New polynomial preserving operators on simplices: direct results

Constrained orthogonal polynomials have been recently introduced in the study of the Hohenberg–Kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean inner product, to a given i-dimensional subspace of polynomials associated with the constraints. In addition, they are mutually orthogonal with respect to a second Euclidean inner product. We recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an Euclidean vector space endowed with distinct inner products. An explicit two step Gram–Schmidt orthogonalization (G-SO) process to determine these bases is given. By definition, the standard block orthogonal (SBO) polynomials are associated with a choice of equal to the subspace of polynomials of degree less than i. We investigate their properties, emphasizing similarities to and differences from the standard orthogonal polynomials. Applications to classical orthogonal polynomials will be given in forthcoming papers.

Here we transfer basic real approximations to corresponding vectorial and fuzzy setting of: Bernstein polynomials, Bernstein-Durrmeyer operators, genuine Bernstein-Durrmeyer operators, Stancu type operators and special Stancu operators. These are convergences to the unit operator with rates. We also give the convergence with rates to zero of the difference of genuine Bernstein-Durrmeyer and special Stancu operators. All approximations involve Jackson type inequalities and moduli of smoothness of various orders. In order to transfer we develop basic and important general results at the vectorial and fuzzy level. Our technique goes from real to vectorial and then to fuzzy setting. This chapter is based on [58].

This paper presents a new family of operators which constitute the link between \(\alpha \)-Bernstein operators and genuine \(\alpha \)-Bernstein–Durrmeyer operators. Some approximation results, which include local approximation and error estimation in terms of the modulus of continuity are given. Finally, a quantitative Voronovskaya type theorem is established and some Gruss type inequalities are obtained.

Bernstein–Durrmeyer operators with respect to arbitrary measure, II: Pointwise convergence

In this paper, we study an asymptotic expansion for a general family of Bernstein–Durrmeyer type operators $$L_n^{j,i}$$ L n j , i , $$i,j\in {\mathbb {N}}_0$$ i , j ∈ N 0 . For $$i=j=1$$ i = j = 1 , these operators reduce to the genuine Bernstein–Durrmeyer operators, while for $$i=j=0$$ i = j = 0 they coincide with the classical Bernstein–Durrmeyer operators. We derive an explicit asymptotic expansion for $$L_n^{j,i}$$ L n j , i under suitable smoothness conditions on the approximated function.

We introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval [a, b] into the space of compact non-empty subsets of Rd. All operators are adapted by replacing the Riemann integral for real-valued functions by the weighted metric integral for SVFs of bounded variation with compact graphs. For such a SVF F, we obtain pointwise error estimates for sequences of integral operators at points of continuity, leading to convergence at such points to F. At points of discontinuity of F, we derive estimates, which yield the convergence to a certain set described in terms of the metric selections of F. To obtain these estimates we refine and extend known results on approximation of real-valued functions by integral operators. Our analysis uses recently defined one-sided local quasi-moduli at points of discontinuity and several notions of local Lipschitz property at points of continuity. We also provide a global approach for error bounds. A multifunction F is represented by the set of all its metric selections, while its approximation (its image under the operator) is represented by the set of images of these metric selections under the operator. A bound on the Hausdorff distance between these two sets of single-valued functions in L1 provides our global estimates. The theory is applied to concrete operators: the Bernstein–Durrmeyer operator and the Kantorovich operator.

The convergence of iterated Boolean sums for different sequences of operators has been studied keeping fixed the number M of iterations or recently letting for a fixed operator. Here we consider some convergence properties of iterated Boolean-type sums of the classical Bernstein and Bernstein–Durrmeyer operators in the case where both the operators and their order of iterations are not fixed. In the case of Bernstein–Durrmeyer operators, we obtain a general solution of this problem. We can also state some alternative expressions of the semigroups and resolvent operators associated with the Voronovskaja’s formula for these sequences of operators.

