Cardinal polysplines of order
p
on annuli are functions in
C
2
p
-
2
R
n
⧹
0
which are piecewise polyharmonic of order
p
such that
Δ
p
-
1
S
may have discontinuities on spheres in
R
n
, centered at ...the origin and having radii of the form
e
j
,
j
∈
Z
. The main result is an interpolation theorem for cardinal polysplines where the data are given by sufficiently smooth functions on the spheres of radius
e
j
and center 0 obeying a certain growth condition in
j
. This result can be considered as an analogue of the famous interpolation theorem of Schoenberg for cardinal splines.
Craig interpolation theorem (which holds for intuitionistic logic) implies that the derivability of
X,X′
⇒
Y′
implies existence of an interpolant
I in the common language of
X and
X′
⇒
Y′
such that ...both
X
⇒
I
and
I,X′
⇒
Y′
are derivable. For classical logic this extends to
X,X′
⇒
Y,Y′
, but for intuitionistic logic there are counterexamples. We present a version true for intuitionistic propositional (but not for predicate) logic, and more complicated version for the predicate case.
We define generalized Lorentz‐Zygmund spaces and obtain interpolation theorems for quasilinear operators on such spaces, using weighted Hardy inequalities. In the limiting cases of interpolation, we ...discover certain scaling property of these spaces and use it to obtain fine interpolation theorems in which the source is a sum of spaces and the target is an intersection of spaces. This yields a considerable improvement of the known results which we demonstrate with examples. We prove sharpness of the interpolation theorems by showing that the constraints on parameters are necessary for the interpolation theorems.
Semiregular Hermite Tetrahedral Finite Elements Zenisek, Alexander; Hoderova-Zlamalova, Jana
Applications of Mathematics/Applications of mathematics,
08/2001, Letnik:
46, Številka:
4
Journal Article
Recenzirano
Odprti dostop
Tetrahedral finite C^sup 0^-elements of the Hermite type satisfying the maximum angle condition are presented and the corresponding finite element interpolation theorems in the maximum norm are ...proved.PUBLICATION ABSTRACT
Interpolation by a Game Kraíček, Jan
Mathematical logic quarterly,
1998, 1998-01-00, Letnik:
44, Številka:
4
Journal Article
Recenzirano
We introduce a notion of a real game (a generalisation of the Karchmer‐Wigderson game (cf. 3) and of real communication complexity, and relate this complexity to the size of monotone real formulas ...and circuits. We give an exponential lower bound for tree‐like monotone protocols (defined in 4, Definition 2.2) of small real communication complexity solving the monotone communication complexity problem associated with the bipartite perfect matching problem. This work is motivated by a research in interpolation theorems for prepositional logic (by a problem posed in 5, Section 8, in particular). Our main objective is to extend the communication complexity approach of 4, 5 to a wider class of proof systems. In this direction we obtain an effective interpolation in a form of a protocol of small real communication complexity. Together with the above mentioned lower bound for tree‐like protocols this yields as a corollary a lower bound on the number of steps for particular semantic derivations of Hall's theorem (these include tree‐like cutting planes proofs for which an exponential lower bound was demonstrated in 2).
Two approaches to deriving the interpolation theorem for narrow quadrilateral isoparametric finite elements not satisfying the condition
ρ
K
h
K
⩾ ρ
0 > 0
,
ρ
K
and
h
K
being the radius of inscribed ...circle and the diameter of the quadrilateral
K, respectively, are presented. The first one, using the Bramble-Hilbert lemma, is successful only in deriving the
L
2(
K)-estimate. The nonapplicability of the standard approach via the Bramble-Hilbert lemma in the case of
H
1(
K)-estimate is presented and a fully efficient method giving the optimum rate of convergence O(
h) in the
H
1(
K)-norm is described. In the end, the dependence of the interpolation error on the geometry of a quadrilateral is demonstrated by an example.
We prove a result related to work by A. Greenleaf and G. Uhlmann concerning Sobolev estimates for operators given by averages over cones. This is done using the almost orthogonality lemma of Cotlar ...and Stein, and the van der Corput lemma on oscillatory integrals.
Various triangular finite C0-elements of Hermite type satisfying the maximum-angle condition are presented and corresponding finite element interpolation theorems are proved. The paper contains also ...a proof that very general hypotheses due to Jamet are not necessary for such finite elements.
We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner ...solutions, we prove that the orthogonalized Calderón projections of the underlying family of elliptic operators form a continuous family of projections. Hence, its images (the Cauchy data spaces) form a continuous family of closed subspaces in the relevant Sobolev spaces. We use only elementary tools and classical results: basic manipulations of operator graphs and other closed subspaces in Banach spaces, elliptic regularity, Green's formula and trace theorems for Sobolev spaces, well-posed boundary conditions, duality of spaces and operators in Hilbert space, and the interpolation theorem for operators in Sobolev spaces.
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