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The Jackson inequality for the best \(L^2\)-approximation of functions on [0,1] with the weight \(x\). (English) Zbl 1174.41338
Summary: 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 \(J_0(\mu_k x)\), 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 \(E(f,\mathcal{L}_n)\) be the error of the best \(L^2([0,1],x)\), i.e., approximation of \(f\) by elements of \(\mathcal{L}_n\). The shift operator of \(f\) at point \(x\in[0,1]\) with step \(t\in[0,1]\) is defined by \[ T(t)f(x)=\frac{1}{\pi}\int^\pi_0f(\sqrt{x^2+t^2-2xt\cos\theta})\,d\theta. \] The differences \((I-T(t))^{r/2}f=\sum^\infty_{j=0}(-1)^j(^{r/2}_{\;j})T^j(t)f\) 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 \(E(f,\mathcal{L}_n)\) and \(\omega_r(f,\tau)\) for some cases of \(r\) and \(\tau\). More precisely, we find the smallest constant \(\mathcal{K}_n(\tau,r)\) with depends only on \(n\), \(r\), and \(\tau\), such that the inequality \(E(f,\mathcal{L}_n)\leq \mathcal{K}_n(\tau,r)\omega_r(f,\tau)\) is valid.

41A50 Best approximation, Chebyshev systems