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- W2364193864 abstract "Let g∈C~q[-1, 1] be such that g~((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomial of degree at most n, such that P_n~((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivatives P_n~((k)) for k≤q well approximate g and its respective derivatives, provided only that P_n well approxi- mates g itself in the weighted norm ‖g(x)-P_n(x) (1-x~2)~(1/2)~q‖ This result is easily extended to an arbitrary f∈C~q[-1, 1], by subtracting from f the polynomial of minnimal degree which interpolates f~((0))…,f~((q)) at±1. As well as providing easy criteria for judging the simultaneous approximation properties of a given Polynomial to a given function, our results further explain the similarities and differences between algebraic polynomial approximation in C~q[-1, 1] and trigonometric polynomial approximation in the space of q times differentiable 2π-periodic functions. Our proofs are elementary and basic in character, permitting the construction of actual error estimates for simultaneous approximation proedures for small values of q." @default.
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- W2364193864 date "1994-01-01" @default.
- W2364193864 modified "2023-09-26" @default.
- W2364193864 title "ON APPROXIMATION OF A FUNCTION AND ITS DERIVATIVES BY A POLYNOMIAL AND ITS DERIVATIVES" @default.
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