In this paper,the author give the spectra characterizations of nonsigular feedback polynomials and local permutation polynomials over finite fields and residue class rings,and simplify existing results over prime fields,and provide a new proof method for the spectra characterization of correlation immune functions over finite fields and residue class rings.

In this paper we investigate the asymptotic properties of local polynomial estimators in semiparametric regression model.

In this paper, we derive estimators of parametric and nonparametric components in semiparametric regression model by using local polynomial fitting and LS method.

The local polynomial estimator of the function coefficient vectorβ(.

Permutation polynomials play an important role in communication field.

Dickson polynomials are of special source of permutation polynomials over finite fields.

With some results of polynomial theory in finite field, a criterion theorem for a permutation polynomial to be an orthormorphic permutation polynomial is presented.

By analytic viewpoint, the author gives a proof of the result that the inverse of any permutation polynomial over Z/pnZ can be represented by a polynomial and so all permutation polynomials over Z/pnZ form a group according to the operation of composite of maps and reduction mod J.

The author studies a class of singular permutation polynomials in n(n≥3) indeterminates modulo p, gives a sufficient condition for them to be permutation polynomials modulo p l(l>1).

This dissertation is based onρ~ -mixing samples which is a kind of broad dependent mixing samples, it studies the local polynomial estimation of unknown function in fixed design nonparametric regression model and unknown parametric and unknown function in semiparametric regression model.