We present two partitioning algorithms that allow a sum of piecewise linear polynomials over a number of overlaying convex partitions of the unit cube Ω in Rd to approximate a function f ∈ W 3 p (Ω) with the order N−6/(2d+1) in Lp-norm, where N is the total number of cells of all partitions, which makes a marked improvement over the N−2/d order achievable on a single convex partition. The gradi...

This overview is intended to provide an “ atlas ” of what is known about approximations of the cone of positive polynomials (on a semialgebraic set KS) by various preorderings (or the corresponding module versions). These approximations depend on the description S of KS, the dimension of the semi-algebraic set KS, intrinsic geometric properties of KS (e.g. compact or unbounded), and special pro...

In a previous work, it was shown that Faber-Pandharipande-Zagier and Miki’s identities can be derived from a polynomial identity which in turn follows from a Fourier series expansion of sums of products of Bernoulli functions. Motivated by this work, we consider three types of sums of finite products of Genocchi functions and derive Fourier series expansions for them. Moreover, we will be able ...

Many applications in electrical engineering, signal processing, and mathematical physics lead to following approximation problem: Let h be a short linear combination of nonincreasing exponentials with complex exponents. Determine all exponents, all coefficients, and the number of summands from finitely many equispaced sampled data of h. This is a nonlinear inverse problem. This paper is an exte...

We introduce a new approach, and associated algorithms, for the efficient approximation of functions and sequences by short linear combinations of exponential functions with complex-valued exponents and coefficients. These approximations are obtained for a finite but arbitrary accuracy and typically have significantly fewer terms than Fourier representations. We present several examples of thes...

The null hypothesis that all of a function’s Fourier coefficients are 0 is tested in frequentist fashion using as test statistic a Laplace approximation to the posterior probability of the null hypothesis. Testing whether or not a regression function has a prescribed linear form is one application of such a test. In contrast to BIC, the Laplace approximation depends on prior probabilities, and ...

‎The main purpose of this note is to establish some bounds in Poisson approximation for row-wise arrays of independent geometric distributed random variables using the operator method‎. ‎Some results related to random sums of independent geometric distributed random variables are also investigated.