Abstract splines in Krein spaces

Abstract

splines Oblique projections We present generalizations to Krein spaces of the abstract interpolation and smoothing problems proposed by Atteia in Hilbert spaces: given a Krein space K and Hilbert spaces H and E (bounded) surjective operators T : H → K and V : H → E, ρ > 0 and a fixed z 0 ∈ E, we study the existence of solutions of the problems argmin{[T x, T x] K : V x = z 0 } and argmin{[T x, T x] K + ρ V x − z 0 2 E : x ∈ H}.

Indefinite Abstract Splines with a Quadratic Constraint

“…Finally, Section 4 is devoted to analyzing the close relationship between the indefinite abstract smoothing problem and a particular version of the indefinite abstract interpolation problem, where the quadratic constraint in (1.1) is replaced by a linear constraint. This problem coincides with the one previously considered in [17]. We focus our attention on the situations in which a set of interpolating splines is also a subset of the solutions to a certain smoothing interpolation problem, or, even further, in which an indefinite interpolation problem can be posed as an indefinite smoothing problem, and viceversa.…”

“…Since T and V are fixed along this work, sp(T, V, z 0 ) is shortened to sp(z 0 ). Necessary and sufficient conditions for the existence of solutions to Problem 3 are given in the following proposition, which is an immediate consequence of [17,Lemma 3.4].…”

“…In [17,Prop. 3.8] necessary and sufficient conditions for the existence of interpolating splines to every vector of the space were provided: sp(z 0 ) = ∅ for every z 0 ∈ E if and only if T (N (V )) is a (closed) uniformly positive subspace of K. This is the case if R(L) is a (positive) regular subspace of (K × E, [ •, • ] ρ ), see [17,Thm. 4.7].…”

“…Hence, (1.1) can be restated as: analyze the existence of min and if the minimum exists, find the set of arguments at which it is attained. If V # V is a positive (or negative) semidefinite operator in H then C V coincides with N (V ), and (1.1) becomes the interpolation problem previously studied in [17]. But, if V # V is indefinite, the set C V is strictly larger than N (V ).…”

Construção De Espaços De Cor Euclidianos E Perceptualmente Uniformes Com Base Na Fórmula Ciede2000

“…Por usar uma formulação singular, o uso prático de espaços pseudoeuclidianos exige uma abordagem diferente. Desde pequenas alterações nas equações (3-20) a (3-23), até o uso de uma formulação específica de splines, para obtermos interpolações contínuas nestes espaços [73]. Em face destas dificuldades, acreditamos que o uso da quarta dimensão será muito restrito, apenas para aplicações com exigências muito grandes de precisão, provavelmente voltadas a regiões específicas do plano ab com maior sensibilidade a essa dimensão.…”