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Ridge approximation, Chebyshev-Fourier analysis and optimal quadrature formulas

A 1997 Preprint by K. Oskolkov

  • 1997:09
  • Free (non linear) ridge $L^2$ approximation $NRA _ n(f),n=1,2,\ldots,$ of a function $f(x)=f(x _ 1,x _ 2)$ in the unit disc $I\!\!B^2$ is considered:

    $\left\|f-\sum^n _ 1F _ j(x\cdot\xi _ j);L^2(I\!\!B^2)\right\|\Longrightarrow\inf$ in $\{F _ j(t)\}^n _ 1$ and $\{\xi _ j\}^n _ 1\subset{S^1}$ ,

    Where $\{F _ j(t)\}^n _ 1$ denotes a set of n single variate functions. Geometrically, the $NRA$ problem means approximation of f by a linear combination of n planar waves of arbitrary shapes $F _ j$ and directions of propagation (wave vectors) $\xi _ j$. A duality relation is established between the $NRA$ problem and that of optimal quadrature formulas, in the sense of Kolmogorov – Nikol’skii, for classes of trigonometric polynomials. On the base of this duality and lower estimates of errors of quadrature formulas, it is proved that if $f(x)$ is radial, $f(x)=f(|x|)$, then algebraic polynomials in two variables provide “almost best” tool for ridge approximation:

    $\frac{1}{c _ 0}PA _ {3n}(f)\leq{NRA _ n}(f)\leq{PA _ {n-1}(f),\ \ \ n=1,2,\ldots,$

    Where $c _ 0$ is an absolute positive constant, and $PA _ n(f)$ denotes the n-th best algebraic polynomial approximation of f in $L^2(I\!\!B^2)$:

    $PA _ n(f):=\min _ {p(x)\in{P}^2 _ n}\left\|f-p;L^2(I\!\!B^2)\right\|;\ \ \ P _ n^2:=S\!p\!a\!n\left\{x^k _ 1x^l _ 2\right\} _ {k+l\leq{n}}$ .

    It is known that algebraic polynomials of degree n in two variables can be represented as linear combinations of $n+1$ planar wave polynomials. Radon – Fourier analysis via Chebyshev ridge polynomials is crucial in the proof.

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