De la Vallée-Poussin singular integral

An integral of the form

$$V_n(f;x)=\frac{1}{2\pi}\frac{(2n)!!}{(2n-1)!!}\int\limits_{-\pi}^\pi f(x+t)\cos^{2n}\frac t2dt$$

(see also de la Vallée-Poussin summation method). The sequence $V_n(f;x)$ converges uniformly to $f(x)$ for functions $f$ which are continuous and $2\pi$-periodic on $(-\infty,\infty)$ [1]. If

$$\left(\int\limits_{-\pi}^xf(t)dt\right)_x'=f(x)$$

at a point $x$, then $V_n(f;x)\to f(x)$ as $n\to\infty$. The following equality is valid [2]:

$$V_n(f;x)-f(x)=\frac{f''(x)}{n}+o\left(\frac1n\right).$$

References

[1]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[2]	I.P. Natanson, "Constructive function theory" , 1 , F. Ungar (1964) (Translated from Russian)

Comments

The notation $(2m)!!$ stands for $2m\cdot(2m-2)\cdots2$ ($m$ terms), and $(2m-1)!!=(2m-1)(2m-3)\cdots3\cdot1$ (also $m$ terms). Thus,

$$\frac{(2n)!!}{(2n-1)!!}=\frac{2^{2n}(n!)^2}{(2n)!}.$$