AbstractWe provide a new upper bound for sampling numbers $$(g_n)_{n\in \mathbb {N}}$$
(
g
n
)
n
∈
N
associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants $$C,c>0$$
C
,
c
>
0
(which are specified in the paper) such that $$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$
g
n
2
≤
C
log
(
n
)
n
∑
k
≥
⌊
c
n
⌋
σ
k
2
,
n
≥
2
,
where $$(\sigma _k)_{k\in \mathbb {N}}$$
(
σ
k
)
k
∈
N
is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding $$\mathrm {Id}:H(K) \rightarrow L_2(D,\varrho _D)$$
Id
:
H
(
K
)
→
L
2
(
D
,
ϱ
D
)
. The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of $$H^s_{\text {mix}}(\mathbb {T}^d)$$
H
mix
s
(
T
d
)
in $$L_2(\mathbb {T}^d)$$
L
2
(
T
d
)
with $$s>1/2$$
s
>
1
/
2
. We obtain the asymptotic bound $$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$
g
n
≤
C
s
,
d
n
-
s
log
(
n
)
(
d
-
1
)
s
+
1
/
2
,
which improves on very recent results by shortening the gap between upper and lower bound to $$\sqrt{\log (n)}$$
log
(
n
)
. The result implies that for dimensions $$d>2$$
d
>
2
any sparse grid sampling recovery method does not perform asymptotically optimal.
On Polynomial Approximation of Square Integrable Functions on a Subarc of the Unit Circle
