Uniform approximation by positive operators on infinite intervals

В работе устанавливае тся оценка (*) |L _n (???| ≦Kω _? (?;α _n) для положительных оп ераторов, определенн ых на конечном или бесконе чном интервале (a,b), гдеL _n(1,χ)≡1,L _n((t?χ)²;χ)≦K? ²(χ)α _n ² (x∈(a,b)) ;и \(\omega _\varphi (f;\delta ) = \mathop {\sup }\limits_{0 \leqq h \leqq \delta ,x \pm h\varphi (x) \in (a,b)} \left| {f(x - h\varphi (x)) - 2f(x) + f(x + h\varphi (x))} \right|\) модуль гладкости?, св язанный с ? (функция? удовлетворяет некот орым условиям регуля рности). С помощью (*) для некотор ых {L _n } получена характеристика тех ф ункций?, для которыхL _n (?)??=o(1) равном ерно на (a, b). Наконец, рассматриваются слу чай насыщения и случай так называем ой неоптимальной апп роксимации. Результаты применяю тся к операторам Саса —Миракяна, Баскакова, Мейер-Кëни га и Целлера, гамма и бета операторам, а также к н екоторым операторам типа свер тки.     

On Markoff's Inequality

For compact subsets of the real line the Markoff factors increase at least as fast as n ² . In this paper it is shown that there are sets of measure zero with Markoff factors of order n ² . We shall also show that this cannot happen for compact sets of logarithmic capacity zero. In connection with Markoff's inequality for polynomials of two variables we show a set E

Saturation of Kantorovich type operators

It is proved that an integrable functionf can be approximated by the Kantorovich type modification of the Szász—Mirakjan and Baskakov operators inL ¹ metric in the optimal order {n ^–1} if and only if ² f is of bounded variation where and , respectively.     

Ingham-Beurling type theorems with weakened gap conditions

Completing a series of works begun by Wiener [34], Paley and Wiener [28] and Ingham [9], a far-reaching generalization of Parseval"s identity was obtained by Beurling [4] for nonharmonic Fourier series whose exponents satisfy a uniform gap condition. Later this gap condition was weakened by Ullrich [33], Castro and Zuazua [5], Jaffard, Tucsnak and Zuazua [11] and then in [2] in some particular cases. In this paper we prove a general theorem which contains all previous results. Furthermore, applying a different method, we prove a variant of this theorem for nonharmonic Fourier series with vector coefficients. This result, partly motivated by control-theoretical applications, extends several earlier results obtained in [15] and [2]. Finally, applying these results we obtain an optimal simultaneous observability theorem concerning a system of vibrating strings. This revised version was published online in June 2006 with corrections to the Cover Date.     

Sharpening of Hilbert’s lemniscate theorem

In this paper, we extend Hilbert’s lemniscate theorem to touching systems of curves. The result allows finding sharp constants in Bernstein type inequalities. Supported by OTKA TS44782. Supported by NSF grant DMS-0097484 and by OTKA T/034323, TS44782.     

Summability in the classes Hω

Mathematical Notes -     

Rapid analysis of isomeric exogenous metabolites by differential mobility spectrometry-mass spectrometry

The direct separation of isomeric glucuronide metabolites from propranolol dosed tissue extracts by differential mobility spectrometry-mass spectrometry (DMS-MS) with the use of the polar gas-phase chemical modifier acetonitrile was demonstrated. The DMS gas-phase separation was able to resolve the isomeric metabolites with separation times on the order of milliseconds instead of minutes which is typically required when using pre-ionization chromatographic separation methods. Direct separation of isomeric metabolites from the complex tissue extract was confirmed by implementing a high-performance liquid chromatography (HPLC) separation prior to the DMS-MS analysis to pre-separate the species of interest. The ability to separate isomeric exogenous metabolites directly from a complex tissue extract is expected to facilitate the drug development process by increasing analytical throughput without the requirement for pre-ionization cleanup or separation strategies.