Accuracy of the approximation of the characteristic functions by polynomials

In the paper one obtains a series of statements allowing us to estimate the accuracy of the approximation of the characteristic function $f(t)=\int e^{itx}dV(x)$ by a polynomial of integer powers of $it$. For example,
$$ C_1\Gamma(b)\le\sup_{|t|\le b}|f(t)-1-\sum_{l=1}^{2M-1}\frac{(it)^l}{l!}d_l|\le C_2\Gamma(b) $$
where the positive constants 
$$, $$
depend only on $M$, $M\ge1$ is an integer, $b>0,$
$$ \Gamma(b)=\int_{-\infty}^{\infty}\min\Big(1, (xb)^{2M}\Big)dV(x)+\max_{1\le l\le2M}b^2|d_l-\int_{|xb|\le1}x^ldV(x)|. $$