Penalized Least Squares Fitting

* Supported by the Army Research Office under grant DAAD-19-02-10059.Bounds on the error of certain penalized least squares data fitting methods are derived. In addition to general results in a fairly abstract setting, more detailed results are included for several particularly interesting special cases, including splines in both one and several variables

Approximation by incomplete polynomials

AbstractFor any Θ with 0 < Θ < 1, it is known that the set of all incomplete polynomials of form Pn(x)=∑k=νnakxk, μ⩾φ·n is not dense in Co[a, 1]: = {fϵ C[a, 1]:f(a) = 0} if a < Θ2. In this paper, we prove that the set (1) of incomplete polynomials is dense in Co[a, 1]if a ⩾ Θ2 and even has the Jackson property on [a, 1]if a > Θ2