Abstract. The exact Fourier coefficients c,(PM/) are proportional to the discrete Fourier coefficients dj-"\f) if Pn is a translation invariant operator which depends only on the values of/on an equidistant mesh of width 2it/n. The proportionality factors which depend only on P" but not on / are called attenuation factors and have been calculated for several operators P" of spline type. Here we analyze first the interpolation problem which is produced by the functions

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Summary. Usually the error R n (f) of a quadrature formula is estimated with the aid or the Lz-norm of the Peano kernel. It is shown that this term may be estimated rather sharp using the norm I]Qnl[ of the quadrature rule. Then it follows that formulas with non-negative weights are favourable also in the sense of minimizing the Lx-norm of the kernel. A remainder term of the type R, (]) = c/(n+l) (~) is possible iff the kernel is definite. In the ease of an interpolatory formula this definiteness is usually shown by an application of the so-called "V-method". We determine the optimal formulas in the sense of this method. Then we analyse the influence of the structure of the mesh on the norm of a formula. We find that on an equidistant mesh with m nodes there exists a rule with a small norm if the order is not greater than cVm.

Summary.By the argument principle the number of zeros inside of the unit circle of a real polynomial p,, p,(x) = ~=o a~ x~, a~lR, a,#O, can be estimated by the variation of the argument of p,(exp(it)) if t varies from 0 to To. This variation has its maximal value ~n iff the n distinct zeros of ~0, :--~"=oa~T~ are separated by the n--1 n -1 distinct zeros of ~p,_ 1:= ~v=oa,,+~U~. Now from Sturm sequence techniques in connection with special properties of the Chebyshev polynomials there results a very simple stability test.