This paper introduces a new approach to prediction by bringing together two different nonparametric ideas: distribution free inference and nonparametric smoothing. Specifically, we consider the problem of constructing nonparametric tolerance/prediction sets. We start from the general conformal prediction approach and we use a kernel density estimator as a measure of agreement between a sample p...

In the paper titled as “Jackson-type inequality on the sphere” 2004 , Ditzian introduced a spherical nonconvolution operator Ot,r , which played an important role in the proof of the wellknown Jackson inequality for spherical harmonics. In this paper, we give the lower bound of approximation by this operator. Namely, we prove that there are constants C1 and C2 such that C1ω2r f, t p ≤ ‖Ot,rf − ...

We introduce new moduli of smoothness for functions f ∈ Lp[−1, 1]∩Cr−1(−1, 1), 1 ≤ p ≤ ∞, r ≥ 1, that have an (r − 1)st locally absolutely continuous derivative in (−1, 1), and such that φrf (r) is in Lp[−1, 1], where φ(x) = (1 − x2)1/2. These moduli are equivalent to certain weighted DT moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted ...

Several results on equivalence of moduli of smoothness of univariate splines are obtained. For example, it is shown that, for any 1 ≤ k ≤ r+1, 0 ≤ m ≤ r − 1, and 1 ≤ p ≤ ∞, the inequality n−νωk−ν(s(ν), n−1)p ∼ ωk(s, n −1)p, 1 ≤ ν ≤ min{k,m + 1}, is satisfied, where s ∈ Cm[−1, 1] is a piecewise polynomial of degree ≤ r on a quasi-uniform (i.e., the ratio of lengths of the largest and the smalles...

We prove two-sided inequalities between the integral moduli of smoothness of a function on R d /T d and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is given by the equivalence results for functions satisfying certain regular conditions. Applications include a quantitative form of the Riemann–Lebesgue lemma as well as several othe...

Several inverse problems exist in the atmospheric sciences that are computationally costly when using traditional gradient based methods. Unfortunately, many standard evolutionary algorithms do not perform well on these problems. This paper investigates why the temperature inversion problem is so difficult for heuristic search. We show that algorithms imposing smoothness constraints find more c...

We study irregularity properties of generic Peano functions; we apply these results to the determination of the pointwise smoothness of a Peano function introduced by Lebesgue and of some related functions, showing that they are either monohölder or multifractal functions. We test on these examples several numerical variants of the multifractal formalism, and we show how a change of topology on...

The Sauer-Shelah lemma has been instrumental in the analysis of algorithms in many areas including learning theory, combinatorial geometry, graph theory. Algorithms over discrete structures, for instance, sets of Boolean functions, often involve a search over a constrained subset which satisfies some properties. In this paper we study the complexity of classes of functions h of finite VCdimensi...

applying 2d algorithms for inverting the potential field data is more useful and efficient than their 3d counterparts, whenever the geologic situation permits. this is because the computation time is less and modeling the subsurface is easier. in this paper we present a 2d inversion algorithm for interpreting gravity data by employing a set of constraints including minimum distance, smoothness,...