# De la Vallée-Poussin sum

The expression

$$\tag{* } V _ {n, p } ( f, x) = \ \frac{1}{p + 1 } \sum _ {k = n - p } ^ { n } S _ {k} ( f, x),$$

$$p = 0 \dots n; \ n = 0, 1 \dots$$

where $S _ {k} ( f, x)$, $k = 0, 1 \dots$ are the partial sums of the Fourier series of a function $f$ with period $2 \pi$. If $p = 0$, the de la Vallée-Poussin sums become identical with the partial Fourier sums, and if $p = n$, they become identical with the Fejér sums (cf. Fejér sum). Ch.J. de la Vallée-Poussin ,  was the first to study the method of approximating periodic functions by polynomials of the form (*); he also established the inequality

$$| f ( x) - V _ {n, p } ( f, x) | \leq \ 2 \frac{n + 1 }{p + 1 } E _ {n - p } ( f ),$$

$$p = 0 \dots n,$$

where $E _ {m} ( f )$ is the best uniform approximation of the function $f \in C _ {2 \pi }$ using trigonometric polynomials of order not greater than $m$. If $p = [ cn]$, $0 < c < 1$ and $[ a]$ is the integer part of the number $a$, the polynomials $V _ {n,[ cn] } ( f, x)$ realize an approximation of order $O( E _ {[( 1- c) n] } ( f ))$. The polynomials $V _ {n,[ cn] } ( f, x)$ yield the best order approximations of continuous functions of period $2 \pi$, with an estimate $E _ {[ \theta n] } ( f ) = O( E _ {n} ( f ))$ for certain values of $\theta$, $0 \leq \theta < 1$. The de la Vallée-Poussin sums have several properties which are of interest in the theory of summation of Fourier series. For instance, if $p = [ cn]$, $0 < c < 1$, then $| V _ {n,p} ( f, x) | \leq K( c) \max | f( x) |$, and if $f$ is a trigonometric polynomial of order not exceeding $n - p$, then $V _ {n,p} ( f, x) = f( x)$. A de la Vallée-Poussin sum may be written as follows

$$V _ {n, p } ( f, x) =$$

$$= \ \frac{1}{( p + 1) \pi } \int\limits _ {- \pi } ^ \pi \left [ f ( x + t) \sin \frac{2n + 1 - p }{2} t \frac{\sin ( p + 1) t / 2 }{2 \sin ^ {2} {t / 2 } } \right ] dt,$$

where the expressions

$$K _ {n, p } ( t) = \ \frac{\sin (( 2n + 1 - p) t / 2 ) \sin (( p + 1) t / 2 ) }{2 ( p + 1) \sin ^ {2} {t / 2 } } ,$$

$$p = 0 \dots n; \ n = 0, 1 \dots$$

are said to be the de la Vallée-Poussin kernels.

How to Cite This Entry:
De la Vallée-Poussin sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_la_Vall%C3%A9e-Poussin_sum&oldid=46592
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article