Suppose that fN Iuml , Iuml R are integrable and that fN f uniformly on Iuml , Iuml as N a) Prove that ak(fN) ak(f) and bk(fN) bk(f), as N , uniformly in k b) Show that part a) holds under the weaker hypothesis lim If(x) fy(x) dx 0

Question: Suppose that fN: [-Ï, Ï] R are integrable and that fN f uniformly on [-Ï, Ï] as N . a) Prove that ak(fN) ak(f) and

Suppose that fN: [-Ï€, Ï€] †’ R are integrable and that fN †’ f uniformly on [-Ï€, Ï€] as N †’ ˆž.
a) Prove that ak(fN) ak(f) and bk(fN) †’ bk(f), as N †’ ˆž, uniformly in k.
b) Show that part a) holds under the weaker hypothesis

lim If(x)-fy(x)|dx = 0.

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