Fourier-Stieltjes series

A series

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

where for $ n = 0, 1 \dots $

$$ a _ {n} = \ { \frac{1} \pi } \int\limits _ { 0 } ^ { {2 } \pi } \cos nx dF ( x),\ \ b _ {n} = \ { \frac{1} \pi } \int\limits _ { 0 } ^ { {2 } \pi } \sin nx dF ( x) $$

(the integrals are taken in the sense of Stieltjes). Here $ F $ is a function of bounded variation on $ [ 0, 2 \pi ] $. Alternatively one could write

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

If $ F $ is absolutely continuous on $ [ 0, 2 \pi ] $, then (*) is the Fourier series of the function $ F ^ { \prime } $. In complex form the series (*) is

$$ dF ( x) \sim \ \sum _ {n = - \infty } ^ { {+ } \infty } c _ {n} e ^ {inx} , $$

where

$$ c _ {n} = \ { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } e ^ {-} inx dF ( x). $$

Moreover,

$$ F ( x) - c _ {0} x \sim \ C _ {0} + \sum _ { \begin{array}{c} n = - \infty \\ n \neq 0 \end{array} } ^ \infty \frac{c _ {n} }{in } e ^ {inx} , $$

and $ \{ c _ {n} \} $ will be bounded. If $ c _ {n} \rightarrow 0 $, then $ F $ is continuous on $ [ 0, 2 \pi ] $. There is a continuous function $ F $ for which $ c _ {n} $ does not tend to $ 0 $ as $ n \rightarrow + \infty $. The series (*) is summable to $ F ^ { \prime } ( x) $ by the Cesàro method $ ( C, r) $, $ r > 0 $, almost-everywhere on $ [ 0, 2 \pi ] $.

References

[1]	A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)