Universal series induced by approximate identities and some relevant applications

Abstract

We prove the existence of series ∑anψn, whose coefficients (an) are in ∩p>1ℓp and whose terms (ψn) are translates by rational vectors in double-struckRd of a family of approximations to the identity, having the property that the partial sums are dense in various spaces of functions such as Wiener’s algebra W(C0,ℓ1), Cb(Rd), C0(Rd), Lp(Rd), for every p∈[1,∞), and the space of measurable functions. Applying this theory to particular situations, we establish approximations by such series to solutions of the heat and Laplace equations as well as to probability density functions.

Approximation by finite mixtures of continuous density functions that vanish at infinity

“…Denote the class of bounded continuous functions by C b = C ∩ L ∞ . The following theorem was proved in [26]. Utilizing the techniques from [27], [1] proved a similar set of results to Theorem 2, under the restriction that f is a non-negative function with support R, using g = φ (i.e.…”

“…The result was then improved upon, in [26], whereupon the more general space W was taken as a replacement for V, in Theorem 2. Denote the class of bounded continuous functions by C b = C ∩ L ∞ .…”

“…Some recent results in the spirit of Claim 1 have been obtained by [27] and [26], using methods from the study of universal series (see for example in [25]). Let…”

“…be a class of functions with tails decaying at a faster rate than o ( x n 2 ). In [26], it is noted that V ⊂ W. Further, let C c = f ∈ C : ∃ a compact set K, such that 1 K ∁ f = 0 , denote the set of compactly supported continuous functions. The following theorem was proved in [27].…”

Gaussian mixture model decomposition of multivariate signals

“…The ability of GMMs to approximate functions in L p spaces has been investigated previously, see, e.g., [12] where it is noted that any probability density function may be approximated in the sense of L 1 by GMMs. For completeness, we here give a proof of the density of GMMs in the L ∞ -norm.…”