Related papers: Polynomials of the best uniform approximation to s…
A rational approximation by a ratio of polynomial functions is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non- Lipschitz functions, where polynomial…
We present one of the approaches to find the best approximation of the given function by trigonometric polynomials in $L^1$ metric and applied it to find the optimal constants in the Nikolsky's type inequality, concerning approximation of…
The asymptotic expansion of the Touchard polynomials $T_n(z)$ (also known as the exponential polynomials) for large $n$ and complex values of the variable $z$, where $|z|$ may be finite or allowed to be large like $O(n)$, has been recently…
Uniform polynomial approximation, also called minimax approximation or Chebyshev approximation, consists in searching polynomial approximation that minimizes the worst case error. Optimality conditions for the uniform approximation of…
An example of trigonometric polynomials with extremely small uniform norm is given. This example demonstrates the potential limits for extension of Sidon's inequality for lacunary polynomials in a certain direction.
An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the…
For a function $f$ that is piecewise analytic on a quasi-smooth arc $\mathcal{L}$ and any $0<\sigma<1$ we construct a sequence of "near-best" polynomials that converge at a rate $e^{-n^{\sigma}}$ at each point of analyticity of $f$ and are…
We obtain order-exact estimates for uniform approximations by using Zygmund sums $Z^{s}_{n}$ of classes $C^{\psi}_{\beta,p}$ of $2\pi$-periodic continuous functions $f$ representable by convolutions of functions from unit balls of the space…
We characterize the best $L_{2}$ approximation to a multivariate function by linear combinations of ridge functions multiplied by some fixed weight functions. In the special case when the weight functions are constants, we propose explicit…
We obtain the estimates for the best approximation in the uniform metric of the classes of $2\pi $-periodic functions whose $(\psi ,\beta)$-derivatives have a given majorant $\omega$ of the modulus of continuity. It is shown that the…
We present a general approach to the problem of determining tight asymptotic lower bounds for generalized central moments of the optimal alignment score of two independent sequences of i.i.d. random variables. At first, these are obtained…
We establish formulas for the constant factor in several asymptotic estimates related to the distribution of integer and polynomial divisors. The formulas are then used to approximate these factors numerically.
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
For first order differential equations of the form $y'=\sum_{p=0}^P F_p(x)y^p$ and second order homogeneous linear differential equations $y''+a(x)y'+b(x)y=0$ with locally integrable coefficients having asymptotic (possibly divergent) power…
Differential (Ore) type polynomials with "approximate" polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate Greatest…
Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for graph partitioning. Using a special class…
We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real or the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of…
Polynomial meshes (called sometimes "norming sets") allow us to estimate the supremum norm of polynomials on a fixed compact set by the norm on its discrete subset. We give a general construction of polynomial weakly admissible meshes on…
A new method, called the method of self-similar approximants, and its recent developments are described. The method is based on the ideas of renormalization group theory and optimal control theory. It allows for the effective extrapolation…
We introduce polystar bodies: compact starshaped sets whose gauge or radial functions are expressible by polynomials, enabling tractable computations, such as that of intersection bodies. We prove that polystar bodies are uniformly dense in…