A homeomorphism theorem for sums of translates

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.