Jacobi Polynomials, Weighted Sobolev Spaces and Approximation Results of Some Singularities

In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWF). This set of special functions have been defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super-exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

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“…In their study, Nicaise 20 has shown that on H s ,0 , we can also use the norm We have directly that…”

Section: Approximation By the Gpswfs In Weighted Sobolev Spacesmentioning

confidence: 99%

“…In their study, Nicaise 20 has shown that on

H_{α, 0}^{s}

, we can also use the norm

{‖v‖}_{α, s}^{2} = \sum_{n \in ℕ} (1 + {(n (n + 2 α + 1))}^{2 s}) | {\hat{v}}_{n, α} |^{2} where v = \sum_{n = 0}^{\infty} {\hat{v}}_{n, α} {\tilde{P}}_{n}^{(α, α)} .

…”

Section: Approximation By the Gpswfs In Weighted Sobolev Spacesmentioning

confidence: 99%

Further spectral properties of the weighted finite Fourier transform operator and approximation in weighted Sobolev spaces

Bourguiba

Souabni

2020

Math Methods in App Sciences

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“…Secondly, we consider the smoothness spaces based on Legendre polynomials and, more general, on Jacobi polynomials in Examples 3.3 and 3.4, which are related to weighted Sobolev spaces. Such spaces were considered in, e.g., [31]. We discuss one special case where the relation can be directly explained.…”

Section: 3mentioning

confidence: 99%

Embeddings for infinite-dimensional integration and L2-approximation with increasing smoothness

Gnewuch

Hefter

Hinrichs

et al. 2019

Journal of Complexity

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We study integration and L 2 -approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness and appropriate weighted function spaces of fixed smoothness.

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“…At this stage it is relevant to point out that the same result could be obtained with spaces that, with respect to the H k ρ α (B d ) spaces, have a slightly less stringent requirement on the radial derivative of its members but still map to L 2 ρ α (B d ) (k even) or H 1 ρ α (B d ) (k odd) under the action of the ⌊k/2⌋-th power of the operator L (α) . In the one-dimensional case this idea is pursued in [29].…”

Section: Polynomial Eigenfunctions On the Euclidean Unit Ballmentioning

confidence: 99%

Orthogonal Polynomial Projection Error Measured in Sobolev Norms in the Unit Disk

Figueroa

2016

Constr Approx

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Abstract. We study approximation properties of weighted L 2 -orthogonal projectors onto the space of polynomials of degree less than or equal to N on the unit disk where the weight is of the generalized Gegenbauer form x → (1 − |x| 2 ) α . The approximation properties are measured in Sobolev-type norms involving canonical weak derivatives, all measured in the same weighted L 2 norm. Our basic tool consists in the analysis of orthogonal expansions with respect to Zernike polynomials. The sharpness of the main result is proved in some cases and otherwise strongly hinted at by reported numerical tests.A number of auxiliary results of independent interest are obtained including some properties of the uniformly and non-uniformly weighted Sobolev spaces involved, a Markov-type inequality, connection coefficients between Zernike polynomials and relations between the Fourier-Zernike expansions of a function and its derivatives.

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Jacobi Polynomials, Weighted Sobolev Spaces and Approximation Results of Some Singularities

Cited by 13 publications

References 18 publications

Further spectral properties of the weighted finite Fourier transform operator and approximation in weighted Sobolev spaces

Further spectral properties of the weighted finite Fourier transform operator and approximation in weighted Sobolev spaces

Embeddings for infinite-dimensional integration and L2-approximation with increasing smoothness

Orthogonal Polynomial Projection Error Measured in Sobolev Norms in the Unit Disk

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