# Riesz interpolation formula

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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)