Volume 1, Issue 3
The Jackson Inequality for the Best $L^2$-Approximation of Functions on [0,1] with the Weight $x$

Numer. Math. Theor. Meth. Appl., 1 (2008), pp. 340-356.

Published online: 2008-01

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• Abstract

Let $L^2([0,1],x)$ be the space of the real valued, measurable, square summable functions on $[0,1]$ with weight $x$, and let $\mathcal{L}_n$ be the subspace of $L^2([0,1],x)$ defined by a linear combination of $J_0(\mu_kx)$, where $J_0$ is the Bessel function of order $0$ and $\{\mu_k\}$ is the strictly increasing sequence of all positive zeros of $J_0$. For $f\in L^2([0,1],x),$ let $E(f,{\cal L}_n)$ be the error of the best $L^2([0,1],x)$, i.e., approximation of $f$ by elements of ${\cal L}_n.$ The shift operator of $f$ at point $x\in [0, 1]$ with step $t\in [0, 1]$ is defined by$$T(t)f(x)=\frac{1}{\pi}\int_0^{\pi}f\big(\sqrt{x^2+t^2-2xt\cosθ}\big)dθ.$$The differences $(I-T(t))^{r/2}f=\sum_{j=0}^{\infty}(-1)^j{{r/2}\choose{j}}T^j(t)f$ of order $r\in (0, \infty)$ and the $L^2([0,1],x)-$modulus of continuity $\omega_r(f,\tau)=\sup\{\|(I-T(t))^{r/2}f\|:$ $0 \leq t\leq \tau \}$ of order $r$ are defined in the standard way, where $T^0(t)=I$ is the identity operator. In this paper, we establish the sharp Jackson inequality between $E(f, {\cal L}_n)$ and $\omega_r(f,\tau)$ for some cases of $r$ and $\tau$. More precisely, we will find the smallest constant $\mathcal{K}_n(\tau, r)$ which depends only on $n, r,$and $\tau,$ such that the inequality $E(f, {\cal L}_n)\le\mathcal{K}_n(\tau, r)(\tau, r)\omega_r(f,\tau)$ is valid.

• Keywords

Jackson inequality, modulus of continuity, best approximation, Bessel function.

42B25, 47B47, 42B20, 47A

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• TXT
@Article{NMTMA-1-340, author = {}, title = {The Jackson Inequality for the Best $L^2$-Approximation of Functions on [0,1] with the Weight $x$}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2008}, volume = {1}, number = {3}, pages = {340--356}, abstract = {

Let $L^2([0,1],x)$ be the space of the real valued, measurable, square summable functions on $[0,1]$ with weight $x$, and let $\mathcal{L}_n$ be the subspace of $L^2([0,1],x)$ defined by a linear combination of $J_0(\mu_kx)$, where $J_0$ is the Bessel function of order $0$ and $\{\mu_k\}$ is the strictly increasing sequence of all positive zeros of $J_0$. For $f\in L^2([0,1],x),$ let $E(f,{\cal L}_n)$ be the error of the best $L^2([0,1],x)$, i.e., approximation of $f$ by elements of ${\cal L}_n.$ The shift operator of $f$ at point $x\in [0, 1]$ with step $t\in [0, 1]$ is defined by$$T(t)f(x)=\frac{1}{\pi}\int_0^{\pi}f\big(\sqrt{x^2+t^2-2xt\cosθ}\big)dθ.$$The differences $(I-T(t))^{r/2}f=\sum_{j=0}^{\infty}(-1)^j{{r/2}\choose{j}}T^j(t)f$ of order $r\in (0, \infty)$ and the $L^2([0,1],x)-$modulus of continuity $\omega_r(f,\tau)=\sup\{\|(I-T(t))^{r/2}f\|:$ $0 \leq t\leq \tau \}$ of order $r$ are defined in the standard way, where $T^0(t)=I$ is the identity operator. In this paper, we establish the sharp Jackson inequality between $E(f, {\cal L}_n)$ and $\omega_r(f,\tau)$ for some cases of $r$ and $\tau$. More precisely, we will find the smallest constant $\mathcal{K}_n(\tau, r)$ which depends only on $n, r,$and $\tau,$ such that the inequality $E(f, {\cal L}_n)\le\mathcal{K}_n(\tau, r)(\tau, r)\omega_r(f,\tau)$ is valid.

}, issn = {2079-7338}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/nmtma/10110.html} }
TY - JOUR T1 - The Jackson Inequality for the Best $L^2$-Approximation of Functions on [0,1] with the Weight $x$ JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 340 EP - 356 PY - 2008 DA - 2008/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/nmtma/10110.html KW - Jackson inequality, modulus of continuity, best approximation, Bessel function. AB -

Let $L^2([0,1],x)$ be the space of the real valued, measurable, square summable functions on $[0,1]$ with weight $x$, and let $\mathcal{L}_n$ be the subspace of $L^2([0,1],x)$ defined by a linear combination of $J_0(\mu_kx)$, where $J_0$ is the Bessel function of order $0$ and $\{\mu_k\}$ is the strictly increasing sequence of all positive zeros of $J_0$. For $f\in L^2([0,1],x),$ let $E(f,{\cal L}_n)$ be the error of the best $L^2([0,1],x)$, i.e., approximation of $f$ by elements of ${\cal L}_n.$ The shift operator of $f$ at point $x\in [0, 1]$ with step $t\in [0, 1]$ is defined by$$T(t)f(x)=\frac{1}{\pi}\int_0^{\pi}f\big(\sqrt{x^2+t^2-2xt\cosθ}\big)dθ.$$The differences $(I-T(t))^{r/2}f=\sum_{j=0}^{\infty}(-1)^j{{r/2}\choose{j}}T^j(t)f$ of order $r\in (0, \infty)$ and the $L^2([0,1],x)-$modulus of continuity $\omega_r(f,\tau)=\sup\{\|(I-T(t))^{r/2}f\|:$ $0 \leq t\leq \tau \}$ of order $r$ are defined in the standard way, where $T^0(t)=I$ is the identity operator. In this paper, we establish the sharp Jackson inequality between $E(f, {\cal L}_n)$ and $\omega_r(f,\tau)$ for some cases of $r$ and $\tau$. More precisely, we will find the smallest constant $\mathcal{K}_n(\tau, r)$ which depends only on $n, r,$and $\tau,$ such that the inequality $E(f, {\cal L}_n)\le\mathcal{K}_n(\tau, r)(\tau, r)\omega_r(f,\tau)$ is valid.

Jian Li & Yongping Liu. (2020). The Jackson Inequality for the Best $L^2$-Approximation of Functions on [0,1] with the Weight $x$. Numerical Mathematics: Theory, Methods and Applications. 1 (3). 340-356. doi:
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