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)