Related papers: Multidimensional integer trigonometry
In this paper, we propose a new trigonometric interpolation algorithm and establish relevant convergent properties. The method adjusts an existing trigonometric interpolation algorithm such that it can better leverage Fast Fourier Transform…
The paper deals with two fundamental types of trigonometric polynomials and splines on uniform grids, which allow us to construct interpolation approximations that depend linearly on the values of the interpolated function. Fundamental on…
The development of the trigonometric functions in introductory texts usually follows geometric constructions using right triangles or the unit circle. While these methods are satisfactory at the elementary level, advanced mathematics…
Integral transforms are invaluable mathematical tools to map functions into spaces where they are easier to characterize. We introduce the hyperdimensional transform as a new kind of integral transform. It converts square-integrable…
Here we introduce a way to construct generalized trigonometric functions associated with any complex polynomials, and the well known trigonometric functions can be seen to associate with polynomial $x^2-1$. We will show that those…
Trigonometry is the study of circular functions, which are functions defined on the unit circle $x^2+y^2 =1$, where distances are measured using the Euclidean norm. When distances are measured using the $L_p$-norm, we get generalized…
We develop the basic formulae of hyperspherical trigonometry in multidimensional Euclidean space, using multidimensional vector products, and their conversion to identities for elliptic functions. We show that the basic addition formulae…
When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given…
In our previous paper, Real Polynomials with a Complex Twist [see http://archives.math.utk.edu/ICTCM/VOL28/A040/paper.pdf], we used advancements in computer graphics that allow us to easily illustrate more complete graphs of polynomial…
Pointwise tangential dimensions are introduced for metric spaces. Under regularity conditions, the upper, resp. lower, tangential dimensions of X at x can be defined as the supremum, resp. infimum, of box dimensions of the tangent sets, a…
Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is…
The intrinsic dimensionality refers to the ``true'' dimensionality of the data, as opposed to the dimensionality of the data representation. For example, when attributes are highly correlated, the intrinsic dimensionality can be much lower…
Multicomplex numbers of order n have an associated trigonometry (multisine functions with (n-1) parameters) leading to a natural extension of the sine-Gordon model. The parameters are constrained from the requirement of local current…
To represent positive integers by regular patterns on a plane or in three-dimensional space may be traced back to the Pythagoreans. The aim of the present article is to explore the possibility of extending the representation framework for…
A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method encapsulates trigonometry for all these spaces into a single basic trigonometric group…
The article presents an algebra to represent two dimensional patterns using reciprocals of polynomials. Such a representation will be useful in neural network training and it provides a method of training patterns that is much more…
The methods of approximation, regularization and smoothing of trigonometric interpolation splines are considered in the paper. It is shown that trigonometric splines can be considered from two points of view - as a trigonometric Fourier…
Generalizations of some known results on the best, best linear and best one-sided approxima- tions by trigonometric polynomials of the classes of 2\pi - periodic functions presented in the form of convolutions to the case of set-valued…
A dual approach to defining the triangle sequence (a type of multidimensional continued fraction algorithm, initially developed in NT/9906016) for a pair of real numbers is presented, providing a new, clean geometric interpretation of the…
We revisit the classical kernel method of approximation/interpolation theory in a very specific context motivated by the desire to obtain a robust procedure to approximate discrete data sets by (super)level sets of functions that are merely…