The expression
 | (*) |
where 
, 
are the partial sums of the Fourier series of a function 
with period 
. If 
, the de la Vallée-Poussin sums become identical with the partial Fourier sums, and if 
, 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
 |
 |
where 
is the best uniform approximation of the function 
using trigonometric polynomials of order not greater than 
. If 
, 
and 
is the integer part of the number 
, the polynomials 
realize an approximation of order 
. The polynomials 
yield the best order approximations of continuous functions of period 
, with an estimate 
for certain values of 
, 
. The de la Vallée-Poussin sums have several properties which are of interest in the theory of summation of Fourier series. For instance, if 
, 
, then 
, and if 
is a trigonometric polynomial of order not exceeding 
, then 
. A de la Vallée-Poussin sum may be written as follows
where the expressions
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 |
The de la Vallée-Poussin kernels are also given by the following formula, which in a way most clearly reveals their structure:
Here the 
(
) are the Dirichlet kernels (cf. Dirichlet kernel).
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=23250
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
