Lower bounds of some singular integrals and their applications

Abstract.

Let ${\beta > 1}$ and ${f \in C(\mathbb{R})}$ . We are interested in the lower bounds of the integral:

$$ \int\limits^{\infty}_{h}\frac{\Delta^{2}_{t}f(x)}{t^{\beta}}dt, $$

where h > 0 and ${{\Delta^{2}_{t}f(x)} = f(x+t)+f(x-t)-2f(x)}$ . Using the lower bounds for these integrals we obtain in particular for the so-called Fejér operator ${\sigma_{n}(f)}$ of ${f \in C_{2\pi}}$ the following asymptotic expression

$$ \| \sigma_{n}(f)- f \| \asymp \frac{1}{n} \| \int\limits^{\infty}_{\frac{1}{n}} \frac{\Delta^{2}_{t}f}{t^{2}}dt \|, $$

which essentially improves the results concerning the approximation behavior of this operator.