Akash Anand

Akash Anand

It is well known that approximation of functions on $[0,1]$ whose periodic extension is not continuous fail to converge uniformly due to rapid Gibbs oscillations near the boundary. Among several approaches that have been proposed toward the resolution of Gibbs phenomenon in recent years, a Fourier continuation (FC) based approximation scheme has been suggested by Bruno and collaborators in the context of certain PDE solvers where approximation grids used are equispaced. While the practical efficacy of FC based schemes in obtaining a high-order numerical solution of PDEs is well known, theoretical convergence analyses largely remain unavailable. The primary objective of this paper is to take a step in this direction where we analyze the convergence rates of a Fourier continuation framework for approximations based on discrete functional data coming from equispaced grids. In this context, we explore a certain two-point Hermite interpolation strategy for constructing Fourier continuations that, not only simplifies the implementation of such approximations but also makes possible a rigorous analysis of its numerical properties. In particular, we show that the approximations converge with order $r+1$ for functions coming from a subspace of $C^{r,1}([0,1])$, the space of $r$-times continuously differentiable function whose $r$th derivative is Lipschitz continuous. We also demonstrate that theoretical rates are indeed achieved in practice, through a variety of numerical experiments.

Prakash Nainwal, Akash Anand

The FC-Gram algorithm approximates non-periodic functions to high order by constructing a periodic extension with controlled boundary behavior and applying trigonometric interpolation. In this paper we introduce a generalized FC-Gram framework (GenFC), which provides greater flexibility in the construction of the blending continuation of Gram polynomials. This flexibility gives better control over the shape of the periodic extension and leads to improved approximation accuracy. We establish a convergence theorem showing that the trigonometric interpolant converges at the rate $\mathcal{O}(n^{-\min(r+β,\,d)})$ in the supremum norm on the original interval, where $r$ is the smoothness of the target function, $d$ the number of Gram polynomials, and $β\in [0,1]$ a Fourier-decay parameter. The framework and its analysis are developed so that the modified FC-Gram method of [J. Sci. Comput., 105(1):8, 2025] is recovered as a particular case. Numerical experiments confirm the predicted convergence rates and show that the added flexibility of the GenFC framework leads to improved approximation accuracy, with the gains carrying over to a Fourier continuation solver for two-point boundary value problems.

Akash Anand, Awanish Kumar Tiwari

Computationally efficient numerical methods for high-order approximations of convolution integrals involving weakly singular kernels find many practical applications including those in the development of fast quadrature methods for numerical solution of integral equations. Most fast techniques in this direction utilize uniform grid discretizations of the integral that facilitate the use of FFT for $O(n\log n)$ computations on a grid of size $n$. In general, however, the resulting error converges slowly with increasing $n$ when the integrand does not have a smooth periodic extension. Such extensions, in fact, are often discontinuous and, therefore, their approximations by truncated Fourier series suffer from Gibb's oscillations. In this paper, we present and analyze an $O(n\log n)$ scheme, based on a Fourier extension approach for removing such unwanted oscillations, that not only converges with high-order but is also relatively simple to implement. We include a theoretical error analysis as well as a wide variety of numerical experiments to demonstrate its efficacy.

Manish Sharma, Akash Anand, Rupika Srivastava et al.

Wearables like smartwatches which are embedded with sensors and powerful processors, provide a strong platform for development of analytics solutions in sports domain. To analyze players' games, while motion sensor based shot detection has been extensively studied in sports like Tennis, Golf, Baseball; Table Tennis and Badminton are relatively less explored due to possible less intense hand motion during shots. In our paper, we propose a novel, computationally inexpensive and real-time system for shot detection in table tennis, based on fusion of Inertial Measurement Unit (IMU) and audio sensor data embedded in a wrist-worn wearable. The system builds upon our presented methodology for synchronizing IMU and audio sensor input in time using detected shots and achieves 95.6% accuracy. To our knowledge, it is the first fusion-based solution for sports analysis in wearables. Shot detectors for other racquet sports as well as further analytics to provide features like shot classification, rally analysis and recommendations, can easily be built over our proposed solution.