Abstract
AbstractIt is shown that already the knowledge of the saturation class $$\, {{\mathfrak {F}}}(J_t, L^p) $$
F
(
J
t
,
L
p
)
of a saturated convolution approximation process $$\, \{J_t\}$$
{
J
t
}
on $$\, L^p({{\mathbb {R}}}^n),\, 1\le p< \infty ,$$
L
p
(
R
n
)
,
1
≤
p
<
∞
,
completely determines its norm approximation behavior. This is achieved by using the $$\, K$$
K
-functional $$\, K(t,f;L^p,{{\mathfrak {F}}}(J_t, L^p)) $$
K
(
t
,
f
;
L
p
,
F
(
J
t
,
L
p
)
)
as a comparison scale, which relates on the one hand the approximation process and on the other appropriate moduli of smoothness. This implies that simultaneously one gets the so-called direct and inverse theorems. There are open problems if the saturation order is slightly perturbed, e.g., by a $$\, |\log t|^\lambda $$
|
log
t
|
λ
-factor, $$\, \lambda >0.$$
λ
>
0
.
Proofs are mainly based on the Fourier transformation.
Funder
Technische Universität Darmstadt
Publisher
Springer Science and Business Media LLC
Subject
Computational Mathematics,Radiology, Nuclear Medicine and imaging,Signal Processing,Algebra and Number Theory,Analysis
Reference24 articles.
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