If g is piecewise continuous on the interval [a,b], then
∫ g(t) sin(ωt) dt -> 0 as ω->∞
[this is quoted directly from my textbook]
(i) Now assuming this result, is it possible to prove from this result that
∫ g(t) cos(ωt) dt -> 0 as ω->∞ ??
I think it also works for cosine becuase cosine is just a horizontal shift of sine, but does this imply that the second result(with cos) is an immediate consequence of the first(with sin)? How can we prove this?
(ii) Also, if we have
, is the lemma above still true?
Any help is appreciated!
[note: also under discussion in s.o.s. math cyberboard]