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) |
Formal product of trigonometric series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Formal_product_of_trigonometric_series&oldid=46956