Fourier Analysis: A New Result
This article contains a new result in Fourier analysis concerning jump type discontinuities.
math.GM↗
arXiv subjects
Rajesh Dachiraju
Publications and source records attributed to Rajesh Dachiraju.
Approximation of a Multivariate Function of Bounded Variation from its Scattered Data
This paper addresses the problem of approximating a function of bounded variation from its scattered data. Radial basis function(RBF) interpolation methods are known to approximate only functions in their native spaces, and to date, there has been no known proof that they can approximate functions outside the native space associated with the particular RBF being used. In this paper, we describe a scattered data interpolation method which can approximate any function of bounded variation from its scattered data as the data points grow dense. As the class of functions of bounded variation is a much wider class than the native spaces of the RBF, this method provides a crucial advantage over RBF interpolation methods.
math.NA↗
A Function Fitting Method
In this article, we describe a function fitting method that has potential applications in machine learning and also prove relevant theorems. The described function fitting method is a convex minimization problem and can be solved using a gradient descent algorithm. We also provide qualitative analysis on fitness to data of this function fitting method. The function fitting problem is also shown to be a solution of a linear, weak partial differential equation(PDE). We describe a way to fit a Sobolev function by giving a method to choose the optimal $λ$ parameter. We describe a closed-form solution to the derived PDE, which enables the parametrization of the solution function. We describe a simple numerical solution using a gradient descent algorithm, that converges uniformly to the actual solution. As the functional of the minimization problem is a quadratic form, there also exists a numerical method using linear algebra. Lastly, we give some numerical examples and also numerically demonstrate its application to a binary classification problem.
math.AP↗