The k degree V-system is a class of orthogonal piecewise polynomial functions which is also named as high order Haar functions. V-system is defined on the uniform partition of interval ［0

1］and obtained by multiscale squeezing and shifting operations on the so-called generators. The V-system to the case of non-uniform partition is generalized

and the corresponding result is named as high order non-uniform Haar functions. For any given partition on the interval ［0

1］

a set of truncated monomials was firstly defined. It is proved that the non-uniform Haar functions can be obtained through the Gram-Schmidt orthogonalization process. The orthogonality

reproducibility and convergence of the proposed functions are proved

and a specific constructive example is also given.