A property of approximation operators and applications to Tauberian constants (original) (raw)
Approximation operators and tauberian constants
Israel Journal of Mathematics, 1969
The explicit expression of the smallest constantC satisfying \mathop {lim}\limits_{\lambda \to \infty } \left| {t_{n(\lambda )}^{(1)} - t_{m(\lambda )}^{(2)} } \right| \leqq C. \mathop {lim sup}\limits_{n \to \infty } \left| {d_n } \right|$$ for all sequences {s n} satisfying lim supn→∞ |d n| <∞, where {t n (1)}, {t n (2)} are two generalised Hausdorff transforms of {sn }, {d n} is the generalised (C, α)-transform (0≦α≦1) of {λ na n} andn(λ, m(λ) are suitably related, is obtained. These results are obtained by using new properties of positive approximation operators and generalised Bernstein approximation operators.
J i = I 2i ∪ I 2i+1 , i = 1, 2, . . . , N (ε)/2, for even N (ε) and J i = I 2i ∪ I 2i+1 , i = 1, 2, . . . , (N (ε) − 3)/2, ε) for odd N (ε).
On a Theorem of Piatetsky-Shapiro and Approximation of Multiple Integrals
Mathematics of Computation, 1969
Let / be a function of s real variables which is of period 1 in each variable, and let the integral I of / over the unit cube in s-space be approximated by Qif) =-TF L /("0 (where x = x(A0 is a point in s-space). For certain classes of f's, defined by conditions on their Fourier coefficients, it is shown using methods of N. M. Korobov, that x's can be found for which error bounds of the form [Iif)-Q(f)\ < K0f)N~p will be true. However, for the class of all f's with absolutely convergent Fourier series, it is shown that there are no x's for which a bound of the form \I(f)-Qif)\ = OiFiN)) will hold, for any F(N) which approaches zero as N goes to infinity. ■