The Bramble--Hilbert lemma is a fundamental result on multivariate polynomial approximation. It frequently applied in the analysis of finite elements methods (FEM) used for numerical solutions PDEs. However, this classical estimate depends geometry domain and may "blow up" simple examples such as sequence triangles equivalent diameter that become thinner thinner. Thus, FEM applications one usually requires mesh has ``quasi-uniform" geometry. This assumption perhaps too restrictive when tries...

For a convex function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper C left-bracket negative 1 comma right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−<!-- − <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f \in C[ -...

There are two different ways by which one obtains representation theorems for the Laplace transform. One way is to impose integral conditions on inverse operator; and other summation without referring operator. Representation convolution transform have hitherto been obtained imposing operator, no attempt has made conditions. We obtain here some theorems, involve conditions, transforms with kernels in Class lί. A theorem of II determining functions bounded variation (—°°, °°), given. Also,...

We survey developments, over the last thirty years, in theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a finite interval. In this article, "shape" refers to (finitely many changes of) monotonicity, convexity, or q-monotonicity function (for definition, see Section 4). It is rather well known that it possible approximate preserve its shape (i.e., Weierstrass approximation theorem valid for SPA). At same time, degree SPA much worse than best unconstrained some cases,...

The binary space partition (BSP) technique is a simple and efficient method to adaptively an initial given domain match the geometry of input function. As such, BSP has been widely used by practitioners, but up until now no rigorous mathematical justification for it offered. Here we attempt put on sound foundations, offer enhancement algorithm in spirit what are going call geometric wavelets. This new approach sparse representation based recent developments theory multivariate nonlinear...

Abstract Jackson type theorems are obtained for the comonotone approximation of piecewise monotone functions by polynomials.

Abstract We construct trigonometric polynomials that fast decrease towards $$\pm \pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> . apply them to a polynomial the derivative of which interpolates given $$2\pi <mml:mn>2</mml:mn> -periodic function, at some prescribed distinct points in $$[-\pi ,\pi )$$ <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mo>,</mml:mo> <mml:mo>)</mml:mo> , and vanishes other...