Weighted Weierstrass' theorem with first derivatives

Abstract

We characterize the set of functions which can be approximated by continuous functions with the norm f L ∞ (w) for every weight w. This fact allows to determine the closure of the space of polynomials in L ∞ (w) for every weight w with compact support. We characterize as well the set of functions which can be approximated by smooth functions with the normfor a wide range of (even non-bounded) weights w j 's. We allow a great deal of independence among the weights w j 's.

OnW1,p-convergence of Fourier–Sobolev expansions

“…Following the ideas of [6] (see also, [7][8][9][10], the authors in [11,12] introduced general classes of Sobolev spaces appearing in the context of orthogonal polynomials on the real line. We will use the approach given in these papers to establish the Kufner-Opic type property as follows.…”

On Hilbert extensions of Weierstrass′ theorem with weights

“…Unfortunately, the last equality in (2.4) does not allow to obtain information on local behavior of the functions f ∈ L ∞ (I, w) which can be approximated. Furthermore, if f ∈ L ∞ (I, w), then in general f w is not continuous function, since its continuity also depends of the singularities of weight w (see [6], [16], [17], [28], [24], [27]).…”

“…The next definition presents the classification of the singularities of a scalar weight w done in [27] to show the results about density of continuous functions in the space L ∞ (supp(w), w). We say that a singularity a of w is of type 1 if ess lim x→a w(x) = 0.…”

“…Considering weighted norms has been proved to be interesting mainly because of two reasons: first, they allow to wider the set of approximable functions (since the functions in L ∞ (I, w) can have singularities where the weight tends to zero); and, second, it is possible to find functions which approximate a given function f , whose qualitative behavior is similar to qualitative behavior of f at those points where the weight tends to infinity. The reader may find in [24], [26], [27] and [28] recent and detailed studies about such subject.…”

“…Another special kind of approximation problems arises when we consider simultaneous approximation which includes derivatives of certain functions; this is the case of versions of Weierstrass' theorem in weighted Sobolev spaces. With respect to recent developments about this subject we refer to [26] and [27]: In the first paper, under enough general conditions concerning the vector weights (the so called type 1) defined in a compact interval I, the authors characterize to the closure in the weighted Sobolev space W (k,∞) (I, w) of the spaces of polynomials, k-differentiable functions, and infinite differentiable functions, respectively. In the second paper, the reader may find shaper results for the case k = 1.…”