Introduction. This article considers a fairly general class of operators on sections of a vector bundle over a compact manifold, including the differential operators and singular integral operators. The members of this class share many of the properties of differential operators, particularly the elliptic ones. Two general advantages have motivated the development. First, it leads to transparent proofs of the familiar results for elliptic equations, on regularity, the Fredholm alternative, and eigenfunction expansions; and for a larger class than the differential operators. These proofs are not new; rather some of the techniques used in the case of differential operators appear here as general properties of the class of integro-differential operators considered. A second advantage of the larger system, not extensively exploited in this article, is topological. Homotopies (in the class of smooth functions) of the characteristic polynomial of a differential operator can be lifted to homotopies of the operator itself in the class of integro-differential operators considered, but not (generally) in the class of differential operators. This is an important help in treating some questions raised by Gelfand [6]; some of the questions concerning the index have now been answered by Atiyah and Singer [1]. To find the notation and main results, one can read ?1-?3 (except for proofs), ?6, and the definitions and statements of theorems and corollaries from the remaining sections. The paper is organized as follows. ?1 describes the well-known function spaces on Rn that are involved, as well as certain operators on them. ?2 describes the singular integral operators and their symbols. ?3 extends this collection to one that contains the differential operators on R, as well as the inverses of the invertible elliptic operators. The symbol u(A) of an operator A is defined, and the behavior of a under composition of operators is discussed. ?4 considers the behavior of a under coordinate changes. ?5 gives some necessary lemmas from functional analysis. ?6 establishes the notation for vector bundles, and the analogs for bundles of the function spaces of ?1. ?7 defines the singular integral operators on sections of a vector bundle E over a compact manifold X, and their symbols. If A is a singular integral operator from sections of one bundle E

We study the numerical approximation to a nonlocal Volterra integro‐differential equation, in which the integral term is the convolution product of a positive‐definite kernel and a nonlocal peridynamic differential operator (PDDO). Compared with the classical differential operators, the nonlocal PDDOs describe, for example, discontinuities and have demonstrated more widespread applications. The equation is discretized in space by the Galerkin finite element method, and we accordingly prove its error estimate. We then discretize the equation in time by the backward Euler method, and a positive quadrature rule is combined to approximate the convolution term. The convergence rate of the fully‐discrete finite element scheme is proved, and numerical experiments are carried out to substantiate the theoretical findings.

$G$-operators, a class of differential operators containing the differential operators of minimal order annihilating Siegel's $G$-functions, satisfy a condition of moderate growth called Galochkin condition, encoded by a $p$-adic quantity, the size. Previous works of Chudnovsky, Andr\'e and Dwork have provided inequalities between the size of a $G$ -operator and certain computable constants depending among others on its solutions. First, we recall Andr\'e's idea to attach a notion of size to differential modules and detail his results on the behavior of the size relatively to the standard algebraic operations on the modules. This is the corner stone to prove a quantitative version of Andr\'e's generalization of Chudnovsky's Theorem: for $f(z)=\sum_{\alpha, k,\ell} c_{\alpha, k,\ell} z^{\alpha} \log(z)^k f_{\alpha, k,\ell}(z)$, where $f_{\alpha, k,\ell}(z)$ are $G$-functions, we can determine an upper bound on the size of the minimal operator $L$ over $\overline{\mathbb{Q}}(z)$ of $f(z)$ in terms of quantities depending on the $f_{\alpha, k,\ell}(z)$, the rationals $\alpha$ and the integers $k$. We give two applications of this result: we estimate the size of a product of two $G$-operators in function of the size of each operator; we also compute a constant appearing in a Diophantine problem encountered by the author.

A linear differential operator P(D) = P(D 1, …, D n ) with constant coefficients is called almost hypoelliptic if all the derivatives D α P of the characteristic polynomial P(ξ 1, …, ξ n ) can be estimated by P. The paper proves that if P is an almost hypoelliptic operator and f is an infinitely differentiable function, square-summable with a definite exponential weight, then any square summable with the same weight solution u of the equation P(D)u = f is again an infinitely differentiable function and P(ξ) → ∞ as ξ → ∞.

A description is given of all matrix differential operators of the form d/dt + tkA, satisfying in L2(Rn) the coercive estimate $$\left\| {\frac{d}{{dt}}u} \right\| \le c\left\| {B\frac{d}{{dt}}u + t^k Au} \right\|$$ and the estimate $$\left\| u \right\| \le c\left\| {B\frac{d}{{dt}}u + t^k Au} \right\|.$$ .

We consider the problem of the a second order singular differential operator with complex coefficients in the discrete spectrum case. The Titchmarsh-Weyl m-function is constructed without the use of nesting circles, and it is then used to give a representation of the resolvent operator. Under conditions on the growth of the coefficients, the resolvent operator is proved to be Hilbert-Schmidt and the root subspaces are shown to be complete in the associated Hilbert space. The operator is considered on both the half line and whole line cases.

In this chapter we introduce some functional spaces, classes of differential operators and a transform that will play the role of the Fourier transform. We investigate the properties of the spaces and operators with respect to this transform.KeywordsPseudodifferential OperatorFundamental DomainTrivial BundleFloquet TheoryStein ManifoldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On the frequency mapping of modal parameters identification