By Siemons J.
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Additional resources for 2-Designs and a differential equation
3 The graphics of functions √ x → f (x) = | sin πx/2| and x → P ∗ (x) for n = 6 (dashed line) and n = 12 (solid line) Fig. 1779. 1260. The corresponding graphics are presented in Figs. 4. 4 Some details on the first Remez algorithm can be found in [397, pp. 10–12]. Several modifications of this algorithm for solving linear and nonlinear Chebyshev approximation problems on compact B ⊂ Rs , as well as their convergence, were studied by Reemtsen . 2) k=1 and Fn (θ ) = 1 n+1 n k=0 which are known as the Dirichlet kernel and the Fejér kernel, respectively.
Kj − 1) we have the lacunary interpolation or Birkhoff interpolation (cf.  and ). 1 Representations and Computation of Interpolation Polynomials In Sect. 2 we have already introduced a polynomial of minimal degree which interpolates a function f at n fixed points x1 , x2 , . . , xn . 4 Interpolation by Algebraic Polynomials 49 as the Lagrange interpolation polynomial and it was discovered in 1795 by Joseph Louis Lagrange. Otherwise, interpolation by polynomials is a very old subject in mathematics.
In this case we have the Lp -Zygmund-Hölder spaces (see ) p Br,∞ = f ∈ Lp sup t>0 ωk (f, t)p < +∞, k > r , tr or equivalently p Er,∞ = f ∈ Lp sup i r Ei∗ (f )p < +∞ . i≥1 Another interesting case, which is useful in several applications, is the case when p = q = 2. Let f ∈ L2 be an arbitrary 2π -periodic function and c0 = a0 , 2 ci2 = ci2 (f ) = ai2 + bi2 (i = 1, 2, . 21), respectively). Then, for all r ≥ 0, we define the space +∞ L2r : = f ∈ L2 ci 2 (1 + i)2r < +∞ , i=0 equipped with the norm 1/2 +∞ f L2r := ci (1 + i) 2r 2 .