Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions

We supplement recent results on a class of Bernstein–Durrmeyer operators preserving linear functions. This is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving the first-order and second-order moduli of smoothness. The results generalize and improve earlier statements for Bernstein and genuine Bernstein–Durrmeyer operators.Отримані нещодавно результати щодо одного класу операторів Бернштейна-Дуррмейєра, які зберігають лінійні функції, доповнено шляхом вивчення двох граничних випадків та доведення кількісних тверджень типу Вороновської, що містять модулі гладкості першого та другого порядків. Результати узагальнюють та покращують попередні твердження для операторів Бернпггейна та справжніх операторів Бернштейна - Дуррмейєра

Approximation by boolean sums of positive linear operators VI: monotone approximation and global smoothness preservation

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A new estimate for Hölder approximation by Bernstein operators

Abstract In this work we discuss the rate of simultaneous approximation of Hölder continuous functions by Bernstein operators, measured by Hölder norms with different exponents. We extend the known results on this topic

A class of discretely defined positive linear operators satisfying DeVore-Gopengauz inequalities

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Six topics on inscribable polytopes

Inscribability of polytopes is a classic subject but also a lively research area nowadays. We illustrate this with a selection of well-known results and recent developments on six particular topics related to inscribable polytopes. Along the way we collect a list of (new and old) open questions.Comment: 11 page

Recent progress on univariate and multivariate polynomial and spline quasi-interpolants

Polynomial and spline quasi-interpolants (QIs) are practical and effective approximation operators. Among their remarkable properties, let us cite for example: good shape properties, easy computation and evaluation (no linear system to solve), uniform boundedness independently of the degree (polynomials) or of the partition (splines), good approximation order. We shall emphasize new results on various types of univariate and multivariate polynomial or spline QIs, depending on the nature of coefficient functionals, which can be differential, discrete or integral. We shall also present some applications of QIs to numerical methods

The K+ Channel Opener 1-EBIO Potentiates Residual Function of Mutant CFTR in Rectal Biopsies from Cystic Fibrosis Patients

BACKGROUND: The identification of strategies to improve mutant CFTR function remains a key priority in the development of new treatments for cystic fibrosis (CF). Previous studies demonstrated that the K⁺ channel opener 1-ethyl-2-benzimidazolone (1-EBIO) potentiates CFTR-mediated Cl⁻ secretion in cultured cells and mouse colon. However, the effects of 1-EBIO on wild-type and mutant CFTR function in native human colonic tissues remain unknown. METHODS: We studied the effects of 1-EBIO on CFTR-mediated Cl⁻ secretion in rectal biopsies from 47 CF patients carrying a wide spectrum of CFTR mutations and 57 age-matched controls. Rectal tissues were mounted in perfused micro-Ussing chambers and the effects of 1-EBIO were compared in control tissues, CF tissues expressing residual CFTR function and CF tissues with no detectable Cl⁻ secretion. RESULTS: Studies in control tissues demonstrate that 1-EBIO activated CFTR-mediated Cl⁻ secretion in the absence of cAMP-mediated stimulation and potentiated cAMP-induced Cl⁻ secretion by 39.2±6.7% (P<0.001) via activation of basolateral Ca²⁺-activated and clotrimazole-sensitive KCNN4 K⁺ channels. In CF specimens, 1-EBIO potentiated cAMP-induced Cl⁻ secretion in tissues with residual CFTR function by 44.4±11.5% (P<0.001), but had no effect on tissues lacking CFTR-mediated Cl⁻ conductance. CONCLUSIONS: We conclude that 1-EBIO potentiates Cl⁻secretion in native CF tissues expressing CFTR mutants with residual Cl⁻ channel function by activation of basolateral KCNN4 K⁺ channels that increase the driving force for luminal Cl⁻ exit. This mechanism may augment effects of CFTR correctors and potentiators that increase the number and/or activity of mutant CFTR channels at the cell surface and suggests KCNN4 as a therapeutic target for CF