De la Vallée-Poussin sum

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

Comments

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).