Best Uniform Rational Approximations of Functions by Orthoprojections

Let C[-1,1] be the Banach space of continuous complex functions $f$ on the interval [-1,1] equipped with the standard maximum norm $\left\| f \right\|$ ; let $\omega \left( \cdot \right) = \omega \left( { \cdot ,f} \right)$ be the modulus of continuity of $f$ ; and let $R_n = R_n \left( f \right)$ be the best uniform approximation of $f$ by rational functions (r.f.) whose degrees do not exceed $n = 1, 2, \ldots $ . The space C[-1,1] is also regarded as a pre-Hilbert space with respect to the inner product given by $\left( {f,g} \right) = \left( {1/\pi } \right)\int_{ - 1}^1 {f\left( x \right)g\left( x \right)} \left( {1 - x^2 } \right)^{ - 1/2} dx$ . Let $z_n = \{ z_1 , z_2 , \ldots z_n \} $ be a set of points located outside the interval [-1,1]. By $F\left( { \cdot ,f,z_n } \right)$ we denote an orthoprojection operator acting from the pre-Hilbert space C[-1,1] onto its ( ${n + 1}$ )-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set $z_n $ . In this paper, we show that if $f$ is not a rational function of degree $ \leqslant n$ , then we can find a set of points $z_n = z_n \left( f \right)$ such that $\left\| {f\left( \cdot \right) - F\left( { \cdot ,f,z_n } \right)} \right\| \leqslant 12R_n ln\frac{3}{{\omega ^{ - 1} \left( {R_n /3} \right)}}.$      

Interphase interactions in a Pd-GaAs system and their effect on electrical properties of schottky-barrier structures I. effect of heat treatment on characteristics of GaAs-Pd/Ni contacts

The electrical characteristics of surface-barrier GaAs-Pd/Ni structures and the physicochemical interaction processes at the metal-semiconductor boundary were comprehensively investigated in relation to heat treatment in various atmospheres. X-Ray structural analysis showed that in the investigated system metallurgical reactions begin at 300–350C: Unstable intermediate phases (presumably Pd₂Ga) are formed. At 400–550C all the palladium is converted to the bound state and the intermetallic compound PdGa is formed. The phase changes have no significant effect on the properties of diode structures fired in a hydrogen atmosphere. Heating in vacuum leads to degradation of the contact parameters at 300C or more. This effect is attributed to penetration of oxygen to the interface during formation of the intermediate phases.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 3–7, April, 1981