Some representation formulae for entire functions of exponential type

We obtain some explicit formulae for series of the typewhere f is an entire function of exponential type τ, bounded on the real exis (and satisfying in the first case). These series are expressed in terms of the derivatives of f and Bernoulli numbers. We examine the case where f is a trigonometric polynomial which lead us, in particular, to a new representation of the associated Fejér mean.

On Approximation by Fejér Means to Periodic Functions Satisfying a Lipschitz Condition

S. M. Nikolski [4, Theorem 1; cf. 3, esp. pp. 144 and 148] considered the remainder term in the approximation by the n-th Fejér mean, σn(x), to a function, f(x), of period 2π satisfying a Lipschitz condition of order α, 0<α≤1. In this connection, he introduced the quantity1where the maximum is taken over all x and the supremum is taken over all functions of period 2π, bounded by 1 (a notational convenience only) and satisfying a Laps chitz condition of order α.