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Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is…
There were established the exact-order estimations of the best uniform approximations by{\psi} the trigonometrical polynoms on the $C^{\psi}_{\beta,p}$ classes of $2\pi$-periodic continuous functions $f$, which are defined by the…
We show a procedure that, given oracle access to a function $f\colon \{0,1\}^n\to\{0,1\}$, produces oracle access to a function $f'\colon \{0,1\}^{n'}\to\{0,1\}$ such that if $f$ is monotone, then $f'$ is monotone, and if $f$ is…
We obtain order estimates for the best uniform orthogonal trigonometric approximations of $2\pi$-periodic functions, whose $(\psi,\beta)$-derivatives belong to unit balls of spaces $L_{p}, \ 1\leq p<\infty$, in case at consequences…
The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in…
We obtain order estimates in the spaces $L_p$ of the best $n$-term trigonometric orthogonal approximations of the classes ${\cal F}_{q}^{\psi}$ of periodic functions, whose Fourier coefficients decrease faster then any power function. We…
A real valued function $f$ defined on a real open interval $I$ is called $\Phi$-monotone if, for all $x,y\in I$ with $x\leq y$ it satisfies $$ f(x)\leq f(y)+\Phi(y-x), $$ where $\Phi:[0,\ell(I)[\,\to\mathbb{R}_+$ is a given nonnegative…
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…
We establish best possible pointwise (up to a constant multiple) estimates for approximation, on a finite interval, by polynomials that satisfy finitely many (Hermite) interpolation conditions, and show that these estimates cannot be…
A result of P\'olya states that every sequence of quadrature formulas $Q_n(f)$ with $n$ nodes and positive numbers converges to the integral $I(f)$ of a continuous function $f$ provided $Q_n(f)=I(f)$ for a space of algebraic polynomials of…
We establish interpolation analogues of Lebesgue type inequalities on the sets of $C^{\psi}_{\beta}L_{1}$ $2\pi$-periodic functions $f$, which are representable as convolutions of generating kernel $\Psi_{\beta}(t) =…
There exist well-known tight bounds on the error between a function $f \in C^{\,n + 1}([-1, 1])$ and its best polynomial approximation of degree $n$. We show that the error meets these bounds when and only when $f$ is a polynomial of degree…
We study the approximation capabilities of two families of univariate polynomials that arise in applications of quantum signal processing. Although approximation only in the domain $[0,1]$ is physically desired, these polynomial families…
A Boolean function f of n variables is said to be q-locally correctable if, given a black-box access to a function g which is "close" to an isomorphism f_sigma(x)=f_sigma(x_1, ..., x_n) = f(x_sigma(1), ..., x_sigma(n)) of f, we can compute…
In some particular cases we give criteria for morphic sequences to be almost periodic (=uniformly recurrent). Namely, we deal with fixed points of non-erasing morphisms and with automatic sequences. In both cases a polynomial-time algorithm…
We find uniform with respect to parameter $p \ (1\leq p\leq\infty)$ upper estimations of best approximations by trigonometric polynomials of classes $C^{\psi}_{\beta,p}$ of periodic functions generated by sequences $\psi(k)$, that decrease…
Several results on constrained spline smoothing are obtained. In particular, we establish a general result, showing how one can constructively smooth any monotone or convex piecewise polynomial function (ppf) (or any $q$-monotone ppf,…
The notion of quasi-Fej\'er monotonicity has proven to be an efficient tool to simplify and unify the convergence analysis of various algorithms arising in applied nonlinear analysis. In this paper, we extend this notion in the context of…
Polynomial approximation is studied in the Sobolev space $W_p^r(w_{\alpha,\beta})$ that consists of functions whose $r$-th derivatives are in weighted $L^p$ space with the Jacobi weight function $w_{\alpha,\beta}$. This requires…
By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in R^d. In this paper we study the following question: does the density hold if we approximate only by…