Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point

For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H-0 : f = 0, (the signal f is zero) against the nonparametric alternative H-1 : f is an element of Lambda(epsilon) where Lambda(epsilon) is a set of functions on R-1 of the form Delta(epsilon) = {f : f is an element of F, phi(f) greater than or equal to C psi(epsilon)}. Here F is a Holder or Sobolev class of functions, phi(f) is either the sup-norm of f or the value of f at a fixed point, C > 0 is a constant, psi(epsilon) is the minimax rate of testing and epsilon --> 0 is the asymptotic parameter of the model. We find exact separation constants C* > 0 such that a test with the given summarized asymptotic errors of first and second type is possible for C > C* and is not possible for C < C*. We propose asymptotically minimax test statistics.