Nowhere differentiable function

Wei-Chi Yang

Radford University

www.radford.edu/~wyang

F:=proc(x,k) sum((1/2)^L*cos((2^L)*Pi*x),L=1..k) end;

F(x,30);

plot(F(x,40),x=-5..5,y=-1..1);

with(plots):

for k from 1 by 1 to 40 do

P||k:= plot (F(x,k), x=-5..5,y=-1..1):

od:

plots[display]([seq(P||k, k=1..40)], insequence=true);

G:=proc(a,b,x,k) sum(a^L*cos((b^L)*Pi*x),L=1..k) end;

Remark: We observe from K\303\266rner, T.W., Fourier Analysis, page 38-41, Cambridge University Press, 1988 that the key to construct a nowhere differentiable function is to make a, b such that ab>1+(3/2)\317\200=5.712388981).

plot(G(1/2,12,x,30),x=-5..5,y=-1..1);

plot(G(1/2,12,x,40),x=-5..5,y=-1..1);