# Riesz interpolation formula

From Encyclopedia of Mathematics

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 is a trigonometric polynomial of degree with real coefficients, then for any real the following equality holds:

where , .

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

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