Projection Filter Regularization Of Ill-Conditioned Problem

The usual regularization theory for posed ill-problem may be called the regularization of Weierstrass type in the context of approximation theory. On the contrary, this paper proposes the regularization of Chebyshev type, named the "projection filter regulariza-tion". In the absence of noise, the projection filter provides the best approximation Pf to the real solution f, where P is the orthogonal projection operator onto a subspace. In the presence of noise, it provides the best approximation to the individual Pf in the average sense with respect to noise. A general form of projection filter is obtained. Properties of projection filter are discussed in detail.