Function Values Are Enough for $$L_2$$-Approximation

Abstract

We study the $$L_2$$ L 2 -approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number $$e_n$$ e n is the minimal worst-case error that can be achieved with n function values, whereas the approximation number $$a_n$$ a n is the minimal worst-case error that can be achieved with n pieces of arbitrary linear information (like derivatives or Fourier coefficients). We show that $$\begin{aligned} e_n \,\lesssim \, \sqrt{\frac{1}{k_n} \sum _{j\ge k_n} a_j^2}, \end{aligned}$$ e n ≲ 1 k n ∑ j ≥ k n a j 2 , where $$k_n \asymp n/\log (n)$$ k n ≍ n / log ( n ) . This proves that the sampling numbers decay with the same polynomial rate as the approximation numbers and therefore that function values are basically as powerful as arbitrary linear information if the approximation numbers are square-summable. Our result applies, in particular, to Sobolev spaces $$H^s_\mathrm{mix}(\mathbb {T}^d)$$ H mix s ( T d ) with dominating mixed smoothness $$s>1/2$$ s > 1 / 2 and dimension $$d\in \mathbb {N}$$ d ∈ N , and we obtain $$\begin{aligned} e_n \,\lesssim \, n^{-s} \log ^{sd}(n). \end{aligned}$$ e n ≲ n - s log sd ( n ) . For $$d>2s+1$$ d > 2 s + 1 , this improves upon all previous bounds and disproves the prevalent conjecture that Smolyak’s (sparse grid) algorithm is optimal.

$$L_2$$-norm sampling discretization and recovery of functions from RKHS with finite trace

“…1] by obtaining the worst-case bound o( √ log n/n) in case of square summable singular values (σ k ) k (finite trace) of the embedding. It seems that, in general, their decay influences the bounds rather weakly (in contrast to the results in [11,13,18]).…”

“…This paper can be seen as a continuation of [11,13]. We study the reconstruction of complex-valued multivariate functions on a domain D ⊂ R d from values at the (randomly sampled) nodes X := (x 1 , .…”

“…In addition, we are interested in the sampling discretization of the squared L 2 -norm of such functions using n random nodes. Both problems recently gained substantial interest, see [11,13,14,[25][26][27]29], and are strongly related as we know from Wasilkowski Communicated by Deguang Han.…”

“…In this paper, the functions are modeled as elements from some reproducing kernel Hilbert space H (K ), which is supposed to be compactly embedded into L 2 (D, D ). Its kernel is a positive definite Hermitian function K : D × D → C. In the papers [11,13,18] authors mainly restrict to the case of separable RKHS [11,18] or Mercer kernels on compact domains [13] with finite trace property to study the quantity sup…”

“…As for the sampling recovery problem we start with a result in the most general situation. A modification of the recovery operator S m X from [11,13], see Algorithm 1 below, has been used to study the situation which is left as an open problem in [11]. The result reads as follows.…”