On Bernstein’s Inequality for Entire Functions of Exponential Type

Rahman, Q. I.; Tariq, Q. M.

doi:10.1007/bf03321639

Cited by 9 publications

(3 citation statements)

References 6 publications

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“…This is because a polynomial p ∈ ℘ n cannot vanish at a point ζ unless it vanishes at 1/ζ also. The sub-class ℘ n of P n has been studied in many papers (see [1,[8][9][10][11][12][13][14][15][16][17]19,26,27]). Since the two sub-classes P * n and ℘ n of P n look somewhat similar it is natural to wonder if (1.4) holds also for polynomials belonging to ℘ n .…”

Section: Bernstein's Inequality For Polynomialsmentioning

confidence: 99%

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On Bernstein's inequality for entire functions of exponential type

Rahman

Tariq

2009

Journal of Mathematical Analysis and Applications

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Section: Bernstein's Inequality For Polynomialsmentioning

confidence: 99%

“…It is useful to recall ( [27,Lemma 3,p. 179]; also see [4]) that the Fourier exponents Λ 1 , Λ 2 , .…”

mentioning

confidence: 99%

On Bernstein's inequality for entire functions of exponential type

Rahman

Tariq

2009

Journal of Mathematical Analysis and Applications

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“…In 1951, S. M. Nilkol'skii gave the following inequality f p C p,q κ 1/q−1/p f q , 1 q p ∞, for any entire function f of exponential type κ belonging to L q (R) [4]. Bernstein inequality was studied also in [5][6][7][8][9][10][11]and Nikol'skii inequality was studied in [3,4,12,13]. Note that the inequalities of Bernstein and Nikol'skii play an important role in the Approximation Theory [2,3,15,16].…”

Section: Introductionmentioning

confidence: 99%

Неравенство Бернштейна - Никольского В Весовых Пространствах Лебега

Bang¹,

Huy²

2020

Владикавказский Математический Журнал

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In this paper, we give some results concerning Bernstein--Nikol'skii inequality for weighted Lebesgue spaces. The main result is as follows: Let $1 < u,p < \infty$, $0<q+ 1/p <v + 1/u <1,$ $v-q\geq 0$, $\kappa >0$, $f \in L^u_v(\R)$ and $\supp\widehat{f} \subset [-\kappa, \kappa]$. Then $D^mf \in L^p_q(\R)$, $\supp\widehat{D^m f}=\supp\widehat{f}$ and there exists a~constant~$C$ independent of $f$, $m$, $\kappa$ such that $\|D^mf\|_{L^p_{q}} \leq C m^{-\varrho} \kappa^{m+\varrho} \|f\|_{ L^u_v}, $ for all $m = 1,2,\dots $, where $\varrho=v + \frac{1}{u} -\frac{1}{p} - q>0,$ and the weighted Lebesgue space $L^p_q$ consists of all measurable functions such that $\|f\|_{L^p_q} = \big(\int_{\R} |f(x)|^p |x|^{pq} dx\big)^{1/p} < \infty.$ Moreover, $ \lim_{m\to \infty}\|D^mf\|_{L^p_{q}}^{1/m}= \sup \big\{ |x|: \, x \in \textnormal{supp}\widehat{f}\big \}.$ The~advantage of our result is that $m^{-\varrho}$ appears on the right hand side of the inequality ($\varrho >0$), which has never appeared in related articles by other authors. The corresponding result for the $n$-dimensional case is also obtained.

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Bernstein Inequality for Multivariate Functions with Smooth Fourier Images

Bang

Huy

2023

Ukr Math J

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On Bernstein’s Inequality for Entire Functions of Exponential Type

Cited by 9 publications

References 6 publications

On Bernstein's inequality for entire functions of exponential type

On Bernstein's inequality for entire functions of exponential type

Неравенство Бернштейна - Никольского В Весовых Пространствах Лебега

Bernstein Inequality for Multivariate Functions with Smooth Fourier Images

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