Namespaces
Variants
Actions

Difference between revisions of "Riesz interpolation formula"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A formula giving an expression for the derivative of a [[Trigonometric polynomial|trigonometric polynomial]] at some point by the values of the polynomial itself at a finite number of points. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082260/r0822601.png" /> is a trigonometric polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082260/r0822602.png" /> with real coefficients, then for any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082260/r0822603.png" /> the following equality holds:
+
<!--
 +
r0822601.png
 +
$#A+1 = 10 n = 0
 +
$#C+1 = 10 : ~/encyclopedia/old_files/data/R082/R.0802260 Riesz interpolation formula
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/r/r082/r082260/r0822604.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082260/r0822605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082260/r0822606.png" />.
+
A formula giving an expression for the derivative of a [[Trigonometric polynomial|trigonometric polynomial]] at some point by the values of the polynomial itself at a finite number of points. If  $  T _ {n} ( x) $
 +
is a trigonometric polynomial of degree  $  n $
 +
with real coefficients, then for any real  $  x $
 +
the following equality holds:
  
Riesz' interpolation formula can be generalized to entire functions of exponential type: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082260/r0822607.png" /> is an entire function that is bounded on the real axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082260/r0822608.png" /> and of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082260/r0822609.png" />, then
+
$$
 +
T _ {n}  ^  \prime  ( x)  =
 +
\frac{1}{4n}
 +
\sum _ { k= } 1 ^ { 2n }  (- 1)  ^ {k+} 1
 +
\frac{1}{\sin  ^ {2}  x _ {k}  ^ {(} n) /2 }
 +
T _ {n} ( x + x _ {k}  ^ {(} n) ),
 +
$$
  
<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/r/r082/r082260/r08226010.png" /></td> </tr></table>
+
where  $  x _ {k}  ^ {(} n) = ( 2k- 1) \pi /2n $,
 +
$  k = 1 \dots 2n $.
 +
 
 +
Riesz' interpolation formula can be generalized to entire functions of exponential type: If  $  f $
 +
is an entire function that is bounded on the real axis  $  \mathbf R $
 +
and of order  $  \sigma $,
 +
then
 +
 
 +
$$
 +
f ^ { \prime } ( x)  =
 +
\frac \sigma {\pi  ^ {2} }
 +
\sum _ {k = - \infty } ^  \infty 
 +
\frac{(- 1)  ^ {k} }{\left ( k+
 +
\frac{1}{2}
 +
\right )  ^ {2} }
 +
f \left ( x + 2k+
 +
\frac{1}{2 \sigma }
 +
\pi \right ) ,
 +
\  x \in \mathbf R .
 +
$$
  
 
Moreover, the series at right-hand side of the equality converges uniformly on the entire real axis.
 
Moreover, the series at right-hand side of the equality converges uniformly on the entire real axis.
Line 15: Line 51:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Riesz,  "Formule d'interpolation pour la dérivée d'une polynôme trigonométrique"  ''C.R. Acad. Sci. Paris'' , '''158'''  (1914)  pp. 1152–1154</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Bernshtein,  "Extremal properties of polynomials and best approximation of continuous functions of a real variable" , '''1''' , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Riesz,  "Formule d'interpolation pour la dérivée d'une polynôme trigonométrique"  ''C.R. Acad. Sci. Paris'' , '''158'''  (1914)  pp. 1152–1154</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Bernshtein,  "Extremal properties of polynomials and best approximation of continuous functions of a real variable" , '''1''' , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Riesz,  "Eine trigonometrische Interpolationsformel und einige Ungleichungen für Polynome"  ''Jahresber. Deutsch. Math.-Ver.'' , '''23'''  (1914)  pp. 354–368</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.F. Timan,  "Theory of approximation of functions of a real variable" , Pergamon  (1963)  pp. Chapt. 4  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)  pp. Chapt. X</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Riesz,  "Eine trigonometrische Interpolationsformel und einige Ungleichungen für Polynome"  ''Jahresber. Deutsch. Math.-Ver.'' , '''23'''  (1914)  pp. 354–368</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.F. Timan,  "Theory of approximation of functions of a real variable" , Pergamon  (1963)  pp. Chapt. 4  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)  pp. Chapt. X</TD></TR></table>

Revision as of 08:11, 6 June 2020


A formula giving an expression for the derivative of a trigonometric polynomial at some point by the values of the polynomial itself at a finite number of points. If $ T _ {n} ( x) $ is a trigonometric polynomial of degree $ n $ with real coefficients, then for any real $ x $ the following equality holds:

$$ T _ {n} ^ \prime ( x) = \frac{1}{4n} \sum _ { k= } 1 ^ { 2n } (- 1) ^ {k+} 1 \frac{1}{\sin ^ {2} x _ {k} ^ {(} n) /2 } T _ {n} ( x + x _ {k} ^ {(} n) ), $$

where $ x _ {k} ^ {(} n) = ( 2k- 1) \pi /2n $, $ k = 1 \dots 2n $.

Riesz' interpolation formula can be generalized to entire functions of exponential type: If $ f $ is an entire function that is bounded on the real axis $ \mathbf R $ and of order $ \sigma $, then

$$ f ^ { \prime } ( x) = \frac \sigma {\pi ^ {2} } \sum _ {k = - \infty } ^ \infty \frac{(- 1) ^ {k} }{\left ( k+ \frac{1}{2} \right ) ^ {2} } f \left ( x + 2k+ \frac{1}{2 \sigma } \pi \right ) , \ x \in \mathbf R . $$

Moreover, the series at right-hand side of the equality converges uniformly on the entire real axis.

This result was established by M. Riesz [1].

References

[1] M. Riesz, "Formule d'interpolation pour la dérivée d'une polynôme trigonométrique" C.R. Acad. Sci. Paris , 158 (1914) pp. 1152–1154
[2] S.N. Bernshtein, "Extremal properties of polynomials and best approximation of continuous functions of a real variable" , 1 , Moscow-Leningrad (1937) (In Russian)
[3] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)

Comments

References

[a1] M. Riesz, "Eine trigonometrische Interpolationsformel und einige Ungleichungen für Polynome" Jahresber. Deutsch. Math.-Ver. , 23 (1914) pp. 354–368
[a2] A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) pp. Chapt. 4 (Translated from Russian)
[a3] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988) pp. Chapt. X
How to Cite This Entry:
Riesz interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_interpolation_formula&oldid=18359
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article