Radial basis function neural networks (RBFNNs) of Hankel translates of order $\mu >-1/2$ with varying widths whose activation function $\sigma$ is a.e. continuous, such that $z^{-\mu -1/2}\sigma (z)$ is locally essentially bounded and not an even polynomial, are shown to enjoy the universal approximation property (UAP) in appropriate spaces of continuous and integrable functions. In this way, the requirement that $\sigma$ be continuous for this kind of networks to achieve the UAP is weakened, and some results that hold true for RBFNNs of standard translates are extended to RBFNNs of Hankel translates.

For $\mu \ge -\frac{1}{2}$, and under appropriate conditions on the sequence $\{M_p{\}}_{p=0}^{\mathrm{\infty }}$ of weights, the elements, the (weakly, weakly^*, strongly) bounded subsets, and the (weakly, weakly^*, strongly) convergent sequences in the dual of a space ${\mathcal{K}}_{\mu }$ of type Hankel- $K\{M_p\}$ can be represented by distributional derivatives of functions and measures in terms of iterated adjoints of the differential operator $x^{-1}{D}_x$ and the Bessel operator $S_{\mu }=x^{-\mu -\frac{1}{2}}{D}_x{x}^{2\mu +1}{D}_x{x}^{-\mu -\frac{1}{2}}$. In this paper, such representations are compiled, and new ones involving adjoints of suitable iterations of the Zemanian differential operator $N_{\mu }=$ $x^{\mu +\frac{1}{2}}{D}_x{x}^{-\mu -\frac{1}{2}}$ are proved. Prior to this, new descriptions of the topology of the space ${\mathcal{K}}_{\mu }$ are given in terms of the latter iterations.

In this paper, we establish novel characterizations of locally uniformly rotund (LUR) Banach spaces. Building on these results, we prove that Zubelevich's relaxed version of Brøndsted's fixed point theorem for uniformly rotund spaces holds in the strictly broader class of LUR spaces. As an illustration, we derive existence results for fixed points of directional operators on Orlicz spaces under conditions where neither Brøndsted's nor Zubelevich's theorems apply.