By Carothers N.L.
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Extra info for A short course on approximation theory (Math682)
Example For m > n, the best approximation to f (x) = A cos mx + B sin mx out of Tn is 0! Proof. We may write f (x) = R cos m(x ; x0 ) for some R and x0 . ) Now we need only display a su ciently large alternating set for f (in some interval of length 2 ). Setting xk = x0 + k =m, k = 1 2 : : : 2m, we get f (xk ) = R cos k = R(;1)k and xk 2 (x0 x0 + 2 ]. Since m > n, it follows that 2m 2n + 2. Example The best approximation to f (x) = a0 + out of Tn is T (x) = a0 + q2 nX +1; k=1 ak cos kx + bk sin kx n ; X k=1 ak cos kx + bk sin kx and kf ; T k = an+1 + b2n+1 in C 2 .
Indeed, the 2 -periodic continuous functions on R may be identi ed with the subspace of C 0 2 ] consisting of those f 's which satisfy f (0) = f (2 ). As an alternate description, it is often convenient to instead identify C 2 with the collection C (T), consisting of all continuous real-valued functions on T, where T is the unit circle in the complex plane C . In this case, we simply make the identi cations ! ei and f ( ) ! f (ei ): . 41. (a) By using the recurrence formulas cos kx + cos(k ; 2)x = 2 cos(k ; 1)x cos x and sin(k + 1)x ; sin(k ; 1)x = 2 cos kx sin x, show that each of the functions cos kx and sin(k + 1)x= sin x may be written as algebraic polynomials of degree exactly k in cos x.
F (x) at each point of continuity of f . 33. (Bohman, Korovkin) Let (Tn) be a sequence of monotone linear operators on C 0 1 ] that is, each Tn is a linear map from C 0 1 ] into itself satisfying Tn(f ) Tn (g) whenever f g. Suppose also that Tn(f0 ) f0, Tn(f1 ) f1 , and Tn(f2 ) f2 . Prove that Tn(f ) f for every f 2 C 0 1 ]. ] 34. Find Bn(f ) for f (x) = x3 . ] The same method of calculation can be used to show that Bn(f ) 2 Pm whenever f 2 Pm and n > m. Polynomials 37 35. Let f be continuously di erentiable on a b ], and let " > 0.
A short course on approximation theory (Math682) by Carothers N.L.