SKEDSOFT

Maths For Engineers - 2

Preposition-4:

If f satiesfies a Lapschitz conditions, then SN f → f uniformly as N →  \infty .

Proof: We have

The last equality follows from the periodicity of the integrated functions. Hence,

and it extends to a continuous functions at t = 0 by giving it the value 1 there. The Lapschitz condition on f says that;

                         

Hence the function lh (t)| is continuous and bounded on [π, 0) and on (0, π] . Thus |h2| is integrable on [- π, π], and we have

                      

The function |k2| is integrable in [0, 2π]and we have