Affiliation:
1. Faculty of Cybernetics , Military University of Technology , Radiowa 22, 01-485 Warsaw , Poland
Abstract
Abstract
It it well known that a Hilbert space V of functions defined on U is a reproducing kernel Hilbert space if and only if for any
z
∈
U
{z\in U}
, in the set
V
z
:=
{
f
∈
V
∣
f
(
z
)
=
1
}
{V_{z}:=\{f\in V\mid f(z)=1\}}
, if non-empty, there is exactly one element with minimal norm and there is a direct connection between the reproducing kernel and such an element. In this paper, we define reproducing kernel Banach space as a space which satisfies this property and the reproducing kernel of it using this relation. We show that this reproducing kernel share a lot of basic properties with the classical one. The notable exception is that in Banach spaces the equality
K
(
z
,
w
)
=
K
(
w
,
z
)
¯
{K(z,w)=\overline{K(w,z)}}
does not have to be true without assumptions that
K
(
z
,
w
)
≠
0
,
K
(
w
,
z
)
≠
0
{K(z,w)\neq 0,K(w,z)\neq 0}
. We give sufficient and necessary conditions for a Banach space of functions to be a reproducing kernel Banach space. At the end, we give some examples including ones which show how reproducing kernel Banach spaces can be used to solve extremal problems of Partial Differential Equations Theory.
Reference7 articles.
1. B.-Y. Chen and S. Fu,
Comparison of the Bergman and Szegö kernels,
Adv. Math. 228 (2011), no. 4, 2366–2384.
2. L. C. Evans,
Partial Differential Equations,
Grad. Stud. Math. 19,
American Mathematical Society, Providence, 1998.
3. J. Gipple,
The volume of n-balls,
Rose-Hulman Undergrad. Math. J. 15 (2014), no. 1, 237–248.
4. S. G. Krantz,
Function Theory of Several Complex Variables,
AMS Chelsea, Providence, 2001.
5. R. R. Lin, H. Z. Zhang and J. Zhang,
On reproducing kernel Banach spaces: Generic definitions and unified framework of constructions,
Acta Math. Sin. (Engl. Ser.) 38 (2022), no. 8, 1459–1483.