5. Let I : R ---- R be periodic on R, integrable on [-11,11], and ak (f)...
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5. Let I : R ---- R be periodic on R, integrable on [-11",11"], and ak
(f) 2: 0 for k = 0,1, ....
(a) Prove that (Skf)(O) 2: (Sjf)(O) for all k 2: j 2: o.
(b) Prove that SN 1(0) :S 2U2N 1(0) for N E N.
(c) Prove that 2::~1 lak(f)1 < 00.
(d) Suppose that I is also even. Prove that I must be continuous and SI converges uniformly and absolutely on R.
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