Related papers: Divided Differences, Falling Factorials, and Discr…
We study a novel spline-like basis, which we name the "falling factorial basis", bearing many similarities to the classic truncated power basis. The advantage of the falling factorial basis is that it enables rapid, linear-time computations…
The novel concept of box spline of complex degree is introduced and several of its properties derived and discussed. These box splines of complex degree generalize and extend the classical box splines. Relations to a class of fractional…
Finite trigonometric Fourier series on a set of discrete equidistant points are considered. A finite system of orthogonal functions that have interpolation and certain differential properties on the period is introduced. Finite Fourier…
In some real world applications, such as spectrometry, functional models achieve better predictive performances if they work on the derivatives of order m of their inputs rather than on the original functions. As a consequence, the use of…
We study additive models built with trend filtering, i.e., additive models whose components are each regularized by the (discrete) total variation of their $k$th (discrete) derivative, for a chosen integer $k \geq 0$. This results in $k$th…
We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339-360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the…
We introduce discrete analogues of the exponential, sine, and cosine functions. Then using a discrete trigonometric version of a non-polynomial divided difference, we define discrete analogues of the trigonometric B-splines. We derive a…
Cubic and quartic non-autonomous differential equations with continuous piecewise linear coefficients are considered. The main concern is to find the maximum possible multiplicity of periodic solutions. For many classes, we show that the…
Fractional differential and integral operators, Dirichlet averages, and splines of complex order are three seemingly distinct mathematical subject areas addressing different questions and employing different methodologies. It is the purpose…
Observations made in continuous time are often irregular and contain the missing values across different channels. One approach to handle the missing data is imputing it using splines, by fitting the piecewise polynomials to the observed…
The paradigm of differentiable programming has significantly enhanced the scope of machine learning via the judicious use of gradient-based optimization. However, standard differentiable programming methods (such as autodiff) typically…
Piecewise smooth systems are intensively studied today in many application areas, such as economics, finance, engineering, biology, and ecology. In this work, we consider a class of one-dimensional piecewise linear discontinuous maps with a…
Partitions of unity in ${\mathbf R}^d$ formed by (matrix) scales of a fixed function appear in many parts of harmonic analysis, e.g., wavelet analysis and the analysis of Triebel-Lizorkin spaces. We give a simple characterization of the…
Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multi-degree…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
This survey gives an overview of three central algebraic themes related to the study of splines: duality, group actions, and homology. Splines are piecewise polynomial functions of a prescribed order of smoothness on some subdivided domain…
The problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure is discussed in the paper. The above problem is studied for elements of finite…
Continuous spline functions are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data…
The concept of permutograph is introduced and properties of integral functions on permutographs are established. The central result characterizes the class of integral functions that are representable as lattice polynomials. This result is…
Additive utility function models are widely used in multiple criteria decision analysis. In such models, a numerical value is associated to each alternative involved in the decision problem. It is computed by aggregating the scores of the…