Riemann function

The Riemann function in the theory of trigonometric series is a function introduced by B. Riemann (1851) (see [1]) for studying the problem of the representation of a function by a trigonometric series. Let a series

(*)

with bounded sequences be given. The Riemann function for this series is the function obtained by twice term-by-term integration of the series:

Riemann's first theorem: Let the series (*) converge at a point to a number . The Schwarzian derivative (cf. Riemann derivative) then equals . Riemann's second theorem: Let as . Then at any point ,

moreover, the convergence is uniform on any interval, that is, is a uniformly smooth function.

If the series (*) converges on to and if , then for each and

Let be real numbers tending to as , let

be finite at a point , and let

Then the upper and lower Schwarzian derivatives and belong to , where is an absolute constant. (The du Bois-Reymond lemma.)

References

[1]	B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" , Gesammelte Math. Abhandlungen , Dover, reprint (1957) pp. 227–264
[2]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)

Comments

See also Riemann summation method.

For Riemann's function in the theory of differential equations see Riemann method.