I wonder whether, if $f:[a,b]\to\mathbb{C}$ is an absolutely continuous function, multiplying it by $\cos\frac{2\pi nx}{b-a}$ or $\sin\frac{2\pi nx}{b-a}$ results in another absolutely continuous function continuous, i.e.: are the functions defined by $g(x)=f(x)\cos\frac{2\pi nx}{b-a}$ or $h(x)=f(x)\sin\frac{2\pi nx}{b-a}$ absolutely continuous? If they are, how can it be seen?
That is what I understand from the fact that Kolmogorov and Fomin seem here to use that fact in integrating the Fourier coefficient of the periodic absolutely continuouws function $f$ by parts, where I think that the equality, valid for an absolutely continuous function, $\int_{[a,b]}f' d\mu=f|_a^b$ is used.
Thank you very much for any answer!!!
The product of two absolutely continuous functions on a closed bounded interval is absolutely continuous.
Hint: $$|f(x) g(x) - f(y) g(y)| \le |f(x)| |g(x) - g(y)| + |g(y)| |f(x) - f(y)|$$