By G.A. Watson
ISBN-10: 3540076107
ISBN-13: 9783540076100
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Example text
On letting R → ρ, we get the required result. 6. Birkhoff Interpolation When we state an interpolation problem where not necessarily consecutive higher order derivatives are prescribed, then we call this lacunary (or HermiteBirkhoff, or simply Birkhoff) interpolation. This type of interpolation was introduced by G. Birkhoff [5] at the beginning of the 19th century, but remained unnoticed for a long time. After the pioneering work of P. Tur´ an in the late 50’s, the theory of lacunary interpolation gained new momentum, and nowadays it is a fruitful field in approximation theory.
35 Theorem 3. 1) tends to zero as n → ∞, when α,β (f ; z), Hpn,rn |z| < ρ max |α| |β| , ρ ρ r−p+1 p . A different, but similar result is the following. If r, are positive integers, m = nq + c, and q ≥ r is a given positive integer, we set ∆α,β rn−1, ,m (f ; z) := hr,rn−1 (f ; α, z) − hr,rn−1 h , m−1 (f ; β, z); α, z . 5) We shall now prove Theorem 4. Let r, , m be positive integers, m = nq + c, q ≥ r and c a constant. 6) where |β| σ := ρ/ max ρ q r |β| ρ , ( −1)q r |α| ρ q+1−r r |α| ρ , q+1−r r .
Proof. , K(t, z) = hp,pn−1 Λ(·); α, z . Here Λ(z) is given by trn − z rn 1 (tn − αn )p − (z n − αn )p − rn · n n p (t − α ) (t − z) t − β rn t − z z rn − β rn (z n − αn )p = rn − n /(t − z) = Λ1,n (z) − Λ2,n (z), rn t −β (t − αn )p Λ(z) = 32 WALSH EQUICONVERGENCE OF COMPLEX INTERPOLATING . . where we set z rn (tn − αn )p − (z n − αn )p trn , (t − z)(trn − β rn )(tn − αn )p β rn [(tn − αn )p − (z n − αn )p ] Λ2,n (z) = . (t − z)(trn − β rn )(tn − αn )p Λ1,n (z) = Since Λ2,n (z) ∈ πnp−1 , we have hp,pn−1 (Λ2,n ; α, z) = Λ2,n (z).
Numerical Analysis by G.A. Watson
by Ronald
4.1