Formal product of trigonometric series

$$ \sum _ {n = - \infty } ^ \infty c _ {n} e ^ {inx} \ \ \textrm{ and } \ \ \sum _ {n = - \infty } ^ \infty \gamma _ {n} e ^ {inx} $$

The series

$$ \sum _ {n = - \infty } ^ \infty K _ {n} e ^ {inx} , $$

where

$$ K _ {n} = \ \sum _ {m = - \infty } ^ \infty c _ {m} \gamma _ {n - m } . $$

If $ c _ {n} \rightarrow 0 $ as $ | n | \rightarrow \infty $, $ \sum _ {n = - \infty } ^ \infty | n \gamma _ {n} | < \infty $, and if

$$ \sum _ {n = - \infty } ^ \infty \gamma _ {n} e ^ {inx} $$

has sum $ \lambda ( x) $, then the series

$$ \sum _ {n = - \infty } ^ \infty ( K _ {n} - \lambda ( x) c _ {n} ) e ^ {inx} $$

has sum zero uniformly on $ [- \pi , \pi ] $. The condition

$$ \sum _ {n = - \infty } ^ \infty | n \gamma _ {n} | < \infty $$

is satisfied if, for example,

$$ \sum _ {n = - \infty } ^ \infty \gamma _ {n} e ^ {inx} $$

is the Fourier series of a three-times differentiable function $ \lambda ( x) $.

References

[1]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[2]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)