In a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element a we associate the eigenvalue vector λ(a) in Rn whose components are the eigenvalues of a written in ...the decreasing order. For any p∈1,∞, we define the spectral p-norm of a to be the p-norm of λ(a) in Rn. In a recent paper, based on the K-method of real interpolation theory and a majorization technique, we described an interpolation theorem for a linear transformation on V relative to the same spectral p-norm. In this paper, using complex function theory methods, we describe a Riesz-Thorin type interpolation theorem relative to two spectral p-norms. We illustrate the result by estimating the norms of certain special linear transformations such as Lyapunov transformations, quadratic representations, and positive transformations.
We study the Sobolev spaces
and
on p.c.f. self-similar sets.
First, for
, we make an exact description of the tangents of functions in
at the boundary, and introduce a countable set of critical ...orders that arises naturally in the boundary behavior of functions. These critical orders are just
in the Euclidean case, but become complicated on fractals. Second, we characterize
as the space of functions in
with tangents of appropriate order, that depend on σ and critical orders, being 0. Last, we extend
to
, and obtain various interpolation theorems with
or
. The interpolation space presents a critical phenomenon when the resulted order
is critical. Moreover, for the interpolation couple
, more than the classical theorem, our interpolation theorem fully covers the teratological case that
contains at least one critical order.
In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for ...intuitionistic predicate logic with constant domains 13. More precisely, we show that there is a valid implication
$\phi \rightarrow \psi $
with no interpolant. Importantly, this result does not contradict the unfortunately named ‘Craig interpolation’ theorem established by Rauszer in 24 since that article is about the property more correctly named ‘deductive interpolation’ (see Galatos, Jipsen, Kowalski and Ono’s use of this term in 5) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent.
A number of statements similar to the Marcinkiewicz interpolation theorem are presented. The difference from the classical forms of this theorem is that the spaces of integrable functions are ...replaced by certain classes of functions that are extensions of various Hardy spaces.
In this paper we are concerned with an inverse problem with Robin boundary conditions, which states that, when the potential on
0
,
1
/
2
and the coefficient at the left end point are known a ...priori, a full spectrum uniquely determines its potential on the whole interval and the coefficient at the right end point. We shall give a new method for reconstructing the potential for this problem in terms of the Mittag-Leffler decomposition of entire functions associated with this problem. The new reconstructing method also deduces a necessary and sufficient condition for the existence issue.
In this article, we investigate the characterizations of some localization operators which are associated with the integral representation of a locally compact group. Furthermore, with the help of ...the Watson transform, we find their relationship with wavelet multipliers. We also discuss the trace class and the Schatten-von Neumann property of the localization operators which we have investigated here.
In this paper, we show the validity of a Riesz-Thorin type interpolation theorem for linear operators acting from variable exponent Lebesgue spaces into variable exponent Campanato spaces of order k.
This paper considers a risk model with random premium income and two types of by-claims, which is an extension to the general delayed claims process where there is only one type of by-claim and the ...premium income is a constant. Assume that each main claim induces one of the two by-claims, the by-claim and its associated main claim occur at the same time when the main claim amount is less than a threshold variable; otherwise, the occurrence of the by-claim will be delayed. An integral equations system for the Gerber–Shiu discounted penalty functions is presented by using auxiliary risk models. Given that the premium size is exponentially distributed, the explicit expression for the Laplace transform of the Gerber–Shiu penalty function is derived. By applying Rouché theorem, the Gerber–Shiu discounted penalty functions with reciprocal of the mean of premium surplus are obtained. According to Lagrange interpolation theorem, we prove that the Gerber–Shiu discounted penalty function satisfies a defective renewal equation. Finally, under the assumption of the claim sizes satisfying exponential distribution, the explicit formula of the ruin probability is derived when the discounted factor equals zero and the penalty function equals one.
•We develop a new delayed model with random premium income and two types of by-claims.•We derive an integral equations system for Gerber–Shiu discounted penalty function.•We get an explicit solution to Laplace transform of the discounted penalty function.•We prove the discounted penalty function satisfying a defective renewal equation.•We get an explicit result of the ruin probability under the exponential distribution.
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