Riesz interpolation formula

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, \ldots, 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)

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