Weighted Sobolev spaces: Markov-type inequalities and duality

Marcellán, Francisco; Quintana, Yamilet; Rodríguez, José M.

doi:10.1007/s13373-017-0104-y

Cited by 4 publications

(5 citation statements)

References 35 publications

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“…First, we replace the function c 1 (α, β)(n + 1) max{α,β}+1 by a smallest one, and we remove the hypothesis max{α, β} ≥ −1/2. As a consequence of Theorem 1.1, we improve some Markov-type inequalities in weighted Sobolev spaces which appear in [21].…”

Section: Lupaş-type Inequality and Applications To Markov-type Inequamentioning

confidence: 92%

“…for every n ∈ N and P ∈ P n , where q = max{α, β} ≥ −1/2 and q = min{α, β}. By using Lupaş' inequality and the asymptotic behavior of Gamma function, the authors showed in [21] that there exists a constant c 1 (α, β), which just depends on α and β, such that…”

Section: Lupaş-type Inequality and Applications To Markov-type Inequamentioning

confidence: 99%

“…Inequality (1.5) is interesting by itself. It has been applied (see [21]) to obtain Markov-type inequalities in weighted Sobolev spaces associated with vector of measures intimately related with the classical weights (normal, gamma and beta distributions). The study of properties of such functional spaces has been done in classical monographs as [2] and [22], while [18] is a basic reference about weighted Sobolev spaces.…”

Section: Lupaş-type Inequality and Applications To Markov-type Inequamentioning

confidence: 99%

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Lupaş-type inequality and applications to Markov-type inequalities in weighted Sobolev spaces

Marcellán

Rodríguez

2020

Bull. Math. Sci.

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Communicated by Ari LaptevWeighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. In particular, analytic properties of such polynomials have been extensively studied, mainly focused on their asymptotic behavior and the location of their zeros. On the other hand, the behavior of the Fourier-Sobolev projector allows to deal with very interesting approximation problems. The aim of this paper is twofold. First, we improve a wellknown inequality by Lupaş by using connection formulas for Jacobi polynomials with different parameters. In the next step, we deduce Markov-type inequalities in weighted Sobolev spaces associated with generalized Laguerre and generalized Hermite weights.

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Section: Lupaş-type Inequality and Applications To Markov-type Inequamentioning

confidence: 92%

Section: Lupaş-type Inequality and Applications To Markov-type Inequamentioning

confidence: 99%

Section: Lupaş-type Inequality and Applications To Markov-type Inequamentioning

confidence: 99%

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Lupaş-type inequality and applications to Markov-type inequalities in weighted Sobolev spaces

Marcellán

Rodríguez

2020

Bull. Math. Sci.

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“…Uniformly convex spaces are reflexive and their dual spaces are uniformly convex, too. Examples of uniformly convex spaces are L p and ℓ p , p ∈ (1, ∞) as well as Sobolev spaces W m p , p ∈ (1, ∞) [36] and Orlicz spaces [30]. Therefore L 1 , L ∞ and C[0, 1] are not uniformly convex.…”

Section: R-uniformly Convex Spacesmentioning

confidence: 99%

On $α$-Firmly Nonexpansive Operators in $r$-Uniformly Convex Spaces

Бeрделлима¹,

Steidl²

2021

Preprint

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We introduce the class of α-firmly nonexpansive and quasi α-firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where α-firmly nonexpansive operators coincide with so-called α-averaged operators. For our more general setting, we show that α-averaged operators form a subset of α-firmly nonexpansive operators. We develop some basic calculus rules for (quasi) α-firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) α-firmly nonexpansive. Moreover, we will see that quasi α-firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder's demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates x n+1 := T x n belong to the fixed point set Fix T whenever the operator T is nonexpansive and quasi α-firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial's property, then these iterates converge weakly to some element in Fix T . Further, the projections P Fix T x n converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in L p , p ∈ (1, ∞)\{2} spaces on probability measure spaces.

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“…In [8], the authors considered the Fourier series for polynomials associated to the Sobolev-type inner product (7) f, g S =…”

Section: Introductionmentioning

confidence: 99%

Fourier series of Jacobi–Sobolev polynomials

Ciaurri

Ceniceros

2018

Integral Transforms and Special Functions

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Let {q (α,β,m) n (x)} n≥0 be the orthonormal polynomials respect to the Sobolev-type inner product f, g α,β,m = m k=0 1 −1We obtain necessary and sufficient conditions for the uniform boundedness of the partial sum operators related to this sequence of polynomials in the Sobolev space W p,m α,β . As a consequence we deduce the convergence of such partial sums in the norm of W p,m α,β .2010 Mathematics Subject Classification. Primary: 42A20. Secondary: 33C47. Key words and phrases. Sobolev-type inner product, Sobolev polynomials, Jacobi polynomials, partial sum operator.

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Weighted Sobolev spaces: Markov-type inequalities and duality

Abstract: Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. The aim of this paper is to prove several important properties of weighted Sobolev spaces: separability, reflexivity, uniform convexity, duality and Markov-type inequalities.Communicated by Ari Laptev.

Cited by 4 publications

References 35 publications

Lupaş-type inequality and applications to Markov-type inequalities in weighted Sobolev spaces

Lupaş-type inequality and applications to Markov-type inequalities in weighted Sobolev spaces

On $α$-Firmly Nonexpansive Operators in $r$-Uniformly Convex Spaces

Fourier series of Jacobi–Sobolev polynomials

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