Fourier coefficients of continuous functions with respect to localized Haar system

We construct a nontrivial example of a continuous function $f^*$ on $[0,1]^2$ which is orthogonal to tensor products of Haar functions supported on intervals of the same length. This example clarifies the possible behaviour of Fourier coefficients of continuous functions with respect to a localized Haar system. The function $f^*$ has fractal structure. We give lower bounds on its smoothness.