We investigate polynomial approximations to functionswhere D is a nonempty compact subset of IR r , rEIN, preferably in the uniform norm, but occasionally also in the quadratic average norm. The function is called multivariate, if r~2. C(D) denotes the space of all continuous functions (1.1) which is provided with the norm 1IFIloo := IIFIID := max{lF(x)I : XED}.Our particular interest is directed to the case where D is one of the following sets:sr := {x E IR r Ilxl::; I}, sr-l := {x E IR r Ilxl = I}, E r := {x E IR r Ix~0, Xl + + X r ::; I}, r-l := {x E IR r Ix~0, Xl + + X r = I}.

dedicated to professor dr. dr. h. c. karl zeller on the occasion of his 75th birthdayWe investigate hyperinterpolation operators based on positive weighted quadrature rules, as introduced by Ian H. Sloan. If the rules are exact of double degree then, independently of the number of their nodes, the operator norms increase at the order of the minimal projections. Academic Press

The use of f i n i t e d i f f e r e n c e methods f o r t h e numerical treatment of i n i t i a l value problems depends on two concepts, namely s t a b i l i t y and degree of approximation. The l a t t e r can be improved s i g n i f i c a n t l y i f d e r i v a t i v e s of higher order a r e used.I n t h i s r e p o r t , the s t a b i l i t y problem i s solved f o r such generalized f i n i t e d i f f e r e n c e schemes,

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