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In this work we present new methods for transforming and solving finite series by using the Laplace transform. In addition we introduce both an alternative method based on the Fourier transform and a simplified approach. The latter allows a…
Research on refinable functions in wavelet theory is mostly focused to localized functions. However it is known, that polynomial functions are refinable, too. In our paper we investigate on conversions between refinement masks and…
This paper proposes a novel localized Fourier extension method for approximating non-periodic functions via domain segmentation. By partitioning the computational domain into subregions with uniform discretization scales, the method…
The purpose of this note is to compare various approximation methods as applied to the inverse of the Bessel function of the first kind, in a given domain of the complex plane.
Projections onto sets are used in a wide variety of methods in optimization theory but not every method that uses projections really belongs to the class of projection methods as we mean it here. Here projection methods are iterative…
Approximation properties of Ces\`{a}ro and Abel-Poisson means of hexagonal Fourier series are studied. The degree of approximation by these means of hexagonal Fourier series of functions, which are continuous and periodic with respect to…
We describe a machine learning method for predicting the value of a real-valued function, given the values of multiple input variables. The method induces solutions from samples in the form of ordered disjunctive normal form (DNF) decision…
In this work, we develop a method for rational approximation of the Fourier transform (FT) based on the real and imaginary parts of the complex error function \[ w(z) = e^{-z^2}(1 - {\rm{erf}}(-iz)) = K(x,y) + iL(x,y), \qquad z = x + iy, \]…
A novel method for predicting time series is described and demonstrated. This method inputs time series data points and outputs multiple "spaghetti" functions from which predictions can be made. Spaghetti prediction has desirable properties…
A new type of combinations of Bernstein operators is given in [1]. Here, we introduce another one, which can be used to approximate the functions with singularities. The direct and inverse results of the weighted approximation of this new…
A Boolean function $g$ is said to be an optimal predictor for another Boolean function $f$, if it minimizes the probability that $f(X^{n})\neq g(Y^{n})$ among all functions, where $X^{n}$ is uniform over the Hamming cube and $Y^{n}$ is…
In some applications, one is interested in reconstructing a function $f$ from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we…
Multivariate function approximation is a fundamental problem in machine learning. Classic multivariate function approximations rely on hand-crafted basis functions (e.g., polynomial basis and Fourier basis), which limits their approximation…
The aim of the paper is to establish a convergence theorem for multi-dimensional stochastic approximation when the "innovations" satisfy some "light" averaging properties in the presence of a pathwise Lyapunov function. These averaging…
Where do objective functions come from? How do we select what goals to pursue? Human intelligence is adept at synthesizing new objective functions on the fly. How does this work, and can we endow artificial systems with the same ability?…
This paper examines the existence and region of convergence of Fourier transform of the functions of bicomplex variables with the help of projection on its idempotent components as auxiliary complex planes. Several basic properties of this…
An approximation method is presented for probabilistic inference with continuous random variables. These problems can arise in many practical problems, in particular where there are "second order" probabilities. The approximation, based on…
In this exploratory article, we draw attention to the common formal ground among various estimators such as the belief functions of evidence theory and their relatives, approximation quality of rough set theory, and contextual probability.…
One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in…
The Dirichlet forms methods, in order to represent errors and their propagation, are particularly powerful in infinite dimensional problems such as models involving stochastic analysis encountered in finance or physics, cf. [5]. Now, coming…