This paper deals with the nonparametric estimation of the diffusion coefficient of a diffusion process in the case when it depends only on time. the estimator is based on the regularization of the quadratic mean through multiscale analysis. We then define estimators of the second derivative of the diffusion coefficient, in order to obtain optimal bandwith and to perform a test of linearity. The results we obtain are quite similar to those of the estimation of the density function of an i.i.d. sequence