Equivalence of Sobolev Spaces

We repair the proof of equivalence of certain L²-Sobolev spaces on manifolds with bounded curvature of all orders from [4]. The results are extended to generalized compatible Dirac operators, fractional order Sobolev spaces and weighted Sobolev spaces. A certain way of doing coordinate free computations is presented.     

Geometry on Probability Spaces

Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However, this avenue has been limited in many areas where calculus is obstructed, as in singular spaces, and in function spaces of functions on a space X where X itself is a function space. Examples of the latter occur in vision and quantum field theory. In vision it would be useful to do analysis on the space of images and an image is a function on a patch. Moreover, in analysis and geometry, the Lebesgue measure and its counterpart on manifolds are central. These measures are unavailable in the vision example and even in learning theory in general. There is one situation where, in the last several decades, the problem has been studied with some success. That is when the underlying space is finite (or even discrete). The introduction of the graph Laplacian has been a major development in algorithm research and is certainly useful for unsupervised learning theory. The approach taken here is to take advantage of both the classical research and the newer graph theoretic ideas to develop geometry on probability spaces. This starts with a space X equipped with a kernel (like a Mercer kernel) which gives a topology and geometry; X is to be equipped as well with a probability measure. The main focus is on a construction of a (normalized) Laplacian, an associated heat equation, diffusion distance, etc. In this setting, the point estimates of calculus are replaced by integral quantities. One thinks of secants rather than tangents. Our main result bounds the error of an empirical approximation to this Laplacian on X.     

The Equivalence Theory of Native Spaces

In this paper some equivalence definitions are given for native spaces which were introduced by Madych and Nelson and have become influential in the theory of radial basis functions. The abstract elements in native spaces are interpreted. Moreover, Weinrich and Iske's theories are unified.     

The Equivalence Theory of Native Spaces

In this paper some equivalence definitions are given for native spaces which were introduced by Madych and Nelson and have become influential in the theory of radial basis functions. The abstract elements in native spaces are interpreted. Moreover, Weinrich and Iske's theories are unified.     

On the Equivalence of Spectra of Linear Partial Differential Operators in Certain Function Spaces

For linear differential operators with coefficients of class C on ⁿ, we prove theorems on the simultaneous invertibility and equivalence of spectra in the Lebesgue space L ^p, Stepanov space M ^p, and in a particular Banach space V ^p L ^p, p 1     

Equivalence of Sharp Trudinger-Moser Inequalities in Lorentz-Sobolev Spaces

Potential Analysis - The critical and subcritical Trudinger-Moser inequalities in Lorentz Sobolev space have been studied by Cassani and Tarsi (Asymptot. Anal. 64(1-2):29–51, 2009), Lu and...     

Isometric Equivalence of Certain Operators on Banach Spaces

A pair of operators on a Banach space X are isometrically equivalent if they are intertwined by a surjective isometry of X. We investigate the isometric equivalence problem for pairs of operators on specific types of Banach spaces. We study weighted shifts on symmetric sequence spaces, elementary operators acting on an ideal I of Hilbert space operators, and composition operators on the Bloch space. This last case requires an extension of known results about surjective isometries of the Bloch space.     

乘积概率空间中的维数结果(英语)

在R~(d_1),R~(d_2)中的分形测度和R~(d_1) r_2中乘积测度之间的关系已经被许多作者发展。本文目的是在概率空间(n_1,_1,叭),(o:,芦2,/J:)和它们的乘积概率空间(nl×xn:,厂1×丁2,P1×,:)中对Hausdorff维数和填充维数探索相应的结果。     

Equivalence Theorems for Incomplete Spaces: An Appraisal

We present a practical appraisal of the functional analysisideas involved in some recent equivalence theorems. We alsoderive some new results which may be of independent interest.     

Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces

In this article, we study conditions for the asymptotic equivalence of differential equations in Hilbert spaces. We also discuss the relationship between the properties of solutions of differential equations of triangular form and those of truncated differential equations. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 329–337, March, 2005.     

Approximating Probability Distributions Using Small Sample Spaces

We formulate the notion of a "good approximation" to a probability distribution over a finite abelian group ?. The quality of the approximating distribution is characterized by a parameter ɛ which is a bound on the difference between corresponding Fourier coefficients of the two distributions. It is also required that the sample space of the approximating distribution be of size polynomial in and 1/ɛ. Such approximations are useful in reducing or eliminating the use of randomness in certain randomized algorithms. We demonstrate the existence of such good approximations to arbitrary distributions. In the case of n random variables distributed uniformly and independently over the range , we provide an efficient construction of a good approximation. The approximation constructed has the property that any linear combination of the random variables (modulo d) has essentially the same behavior under the approximating distribution as it does under the uniform distribution over . Our analysis is based on Weil's character sum estimates. We apply this result to the construction of a non-binary linear code where the alphabet symbols appear almost uniformly in each non-zero code-word. Received: September 22, 1990/Revised: First revision November 11, 1990; last revision November 10, 1997     

On Complementary Spaces of the Lizorkin Spaces

Let S(Rⁿ){\cal S}(R^n) be the Schwartz space on R ⁿ . For a subspace V ì S(Rⁿ)V\subset {\cal S}(R^n), if a subspace W ì S(Rⁿ)W \subset {\cal S}(R^n) satisfies the condition that S(Rⁿ){\cal S}(R^n) is a direct sum of V and W, then W is called a complementary space of V in S(Rⁿ){\cal S}(R^n). In this article we give complementary spaces of two kinds of the Lizorkin spaces in S(Rⁿ){\cal S}(R^n).     

交错阵空间上保秩等价的线性算子

设F是任意域,n≥4是一个正整数.令Kn(F)是F上n×n交错阵空间.对于A,B∈Kn(F),如果rank A=rank B,则称A和B是秩等价的.本文主要刻画Kn(F)上的保秩等价的线性算子,并给出一些应用.     

Banach空间中Mann迭代和Ishikawa迭代的等价性问题

用不同于已有的方法证明了任意实Banach空间中一致Lipschitz强连接伪压缩算子在具误差的修正的Mann迭代和具误差的修正的Ishikawa迭代下收敛和稳定的等价性,其中迭代参数{βn}仅需lim sup n→∞βn〈k/L（L＋1）,这推广和改进了目前需假设lim n→∞ βn=0和两迭代程序初始点的取值需相同条件下的已有结果.     

Equivalence of Differential Operators in Spaces of Analytic Functions over the Tate Field

We obtain conditions for the equivalence of certain differential operators in spaces of analytic functions over the Tate field.     

On the Equivalence of Borel Sets

Mathematical Notes - A theorem on the existence of a definable selector for the relation $$E_{aleph_0}$$ on the $$boldsymbol{Delta}^0_2$$ -sets of the Baire space $$omega^omega$$ is proved.     

Lacunary Measures and Self-similar Probability Measures in Function Spaces

The paper deals with multifractal quantities for some types of Radon measures,especiallyself-similar probability measures,and their relations to Besov spaces.