MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINE

For a class of sets with multiple terms$$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times},\underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots,\underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\},$$having density \(d\) counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers\linebr eak \(\{d_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots ,\mu_n-1\} \) satisfying certain growth conditions, we consider a moment problem of the form $$\int_{-\infty}^{\infty}e^{-2w(t)}t^k e^{\lambda_n t}f(t)\, dt=d_{n,k},\quad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,2,\dots, \mu_n-1,$$ in weighted \(L^2 (-\infty, \infty)\) spaces. We obtain a solution \(f\) which extends analytically as an entire function, admitting a Taylor–Dirichlet series representation $$ f(z)=\sum_{n=1}^{\infty}\Big(\sum_{k=0}^{\mu_n-1}c_{n,k} z^k\Big) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C},\quad\forall\,\, z\in \mathbb{C}. $$ The proof depends on our previous work where we characterized the closed span of the exponential system \(\{t^k e^{\lambda_n t}:\, n\in\mathbb{N},\,\, k=0,1,2,\dots,\mu_n-1\}\) in weighted \(L^2 (-\infty, \infty)\) spaces, and also derived a sharp upper bound for the norm of elements of a biorthogonal sequence to the exponential system. The proof also utilizes notions from Non-Harmonic Fourier series such as Bessel and Riesz–Fischer sequences.

A Black Box Design Approach to Avian-Inspired SMA Coil Actuated Wearable Morphing Angel Wings

Black box design is a constraint driven design approach that distills essential elements of a physical process into inputs and outputs. This paper details the black box design implementation and validation of shape memory alloy (SMA) coil actuators as active members in a Watt I six bar avian-inspired wearable morphing angel wing mechanism. SMA coil actuators leverage the unique characteristics of high energy density SMA wire by providing a compact structural platform for large actuation displacement applications. The moderate force and displacement performance of low spring index coil actuators paired with their virtually silent actuation performance made them an attractive actuator solution to an avian-inspired wearable morphing wing mechanism for the University of Minnesota Department of Theatre Arts and Dance production of ‘Marisol’. The wing design constraints (extended span of 7.5 ft, a closed span of 3 ft) required a tailorable actuator system with capacity to perform at particular target force and strain metrics cyclically. A low spring index parameter study was conducted to facilitate an accelerated phase of design prototyping. The parameter study featured six SMA coil actuator prototypes made with 0.012” diameter Dynalloy Flexinol® wire of varying spring indexes (C = 2.5–4.9). The coil actuators were manufactured through a CNC winding process, shape set in a furnace at 450 °C for 10 minutes, and water quenched for hardening. A series of thermomechanical actuation tests were conducted to experimentally characterize the low spring index actuation performances. The coil actuation characterizations demonstrated increased force and decreased actuator displacement corresponding to decreased spring indexes. Scaling these results aided an accelerated design of an actuator system. The actuator system consisted of four C = 3.05 coil actuators wound with 0.02” diameter SMA that were integrated into each Watt I mechanism. The characterization of the force-displacement profiles for low index SMA coil actuators provides an effective empirical design strategy for scaling actuator performance to mechanical systems requiring moderate force, moderate displacement actuators.