View source for De la Vallée-Poussin sum

The expression

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302704.png" /> are the partial sums of the [[Fourier series|Fourier series]] of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302705.png" /> with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302706.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302707.png" />, the de la Vallée-Poussin sums become identical with the partial Fourier sums, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d0302708.png" />, they become identical with the Fejér sums (cf. [[Fejér sum|Fejér sum]]). Ch.J. de la Vallée-Poussin [[#References|[1]]], [[#References|[2]]] was the first to study the method of approximating periodic functions by polynomials of the form (*); he also established the inequality

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027011.png" /> is the best uniform approximation of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027012.png" /> using trigonometric polynomials of order not greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027016.png" /> is the integer part of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027017.png" />, the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027018.png" /> realize an approximation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027019.png" />. The polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027020.png" /> yield the best order approximations of continuous functions of period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027021.png" />, with an estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027022.png" /> for certain values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027024.png" />. The de la Vallée-Poussin sums have several properties which are of interest in the theory of summation of Fourier series. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027027.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027028.png" /> is a trigonometric polynomial of order not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027030.png" />. A de la Vallée-Poussin sum may be written as follows

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where the expressions

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,   "Leçons sur l'approximation des fonctions d'une variable réelle" , Gauthier-Villars  (1919)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.P. Natanson,   "Constructive function theory" , '''1''' , F. Ungar  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.P. Korovkin,   "Linear operators and approximation theory" , Hindushtan Publ. Comp.  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.B. Stechkin,   "On de la Vallée-Poussin sums"  ''Dokl. Akad. Nauk SSSR'' , '''80''' :  4  (1951)  pp. 545–520  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.D. Shcherbina,   "On a summation method of series, conjugate to Fourier series"  ''Mat. Sb.'' , '''27 (69)''' :  2  (1950)  pp. 157–170  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.F. Timan,   "Theory of approximation of functions of a real variable" , Pergamon  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  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</TD></TR></table>



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

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Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027037.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030270/d03027038.png" />) are the Dirichlet kernels (cf. [[Dirichlet kernel|Dirichlet kernel]]).