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 [1], [2] 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.

References

[1]	Ch.J. de la Vallée-Poussin, "Sur la meilleure approximation des fonctions d'une variable réelle par des expressions d'ordre donné" C.R. Acad. Sci. Paris Sér. I. Math. , 166 (1918) pp. 799–802
[2]	Ch.J. de la Vallée-Poussin, "Leçons sur l'approximation des fonctions d'une variable réelle" , Gauthier-Villars (1919)
[3]	I.P. Natanson, "Constructive function theory" , 1 , F. Ungar (1964) (Translated from Russian)
[4]	P.P. Korovkin, "Linear operators and approximation theory" , Hindushtan Publ. Comp. (1960) (Translated from Russian)
[5]	S.M. Nikol'skii, "Sur certaines méthodes d'approximation au moyen de sommes trigonométriques" Izv. Akad. Nauk SSSR Ser. Mat. , 4 : 6 (1940) pp. 509–520
[6]	S.B. Stechkin, "On de la Vallée-Poussin sums" Dokl. Akad. Nauk SSSR , 80 : 4 (1951) pp. 545–520 (In Russian)
[7]	A.D. Shcherbina, "On a summation method of series, conjugate to Fourier series" Mat. Sb. , 27 (69) : 2 (1950) pp. 157–170 (In Russian)
[8]	A.F. Timan, "Approximation properties of linear methods of summation of Fourier series" Izv. Akad. Nauk SSSR Ser. Mat. , 17 (1953) pp. 99–134 (In Russian)
[9]	A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)
[10]	A.V. Efimov, "On approximation of periodic functions by de la Vallée-Poussin sums" Izv. Akad. Nauk SSSR Ser. Mat. , 23 : 5 (1959) pp. 737–770 (In Russian)
[11]	A.V. Efimov, "On approximation of periodic functions by de la Vallée-Poussin sums" Izv. Akad. Nauk SSSR Ser. Mat. , 24 : 3 (1960) pp. 431–468 (In Russian)
[12]	S.A. Telyakovskii, "Approximation of differentiable functions by de la Vallée-Poussin sums" Dokl. Akad. Nauk SSSR , 121 : 3 (1958) pp. 426–429 (In Russian)
[13]	S.A. Telyakovskii, "Approximation to functions differentiable in Weyl's sense by de la Vallée-Poussin sums" Soviet Math. Dokl. , 1 : 2 (1960) pp. 240–243 Dokl. Akad. Nauk SSSR , 131 : 2 (1960) pp. 259–262

Comments

The de la Vallée-Poussin kernels are also given by the following formula, which in a way most clearly reveals their structure:

$$ K _ {n,p} ( t) = \ { \frac{1}{p + 1 } } \sum _ {k = n - p } ^ { n } D _ {k} ( t) = $$

$$ = \ { \frac{1}{2} } + \sum _ {k = 1 } ^ { {n- } p } \cos kt + \sum _ {k = 1 } ^ { p } \left ( 1-{ \frac{k}{p + 1 } } \right ) \ \cos ( n- p + k) t. $$

Here the $ D _ {k} $( $ k \geq 0 $) are the Dirichlet kernels (cf. Dirichlet kernel).