Related papers: Spline Based Series for Sine and Arbitrarily Accur…
Functions of one or more variables are usually approximated with a basis: a complete, linearly-independent system of functions that spans a suitable function space. The topic of this paper is the numerical approximation of functions using…
Sine illusion happens when the more quickly changing pairs of lines lead to bigger underestimates of the delta between them. We evaluate three visual manipulations on mitigating sine illusions: dotted lines, aligned gridlines, and offset…
Given values of a piecewise smooth function $f$ on a square grid within a domain $\Omega$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of…
We use deep sparsely connected neural networks to measure the complexity of a function class in $L^2(\mathbb R^d)$ by restricting connectivity and memory requirement for storing the neural networks. We also introduce representation system -…
We construct a family of spline affine Riesz bases, i.e. sequences of dilations and translations generated by the special spline functions $\psi_m$, and we prove that their Riesz bounds are independent of $m$. We put $\psi_0=\chi$ as the…
Cyclic boundaries are used in many branches of physics and mathematics, typically to assist the approximation of a large space. We show that when determining the performance of planar, fault-tolerant, topological quantum error correction,…
Smoothing splines have been used pervasively in nonparametric regressions. However, the computational burden of smoothing splines is significant when the sample size $n$ is large. When the number of predictors $d\geq2$, the computational…
We consider a sequence of composite Bernstein operators and the quadrature formulae associated with them. Upper bounds for the approximation error of continuous functions and for the approximation of integrals of continuous functions are…
A New Error Bound for shifted surface spline interpolation is presented. This error bound probably is the most powerful one up to now.
We give new bounds on the reliability function of a typewriter channel with 5 inputs and crossover probability $1/2$. The lower bound is more of theoretical than practical importance; it improves very marginally the expurgated bound,…
We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space $H^s({\mathbb{R}})$ and from the space $C^s({\mathbb{R}})$ with an arbitrary integer $s\ge1$. We find tight upper and lower…
We revisit the problem of computing with noisy information considered in Feige et al. 1994, which includes computing the OR function from noisy queries, and computing the MAX, SEARCH and SORT functions from noisy pairwise comparisons. For…
In our previous publications we have introduced the cosine product-to-sum identity [17] $$ \prod\limits_{m = 1}^M {\cos \left( {\frac{t}{{{2^m}}}} \right)} = \frac{1}{{{2^{M - 1}}}}\sum\limits_{m = 1}^{{2^{M - 1}}} {\cos \left( {\frac{{2m -…
Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are…
By employing non-equispaced grid points near boundaries, boundary-optimized upwind finite-difference operators of orders up to nine are developed. The boundary closures are constructed within a diagonal-norm summation-by-parts (SBP)…
Sliced Inverse Regression (SIR) is an effective method for dimension reduction in high-dimensional regression problems. The original method, however, requires the inversion of the predictors covariance matrix. In case of collinearity…
In areas such as kernel smoothing and non-parametric regression there is emphasis on smooth interpolation and smooth statistical models. Splines are known to have optimal smoothness properties in one and higher dimensions. It is shown, with…
We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load…
We study the complexity of learning and approximation of self-bounding functions over the uniform distribution on the Boolean hypercube ${0,1}^n$. Informally, a function $f:{0,1}^n \rightarrow \mathbb{R}$ is self-bounding if for every $x…
In a previous article the authors determined the best-known upper bound for the cardinality of the image set for several classes of functions, including planar functions. Here, we show that the upper bound cannot be tight for planar…