shunt and series resistance of the cell. It is crucial to properly detect and classify such defects and their amount in the structure. Correlating such defects with the performance of the cell is important both during the R&D stages to optimize processes, and for mass production stages where defects detection is an integral part of the production line. For the recognition of texture variations in the scanning electron microscope images caused by these defects is crucial the definition of a set of features for texture representation. Because the low‐order Zernike moments can represent the whole shape of the image and the high‐order Zernike moments can describe the detail. Then, in our case, the feature of the image can be represented by a small number of Zernike moments. In fact, the feature set extracted and described by the Zernike moments are not sensitive to the noises and are hardly redundant. So it possible concentrate the signal energy over a set of few vectors. Finally, for classification, an elliptical basis function neural networks was used. The results show effectiveness of the proposed methodology. In fact, we obtained correct classification of 89.3% over testing data set.

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Generalized Zernike polynomials: operational formulae and generating functions

Cited by 13 publications

References 25 publications

Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Organic solar cells defects classification by using a new feature extraction algorithm and an EBNN with an innovative pruning algorithm

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