Linear and nonlinear methods of relief approximation (Q2519277)

The effectiveness of free (nonlinear) relief approximation, equidistant reflief approximation, and polynomial approximaton \({\mathcal R}^{\text{fr}}_N[f]\), \({\mathcal R}^{\text{eq}}_N[f]\), and \({\mathcal E}_N[f]\) of an individual function \(f(x)\) in the metric \({\mathcal L}^2(\mathbb{B}^2)\), where \(B^2\) is the unit ball \(|x|\leq 1\) in the plane \(\mathbb{R}^2\) is compared. The notation is the following: \[ R^{\text{fr}}_N[f]:= \underset{R\in{\mathcal W}^{\text{fr}}_N}{}{\text{inf}}\| f-R\|,\quad R^{\text{eq}}_N[f]:= \min_{R\in{\mathcal W}^{\text{eq}}_N}\| f-R\|,\quad {\mathcal E}_N[f]:= \min_{P\in{\mathcal P}^2_{N-1}}\| f- R\|. \] Here \({\mathcal W}^{\text{fr}}_N\) is the set of all \(N\)-term linear combinations of functions of the plane-wave type \[ R(x)= \sum^N_1 W_j(x\cdot\theta_j) \] with arbitrary profiles \(W_j(x)\), \(x\in\mathbb{R}^1\), and transmission directions \(\{\theta_j\}^N_1\); \({\mathcal W}^{\text{eq}}_N\) is the subset of \({\mathcal W}^{\text{fr}}_N\) associated with \(N\) equidistant directions; \[ {\mathcal P}^2_{N-1}:= \text{Span}\{x^k_1 x^l_2\}_{k+l<N} \] denotes the subspace of algebraic polynomials of degree less than or equal to \(N-1\) in two real variables. Obviously, the inequalities \({\mathcal R}^{\text{eq}}_N[f]\leq{\mathcal E}_N[f]\) hold. The following model problem is stated. What are the functions which satisfy the relation \({\mathcal R}^{\text{fr}}_N[f]= o({\mathcal R}^{\text{eq}}_N[f])\), i.e., where is the nonlinear approximation \({\mathcal R}^{\text{fr}}\) more effective than a linear one? This effect has been proved for harmonic functions, namely, for any \(\varepsilon> 0\) there exists \(c_\varepsilon>0\) such that if \(\Delta f(x)= 0\), \(|x|< 1\), and \(f\in{\mathcal L}^2(\mathbb{B}^2)\), then \[ {\mathcal R}^{\text{fr}}_N[f]\leq c_\varepsilon({\mathcal R}^{\text{eq}}_N[f]\exp(- N^\varepsilon)+{\mathcal R}^{\text{eq}}_{N^{2-3\varepsilon}}[f]). \] On the other hand, \({\mathcal R}^{\text{fr}}_N[f]\geq {1\over c}{\mathcal R}^{\text{eq}}_{N^2}[f]\). Thus, \({\mathcal R}^{\text{fr}}_{N^2}[f]\) has an ``almost squared effectiveness' of \({\mathcal R}^{\text{eq}}_N[f]\) for \(f= f_{\text{harm}}\). However, this ultra-high order of approximation is obtained via a collapse of wave vectors. On the other hand, the nonlinearity of \({\mathcal R}^{\text{fr}}\) 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 \(f(x)= f(|x|)\), then \[ {\mathcal E}_{2N}[f]\geq {\mathcal R}^{eq}_N[f]\geq \sqrt{{2\over 3}}{\mathcal E}_{2N}(f)\text{ and }{\mathcal R}^{\text{fr}}_N[f]\geq \sup_{\varepsilon> 0} \sqrt{{\varepsilon\over 3(1+\varepsilon)}}{\mathcal R}^{eq}_{(1+\varepsilon)N}[f]. \] The technique of the Fourier-Chebyshev analysis (which is related to the inverse Radon transform on \(\mathbb{B}^2\)) and a duality between the relief approximation problem and the optimization of quadrature formulas in the sense of Kolmogorov and \textit{S. M. Nikolskii} [Quadrature Formulas, Nauka, Moscow (1974)] for trigonometric polynomial classes are used.