Fourier Analysis and Introduction Chapter 3 15 c

Let $$f$$ be $$2 pi$$-periodic and integrable Rienmann on $$[-pi,pi]$$.
$$hat f (n)$$ is the cofficient fourier.

a) Show that
$$hat f (n) = – frac {1} {2 pi} int _ {- pi} ^ { pi} f (x + pi / n) e ^ {- inx} dx$$
Therefore
$$hat f (n) = frac {1} {4 pi} int _ {- pi} ^ { pi}[f(x)-f(x+pi/n)]e ^ {- inx} dx.$$

(b) Suppose now that $$f$$ satisfies a condition of order of the holder $$alpha$$to know
$$| f (x + h) -f (x) | the C | h | ^ {| alpha |}$$
for some people $$0 < alpha the 1$$, some $$C> 0$$, and all $$x, h$$. Use part a) to show
$$hat f (n) = O ( frac {1} {| n | ^ { alpha}}$$

(c) Prove that the above result can not be improved by showing that the function
$$f (x) = sum_ {k = 0} ^ { infty} 2 ^ {- k alpha} e ^ {i2 ^ {k} x},$$
or $$0 < alpha <1,$$ satisfied
$$| f (x + h) -f (x) | the C | h | ^ { alpha},$$
and $$hat f (N) = 1 / N ^ { alpha}$$ does not matter when $$N = 2 ^ {k}.$$

[Tip:To(c)décomposezlasommecommesuit[Hint:For(c)breakupthesumasfollows[Astuce:Pour(c)décomposezlasommecommesuit[Hint:For(c)breakupthesumasfollows$$f (x + h) -f (x) = sum_ {2 ^ {k} le1 / | h |} + sum_ {2 ^ {k}> 1 / | h |}.$$To estimate the first sum, use the fact that $$| 1-e ^ {i theta} | <| theta |$$ does not matter when $$theta$$ is small. To estimate the second sum, use the obvious inequality $$| e ^ {ix} -e ^ {iy} | 2$$]

I proved a) and b), but when I got to c), I was stuck with proving.
There is what I thought:
$$| f (x + h) -f (x) | = | sum_ {k = 0} ^ { infty} 2 ^ {- k alpha} e ^ {i2 ^ {k} (x + h)} – sum_ {k = 0} ^ { infty} 2 ^ { – k alpha} e ^ {i2 ^ {k} x} | = | sum_ {k = 0} ^ { infty} 2 ^ {-k alpha} e ^ {i2 ^ {k} x} (e ^ {i2 ^ {k} h} -1) |$$

divide that into $$f (x + h) -f (x) = sum_ {2 ^ {k} le1 / | h |} + sum_ {2 ^ {k}> 1 / | h |$$ and use the index:
$$| sum_ {k = 0} ^ { infty} 2 ^ {- k alpha} e ^ {i2 ^ {k} x} (e ^ {i2 ^ {k} h} -1) | le | sum_ {2 ^ {k} le1 / | h |} 2 ^ {- k alpha} e ^ {i2 ^ {k} x} 2 ^ {k} h | + | sum_ {2 ^ {k}> 1 / | h |} 2 ^ {- k alpha} e ^ {i2 ^ {k} x} 2 |$$

$$le sum_ {2 ^ {k} le1 / | h |} 2 ^ {- k alpha} + 2 sum_ {2 ^ {k}> 1 / | h |} 2 ^ {- k alpha}$$
$$le sum_ {2 ^ {k} le1 / | h |} | h | ^ { alpha} +2 sum_ {2 ^ {k}> 1 / | h |} | h | ^ { alpha}$$
But I do not know what the next step is and how does it prove that the result in b) can not be improved.