Generalized Christoffel functions and error of positive quadrature

The author gives some upper and lower bounds for the generalized Christoffel functions related to a Ditzian-Totik generalized weight. As an application, an error estimate of Gauss quadrature formula inL ₁-weighted norm is derived.Dedicated to Prof. Luigi Gatteschi on the occasion of his 70th birthdayWork sponsored by MURST 40%.     

Weighted approximation of functions on the real line by Bernstein polynomials

The authors give error estimates, a Voronovskaya-type relation, strong converse results and saturation for the weighted approximation of functions on the real line with Freud weights by Bernstein-type operators.     

Vilsmeier formylation of 5,10,15-triphenylcorrole: expected and unusual products

5,10,15-Triphenylcorrole (1) reacts with the Vilsmeier reagent (POCl(3)/DMF) to give the corresponding 3-formyl derivative 3 as the major product. The regioselectivity of the reaction was proven by X-ray crystallography and only traces of the 2-formyl isomer were observed. A more polar product is also observed and this compound becomes the major product when an excess of DMF is used for the preparation of the Vilsmeier reagent, while the formation of the 3-formyl isomer is almost completely suppressed. X-ray crystallography allowed us to identify this compound as the fully substituted N-ethane bridged derivative 4, formed from the attack of the Vilsmeier reagent at the inner core of the macrocycle. This compound is unique among porphyrinoid macrocycles, and further confirms the peculiarity of corrole chemistry.     

Enhanced ethanol sensing properties of WO3 modified TiO2 nanorods

Pristine and WO₃ decorated TiO₂ nanorods (NRs) were synthesised to investigate n-n-type heterojunction gas sensing properties. TiO₂ NRs were fabricated via hydrothermal method on fluorine-doped tin oxide coated glass (FTO) substrates. Then, tungsten was sputtered on the TiO₂ NRs and thermally oxidised to obtain WO₃ nanoparticles. The heterostructure was characterised by X-ray diffraction (XRD), scanning electron microscopy (SEM), and energy-dispersive X-ray (EDX) spectroscopy. Fabricated sensor devices were exposed to VOCs such as toluene, xylene, acetone and ethanol, and humidity at different operation temperatures. Experimental results demonstrated that the heterostructure has better sensor response toward ethanol at 200 °C. Enhanced sensing properties are attributed to the heterojunction formation by decorating TiO₂ NRs with WO₃.     

Lagrange Interpolation at Laguerre Zeros in Some Weighted Uniform Spaces

We introduce an interpolatory process essentially based on the Laguerre zeros and we prove that it is an optimal process in some weighted uniform spaces.     

Saturation of the Shepard operators

We discuss the Shepard operators S_n (f; x) in this paper and establish the saturation of the sequence {S_n f} _n-1 ^∞ , as well as investigate some related questions. The research of this author was supported in part by the Hungarian Science Foundation for Research, Grant. N ^o . 1157.     

A converse result for approximation by a weighted Szász-Mirakyan operator

We prove that the weighted error of approximation by the Szász-Mirakyan-type operator introduced in [1] is equivalent to the modulus of smoothness of the function. This result is analogous to previous results for Bernstein-type operators obtained by Ditzian-Ivanov and Szabados. Research supported by Hungarian Scientific Research Fund (OTKA), Grant No. T-049196.     

One-sided convergence conditions for Hermite-Fejér interpolation of higher order of Lagrange type

The authors investigate the Hermite-Fejér interpolation of higher order of Lagrange type for continuous functions on the Jacobi abscissas. A uniform convergence theorem is stated, generalizing a previous result for Lagrange interpolation.