The zero sets of (D + a)g(t) with D = d/dt in the (t, a)plane are investigated for g(t) = te(e−1) and g(t) = e(e+1). The results are used to determine entire interpolations to functions xn+e , which give representations for the best approximation and best one-sided approximation from the class of functions of exponential type η > 0 to xn+e .

As usual, p∗ is called a best approximation (b.a.) to f in (or, by elements of) IPγ,n. To give some examples, let X = Lp[0, 1] and set γ(t) = G(·, t), where G(s, t) is defined on [0, 1] × T . With G Green’s function for a k–th order ordinary linear initial value problem on (0, 1] and T = [0, 1), one has approximation by generalized splines. With G(s, t) = e and T = IR, one has approximation by ...

Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in the approximant are required to align themselves in a certain connected tree sturcture. This makes their positions eas...

The hp-adaptive approximation is formulated as an approximation problem on a full binary tree T , where for each of the leaves ∆ an order p(∆) ≥ 1 is assigned in such a way that the sum of all such p(∆) does not exceed N , called complexity of the approximation. The leaves ∆ correspond to the cells of the partition, while p(∆) is the order of the polynomial used for the local approximation on ∆...