Abstract
AbstractFor a fixed positive integer n consider continuous functions $$K_1,\dots $$
K
1
,
⋯
, $$ K_n:[-1,1]\rightarrow {{\mathbb {R}}}\cup \{-\infty \}$$
K
n
:
[
-
1
,
1
]
→
R
∪
{
-
∞
}
that are concave and real valued on $$[-1,0)$$
[
-
1
,
0
)
and on (0, 1], and satisfy $$K_j(0)=-\infty $$
K
j
(
0
)
=
-
∞
. Moreover, let $$J:[0,1]\rightarrow {{\mathbb {R}}}\cup \{-\infty \}$$
J
:
[
0
,
1
]
→
R
∪
{
-
∞
}
be upper bounded and such that $$[0,1]\setminus J^{-1}(\{-\infty \})$$
[
0
,
1
]
\
J
-
1
(
{
-
∞
}
)
has at least $$n+1$$
n
+
1
elements, but it is arbitrary otherwise. For $$x_0:=0<x_1<\dots < x_n \le x_{n+1}:=1$$
x
0
:
=
0
<
x
1
<
⋯
<
x
n
≤
x
n
+
1
:
=
1
, so called nodes, and for $$t\in [0,1]$$
t
∈
[
0
,
1
]
consider the sum of translates function $$F(x_1,\ldots ,x_n,t):=J(t)+\sum _{j=1}^n K_j(t-x_j)$$
F
(
x
1
,
…
,
x
n
,
t
)
:
=
J
(
t
)
+
∑
j
=
1
n
K
j
(
t
-
x
j
)
, and the vector of interval maximum values $$m_j:=m_j(x_1,\ldots ,x_n):=\max _{t\in [x_j,x_{j+1}]}F(x_1,\ldots ,x_n,t)$$
m
j
:
=
m
j
(
x
1
,
…
,
x
n
)
:
=
max
t
∈
[
x
j
,
x
j
+
1
]
F
(
x
1
,
…
,
x
n
,
t
)
($$j=0,1,\ldots ,n$$
j
=
0
,
1
,
…
,
n
). We describe the structure of the arising interval maxima as the nodes run over the n-dimensional simplex. Applications presented here range from abstract moving node Hermite–Fejér interpolation for generalized algebraic and trigonometric polynomials via Bojanov’s problem to more abstract results of interpolation theoretic flavour.
Funder
DAAD-TKA
Nemzeti Kutatási, Fejlesztési és Innovaciós Alap
Publisher
Springer Science and Business Media LLC