Weighted Markov-Bernstein inequalities for entire functions of exponential type

Lubinsky, D. S.

doi:10.2298/pim1410181l

Cited by 6 publications

(2 citation statements)

References 16 publications

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“…Weighted versions of (1.2) for the so-called weights with doubling properties were established by G. Mastroianni and V. Totik [27] when 1 ≤ p ≤ ∞ and by T. Erdelyi [11] when 0 < p < 1. Recently D. Lubinsky [25] proved the weighted analog of (1.3) for entire functions of exponential type, for all p > 0 and for the same type of doubling weights which, among others, contain those of the form (1+x 2 ) α , α ∈ R.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

A discrete weighted Markov--Bernstein inequality for polynomials and sequences

Dimitrov¹,

Nikolov²

2020

Preprint

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For parameters c ∈ (0, 1) and β > 0, let ℓ 2 (c, β) be the Hilbert space of real functions defined on N (i.e., real sequences), for whichWe study the best (i.e., the smallest possible) constant γn(c, β) in the discrete Markov-Bernstein inequalitywhere Pn is the set of real algebraic polynomials of degree at most n and ∆f (x) := f (x + 1) − f (x) .We prove that(ii) For every fixed c ∈ (0, 1) , γn(c, β) is a monotonically decreasing function of β in (0, ∞) ; (iii) For every fixed c ∈ (0, 1) and β > 0 , the best Markov-Bernstein constants γn(c, β) are bounded uniformly with respect to n. A similar Markov-Bernstein unequality is proved for sequences in ℓ 2 (c, β) . We also establish a relation between the best Markov-Bernstein constants γn(c, β) and the smallest eigenvalues of certain explicitly given Jacobi matrices.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

A discrete weighted Markov--Bernstein inequality for polynomials and sequences

Dimitrov¹,

Nikolov²

2020

Preprint

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“…The next lemma says that if we do not focus on the constants, then the inequalities of Bernstein and Riesz type are simple corollaries of the estimates for convolutions. We note that the Bernstein inequality is true for a wider class of weights, see [21].…”

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confidence: 98%

Direct and inverse theorems of approximation theory in Lebesgue spaces with Muckenhoupt weights

Vinogradov

2023

Ufimsk. Mat. Zh.

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In this work we establish direct and inverse theorems of approximation theory in Lebesgue spaces 𝐿 𝑝,𝑤 with Muckenhoupt weights 𝑤 on the axis and on a period. The classical definition of the modulus of continuity can be meaningless in weighted spaces. Therefore, as the modules of continuity, including non-integer order, we use the norms of powers of deviation of Steklov means. The properties of these quantities are derived, some of which are similar to the properties of usual modules of continuity. In addition to the direct and inverse theorems, we obtain equivalence relations between the modules of continuity and the 𝐾-and 𝑅-functionals.The proofs are based on estimates for the norms of convolution operators and they do not employ a maximal function. This allows us to establish the results for all 𝑝 ∈ [1, +∞) not excluding the case 𝑝 = 1. Previously used methods that employed the maximal function in one form or another are unsuitable for 𝑝 → 1. In addition, by the convolution-based approach we can obtain results simultaneously in the periodic and non-periodic case. With rare exceptions, constants are not specified explicitly, but their dependence on parameters is always tracked. All constants in the estimates depend on [𝑤] 𝑝 (Muckenhoupt characteristics of weight 𝑤), and there is no other dependence on 𝑤 and 𝑝. The norms of convolution operators are estimated explicitly in terms of [𝑤] 𝑝 . The methods of this work can be applied to prove direct and inverse theorems in more general functional spaces.

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A discrete weighted Markov–Bernstein inequality for sequences and polynomials

Dimitrov

Nikolov

2021

Journal of Mathematical Analysis and Applications

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Weighted Markov-Bernstein inequalities for entire functions of exponential type

Cited by 6 publications

References 16 publications

A discrete weighted Markov--Bernstein inequality for polynomials and sequences

A discrete weighted Markov--Bernstein inequality for polynomials and sequences

Direct and inverse theorems of approximation theory in Lebesgue spaces with Muckenhoupt weights

A discrete weighted Markov–Bernstein inequality for sequences and polynomials

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