Transcription of The Weierstrass Function
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The Weierstrass Function
The Weierstrass FunctionBrent NelsonWe let sin,cos :R Rbe defined in the usual geometric way, extended to all ofR. We will assume thefollowing facts about these functions:(a) sin and cos are continuous onR;(b)|sin(x)|,|cos(x)| 1 for allx R;(c) sin(x)x 1 for allx R\{0};(d) cos(x) cos(y) = 2 sin(x+y2)sin(x y2)for allx,y R;(e) cos(x+y) = cos(x) cos(y) sin(x) sin(y) for allx,y can be derived by considering, for example, the power series representations:sin(x) = n=0( 1)n(2n+ 1)!x2n+1andcos(x) = n=0( 1)n(2n)! main goal of these notes is to prove the following theorem:Theorem(Karl Weierstrass , 1872).Leta (0,1)and letbbe an odd integer such thatab >1 +3 2. Then theseriesf(x) = n=0ancos(bn x)converges uniformly onRand defines a continuous but nowhere differentiable Function appearing in the above theorem is called the Weierstrass Function . Before we prove the theorem,we require the following lemma:Lemma(The Weierstrass M-test).Let(E,d)be a metric space, and for eachn Nletfn:E Rbe afunction.
The Weierstrass Function Math 104 Proof of Theorem. Since jancos(bnˇx)j an for all x2R and P 1 n=0 a n converges, the series converges uni-formly by the Weierstrass M-test. Moreover, since the partial sums are continuous (as nite sums of continuous functions), their uniform limit fis …
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