Formalization of Transform Methods using HOL Light

Transform methods, like Laplace and Fourier, are frequently used for analyzing the dynamical behaviour of engineering and physical systems, based on their transfer function, and frequency response or the solutions of their corresponding differential equations. In this paper, we present an ongoing project, which focuses on the higher-order logic formalization of transform methods using HOL Light theorem prover. In particular, we present the motivation of the formalization, which is followed by the related work. Next, we present the task completed so far while highlighting some of the challenges faced during the formalization. Finally, we present a roadmap to achieve our objectives, the current status and the future goals for this project.Comment: 15 Pages, CICM 201

On boundedness of discrete multilinear singular integral operators

Let $m(\xi,\eta)$ be a measurable locally bounded function defined in $\mathbb R^2$. Let $1\leq p_1,q_1,p_2,q_2<\infty$ such that $p_i=1$ implies $q_i=\infty$. Let also $0<p_3,q_3<\infty$ and $1/p=1/p_1+1/p_2-1/p_3$. We prove the following transference result: the operator {\mathcal C}_m(f,g)(x)=\int_{\bbbr} \int_{\bbbr} \hat f(\xi) \hat g(\eta) m(\xi,\eta) e^{2\pi i x(\xi +\eta)}d\xi d\eta initially defined for integrable functions with compact Fourier support, extends to a bounded bilinear operator from L^{p_1,q_1}(\bbbr)\times L^{p_2,q_2}(\bbbr) into L^{p_3,q_3}(\bbbr) if and only if the family of operators {\mathcal D}_{\widetilde{m}_{t,p}} (a,b)(n) =t^{\frac{1}{p}}\int_{-\12}^{\12}\int_{-\12}^{\12}P(\xi) Q(\eta) m(t\xi,t\eta) e^{2\pi in(\xi +\eta)}d\xi d\eta initially defined for finite sequences a=(a_{k_{1}})_{k_{1}\in \bbbz}, b=(b_{k_{2}})_{k_{2}\in \bbbz}, where P(\xi)=\sum_{k_{1}\in \bbbz}a_{k_{1}}e^{-2\pi i k_{1}\xi} and Q(\eta)=\sum_{k_{2}\in \bbbz}b_{k_{2}}e^{-2\pi i k_{2}\eta}, extend to bounded bilinear operators from l^{p_1,q_1}(\bbbz)\times l^{p_2,q_2}(\bbbz) into l^{p_3,q_3}(\bbbz) with norm bounded by uniform constant for all $t>0

The non-coplanar baselines effect in radio interferometry: The W-Projection algorithm

We consider a troublesome form of non-isoplanatism in synthesis radio telescopes: non-coplanar baselines. We present a novel interpretation of the non-coplanar baselines effect as being due to differential Fresnel diffraction in the neighborhood of the array antennas. We have developed a new algorithm to deal with this effect. Our new algorithm, which we call "W-projection", has markedly superior performance compared to existing algorithms. At roughly equivalent levels of accuracy, W-projection can be up to an order of magnitude faster than the corresponding facet-based algorithms. Furthermore, the precision of result is not tightly coupled to computing time. W-projection has important consequences for the design and operation of the new generation of radio telescopes operating at centimeter and longer wavelengths.Comment: Accepted for publication in "IEEE Journal of Selected Topics in Signal Processing