Linear and nonlinear methods of relief approximation

In this paper, we compare the effectiveness of free ( nonlinear ) relief approximation, equidistant relief approximation, and polynomial approximation {ie129-01}, and {ie129-02} of an individual function ƒ( x ) in the metric {ie129-03}, where {ie129-04} is the unit ball | x | ≤ 1 in the plane ℝ 2 . The notation we use is the following: {fx129-01}. Here {ie129-05} is the set of all N -term linear combinations of functions of the plane-wave type {fx129-02} with arbitrary profiles W j ( x ), x ∈ ℝ 1 and transmission directions { θ j } 1 N ; {ie129-06} is the subset of {ie129-07} associated with N equidistant directions; {fx129-03} denotes the subspace of algebraic polynomials of degree less than or equal to N − 1 in two real variables. Obviously, the inequalities {ie129-08} hold. We state the following model problem. What are the functions which satisfy the relation {ie129-09}, i.e., where the nonlinear approximation {ie129-10} is more effective than a linear one? This effect has been proved for harmonic functions, namely, for any ε > 0 there exists c ε > 0 such that if Δƒ( x ) = 0, | x | < 1, and ƒ ∈ {ie129-11}, then {fx129-04}. On the other hand, {ie129-12}. Thus, {ie129-13} has an “almost squared effectiveness” of {ie129-14} for ƒ = ƒ harm . However, this ultra-high order of approximation is obtained via a collapse of wave vectors. On the other hand, the nonlinearity of {ie129-15} which corresponds to the freedom of choice of wave vectors does not much improve the order of approximation, for instance, for all the radial functions. If {ie129-16}, then {ie129-17} and {ie129-18}. The technique we use is the Fourier-Chebyshev analysis (which is related to the inverse Radon transform on {ie129-19}) and a duality between the relief approximation problem and the optimization of quadrature formulas in the sense of Kolmogorov-Nikolskii [14] for trigonometric polynomial classes.