Umkehrsätze beim Trapezvedahren

Characterization of the speed of convergence of the trapezoidal rule

“…However, K does not simply consist of linear combinations of such functions. Indeed, as it was observed by Ching (see Reference [1] in [2]) the continuous (but not absolutely continuous) function (4) f(x)= ~ #(n) eiZ~,x ' n n=l where #(.) denotes the M6bius function used in Number Theory, also belongs to K although it has a nontrivial even part.…”

“…The example (4) shows that the assumption of absolute continuity in Theorems 1-4 cannot be replaced by ordinary continuity. However there are other conditions stronger than continuity which may be used alternatively.…”

“…Loxton and Sanders [8] showed that a function f with an absolutely convergent Fourier series belongs to K if and only if it is of the form f=l+u where l is linear and u is odd on [0, 1]. Earlier Brass [4] had obtained the following result which is pertinent in this connection. [0,1] such that (5) R …”

Comparison theorems for compound quadrature formulas

“…This is due to the fact that each one may invertibly represented by Fourier cosine coefficients (in terms of operators T~). This holds true at least on the (dense) set of (1-periodic) trigonometric polynomials but not necessarily for, e.g., all continuous functions (for representations and comparisons in terms of Fourier coefficients see [3,9,10]). …”

Converse theorems in the theory of approximate integration

“…There are only a few results of this kind in terms of the error of a quadrature formula. For instance, one such characterization is due to D. Gaier [10] [5].…”