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

$$ \tag{* } \frac{a _ {0} }{2} + \sum _ { n= } 1 ^ \infty ( a _ {n} \cos nx + b _ {n} \sin nx ) $$

with bounded sequences $ \{ a _ {n} \} , \{ b _ {n} \} $ be given. The Riemann function for this series is the function $ F $ obtained by twice term-by-term integration of the series:

$$ F( x) = \frac{a _ {0} x ^ {2} }{4} - \sum _ { n= } 1 ^ \infty \frac{1}{n ^ {2} } ( a _ {n} \cos nx + b _ {n} \sin nx ) + Cx + D, $$

$$ C, D = \textrm{ const } . $$

Riemann's first theorem: Let the series (*) converge at a point $ x _ {0} $ to a number $ S $. The Schwarzian derivative (cf. Riemann derivative) $ D _ {2} F( x _ {0} ) $ then equals $ S $. Riemann's second theorem: Let $ a _ {n} , b _ {n} \rightarrow 0 $ as $ n \rightarrow \infty $. Then at any point $ x $,

$$ \lim\limits _ {n \rightarrow \infty } \frac{F( x+ h) + F( x- h) - 2F( x) }{h} = 0; $$

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

If the series (*) converges on $ [ 0, 2 \pi ] $ to $ f( x) $ and if $ f \in L[ 0, 2 \pi ] $, then $ D _ {2} F( x) = f( x) $ for each $ x \in [ 0, 2 \pi ] $ and

$$ F( x) = \int\limits _ { 0 } ^ { x } dt \int\limits _ { 0 } ^ { t } f( u) du + Cx + D. $$

Let $ a _ {n} , b _ {n} $ be real numbers tending to $ 0 $ as $ n \rightarrow \infty $, let

$$ \underline{S} ( x) = \ \lim\limits _ {\overline{ {n \rightarrow \infty }}\; } S _ {n} ( x) \ \textrm{ and } \ \overline{S}\; ( x) = \overline{\lim\limits}\; _ {n \rightarrow \infty } S _ {n} ( x) $$

be finite at a point $ x $, and let

$$ S( x) = \frac{1}{2} ( \underline{S} ( x) + \overline{S}\; ( x)),\ \ \delta ( x) = \frac{1}{2} ( \overline{S}\; ( x) - \underline{S} ( x)). $$

Then the upper and lower Schwarzian derivatives $ \overline{D}\; _ {2} F( x) $ and $ \underline{D} _ {2} F( x) $ belong to $ [ S - \mu \delta , S + \mu \delta ] $, where $ \mu $ 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.