A new computational framework for evaluation of the gamma function $Γ(z)$ over the complex plane is developed. The algorithm is based on interpolation by rational functions, and generalizes the classical methods of Lanczos \cite{Lanczos} and Spouge \cite{Spouge} (which we show are also interpolatory). This framework utilizes the exact poles of the gamma function. By relaxing this condition and allowing the poles to vary, a near-optimal rational approximation is possible, which is demonstrated using the adaptive Antoulous Anderson (AAA) algorithm, developed in \cite{AAA,AAA_2020}. The resulting approximations are competitive with Stirling's formula in terms of overall efficiency.
The paper is devoted to the construction of an optimal interpolation formula in $K_2(P_2)$ Hilbert space. Here the interpolation formula consists of a linear combination $\sum_{β=0}^NC_β(z)φ(x_β)$ of given values of a function $φ$ from the space $K_2(P_2)$. The difference between functions and the interpolation formula is considered as a linear functional called the error functional. The error of the interpolation formula is estimated by the norm of the error functional. We obtain the optimal interpolation formula by minimizing the norm of the error functional by coefficients $C_β(z)$ of the i…
We prove a.s. (almost sure) unisolvency of interpolation by continuous random sampling with respect to any given density, in spaces of multivariate a.e. (almost everywhere) analytic functions. Examples are given concerning polynomial and RBF approximation.
Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently.
A mixed interpolation-regression method for function approximation on certain planar domains
La Laguna University, Spain（拉帕尔马大学）；IMAG and Departamento de Matemática Aplicada, Universidad de Granada, Spain（格拉纳达大学）；Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain（马德里卡洛斯三世大学）
In this present paper, I propose a derivation of unified interpolation and extrapolation function that predicts new values inside and outside the given range by expanding direct Taylor series on the middle point of given data set. Mathematical construction of experimental model derived in general form. Trigonometric and Power functions adopted as test functions in the development of the vital aspects in numerical experiments. Experimental model was interpolated and extrapolated on data set that generated by test functions. The results of the numerical experiments which predicted by derived mod…
Matrices resulting from the discretization of a kernel function, e.g., in the context of integral equations or sampling probability distributions, can frequently be approximated by interpolation. In order to improve the efficiency, a multi-level approach can be employed that involves interpolating the kernel function and its approximations multiple times. This article presents a new approach to analyze the error incurred by these iterated interpolation procedures that is considerably more elegant than its predecessors and allows us to treat not only the kernel function itself, but also its der…
The natural kinship between classical theories of interpolation and approximation is well explored. In contrast to this, the interrelation between interpolation and approximation is subtle and this duality is relatively obscure in the context of fractal interpolation. The notion of $α$-fractal function provides a proper foundation for the approximation theoretic facet of univariate fractal interpolation functions (FIFs). However, no comparable approximation theoretic aspects of FIFs has been developed for functions of several variables. The current article intends to open the door for intrigui…
In this study, the standard IEEE 754 2008 and modulo-based floor functions for rounding non-integers have been presented. Their effects on the accuracy of the bilinear interpolation algorithm have been demonstrated. The improved floor uses the modulo operator in an effort to make each non-integer addend an integer when the remainder between the multiplicative inverse of the fractional factor and the numerator is greater than zero. The experiments demonstrated relatively positive effects with the improved floor function and the alternativeness to the standard round function.
We study in this paper the function approximation error of linear interpolation and extrapolation. Several upper bounds are presented along with the conditions under which they are sharp. All results are under the assumptions that the function has Lipschitz continuous gradient and is interpolated on an affinely independent sample set. Errors for quadratic functions and errors of bivariate linear extrapolation are analyzed in depth.
In this paper, applied strictly monotonic increasing scaled maps, a kind of well-conditioned linear barycentric rational interpolations are proposed to approximate functions of singularities at the origin, such as $x^α$ for $α\in (0,1)$ and $\log(x)$. It just takes $O(N)$ flops and can achieve fast convergence rates with the choice the scaled parameter, where $N$ is the maximum degree of the denominator and numerator. The construction of the rational interpolant couples rational polynomials in the barycentric form of second kind with the transformed Jacobi-Gauss-Lobatto points. Numerical exper…
Interpolation of classes of differentiated functions given on a finite interval by trigonometric splines using the phantom node method is considered. This method consists in supplementing a given sequence of values of an approximate function with an even number of values of a phantom function, which is constructed in such a way as to eliminate gaps in both the function itself and its derivatives up to and including a certain order; in the General case, these gaps occur with the periodic continuation of the function given at a finite interval. The results of calculations on test examples for tr…
For the interpolation of graph signals with generalized shifts of a graph basis function (GBF), we introduce the concept of positive definite functions on graphs. This concept merges kernel-based interpolation with spectral theory on graphs and can be regarded as a graph analog of radial basis function interpolation in euclidean spaces or spherical basis functions. We provide several descriptions of positive definite functions on graphs, the most relevant one is a Bochner-type characterization in terms of positive Fourier coefficients. These descriptions allow us to design GBF's and to study G…
We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions $\widetildeφ_j$ and trigonometric polynomials $φ_j$. The class of such operators includes classical interpolation polynomials ($\widetildeφ_j$ is the Dirac delta function), Kantorovich-type operators ($\widetildeφ_j$ is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on $\widetildeφ_j$ and $φ_j$, we obtain upper and lower bound estimates for the $L_p$-error of a…
