6. Let I : R ---- R be continuous and periodic. The modulus 01 continuity of I is defined by w(f,8) = sup I/(t + h) - l(t)l·

tE[0,271"]

Ihl9

(a) Show that ak

(f) = 2~ I: (/(u) - I (u + ~)) coskudu for kEN.

(b) Prove that for kEN.

(c) Use part

(b) to give a different proof the Riemann-Lebesgue Lemma in the special case when I is periodic and continuous