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Constructive sparse trigonometric approximation for functions with small mixed smoothness. (English) Zbl 1373.42009
The paper is concerned with the order of approximation by trigonometric polynomials in \(L_p\)-norms for classes of functions satisfying some smoothness conditions. The best \(m\)-approximation of a function \(f\) by trigonometric polynomials is defined by \(\sigma_m(f)_p=\inf\{\|f-\sum_{j=1}^mc_j e^{i(\boldsymbol{k}_j,\boldsymbol{x})}\|_p: c_j\in\mathbb{R},\, \boldsymbol{k}_j\in\mathbb{Z}^d\}\) and, for a class \(\boldsymbol{W}\) of functions one puts \(\sigma_m(\boldsymbol{W})_p=\sup\{\sigma_m(f)_p : f\in\boldsymbol{W}\}.\)
For \(r>0\) and \(\boldsymbol{x}=(x_1,\dots,x_d)\in\mathbb{R}^d,\) let \(\boldsymbol{F}_r(\boldsymbol{x})=\prod_{j=1}^dF_r(x_j),\) where \(F_r(t)=1+2\sum_{k=1}^\infty k^{-r}\cos(kt-r\pi/2), \, t\in\mathbb{R}\). The class of functions \(\boldsymbol{W}^r_q\) is defined by \(\boldsymbol{W}^r_q= \{f : f=\varphi * \boldsymbol{F}_r,\, \|\varphi\|_q\leq 1\}, \) the norm of a function \(f=\varphi * \boldsymbol{F}_r\) in \(\boldsymbol{W}^r_q\) being \(\|f\|_{\boldsymbol{W}^r_q}=\|\varphi\|_q.\)
In the present paper one determines the order of approximation by trigonometric polynomials in classes \(\boldsymbol{W}^r_q\) for small order of smoothness. For instance, for \(1< q\leq 2<p,\, \beta=1/p-1/q,\,\beta p'<r<1/q,\) where \(p'=p/(p-1),\) the following evaluation holds: \[ \sigma_m\left(\boldsymbol{W}^r_q\right)_p\asymp m^{-(r-\beta)p/2}\left(\log m\right)^{(d-1)(r(p-1)-\beta p)}\,. \] The upper bounds are achieved by a constructive greedy-type algorithm (Theorem 1.2).
The results from this paper complements those obtained in [the author, Sb. Math. 206, No. 11, 1628–1656 (2015; Zbl 1362.41009); translation from Mat. Sb. 206, No. 11, 131–160 (2015)] for large smoothness: \(1\leq q\leq 2<p<\infty\) and \(r>1/q\).

MSC:
42B05 Fourier series and coefficients in several variables
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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