Penalized hyperbolic-polynomial splines

Stable interpolation with exponential-polynomial splines and node selection via greedy algorithms

“…The global space of the exponential-polynomial splines E X, ( 4, ) is then defined by gluing local patches defined over each interval such that E X,𝛼 ( 4,𝛼 ) ⊂ C 2 ([a, b]) , and any element in this space can be expressed using coefficients b i,k as in (2). In particular, this implies via (1) that…”

“…We remark that in [12] the authors assume that 𝛼 > 0 (instead of ∈ ℝ ) in order to enforce a certain exponential behavior outside of the interpolation interval. Moreover, in the same paper it is additionally assumed that the boundary pieces in (2) are contained in a different space 2,− ∶= span{e − x , x e − x }.…”

“…Scattered data interpolation is one of the most investigated topics in the field of numerical analysis, and it is successfully used in many applications. As a consequence, many methods have been developed, including interpolation with polynomials of total degree (see, e.g., [1]), exponential splines [2], and kernel-based methods (refer, e.g., to [3,4]), with their recent developments in the context of image processing (e.g., [5]) and machine learning [6,7].…”

Reproduction Capabilities of Penalized Hyperbolic-polynomial Splines

“…This short paper investigates the reproduction capabilities of hyperbolicpolynomial penalized splines. HP-splines, were recently introduced in [1] as a generalization of the better known P-splines (see [2,3]), and combine a finite difference penalty with HB-splines that piecewise consist of real exponentials and monomials multiplied by these exponentials. Numerical examples show that the exponential nature of HP-splines may turn out to be useful in applications when the data show an exponential trend [4].…”

“…with the frequency α being an extra parameter to tune the smoother effects. Even though all details concerning their definition and construction can be already find in [1], the analysis of their reproduction capability is there missing. To fill the gap, here we show that these type of penalized splines reproduce function in the space {e −αx , xe −αx }, that is fit exponential data of the latter type exactly.…”