Published online by Cambridge University Press: 05 August 2012
17.1 Let Kn be the Fejér kernel discussed in Chapter 2.
(i) Show that, if κ(n) → ∞ as n → ∞, then ∫|s|≥κ(n)/nKn(s)ds → 0 as n → ∞. (We shall be interested in slowly diverging κ(n) such as κ(n) = log(n + 2).)
(ii) Show that if h is the saw tooth function of Chapter 17 then
Conclude that nothing resembling the Gibbs phenomenon can occur for σn(h, t). Extend this observation to cover well-behaved functions with only a finite number of discontinuities. (If you actually graph σn(h, t) you may find that, in spite of what we have proved, σn(h,) is not a very good copy of h for small n, being a bit ‘round shouldered’ near π. Small degree trigonometric polynomials just do not look very much like discontinuous functions.)
17.2 As C. H. Su has pointed out to me, the result of Theorem 17.1 can be interpreted by using a technique from boundary layer theory (and elsewhere) and ‘blowing up’ the independent variable x near π. More precisely, instead of considering Sn(h, x) we introduce a new variable y = n(π − x).
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