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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.

MSC:
 41A50 Best approximation, Chebyshev systems