Convolution operator in the space of real analytic functions

Translated from Matematicheskie Zametki, Vol. 49, No. 3, pp. 57–65, March, 1991.     

A simple example on informativeness and performance

We present in this note a simple example in which increasing informativeness leads to decreasing performance.The research reported in this paper was made possible through support extended by the Division of Engineering and Applied Physics, Harvard University, by the U.S. Office of Naval Research under the Joint Services Electronics Program, Contract No. N00014-67-A-0298-0006, and by the National Science Foundation, Grant No. GK-31511.     

Convolution equations and the local Pompeiu property on symmetric spaces and on phase space associated to the Heisenberg group

In this paper, we present new uniqueness results related to geometric aspects of mean periodicity on various homogeneous spaces. Among these results, we point out the equivalence of the local and the global Pompeiu property for arbitrary families of compactly supported distributions, the solution of the local Pompeiu problem for the class of non real-analytic functions, and local versions of the two-radii theorem on symmetric spaces of arbitrary rank.     

The method of alternating projections and the method of subspace corrections in Hilbert space

A new identity is given in this paper for estimating the norm of the product of nonexpansive operators in Hilbert space. This identity can be applied for the design and analysis of the method of alternating projections and the method of subspace corrections. The method of alternating projections is an iterative algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces by alternatively computing the best approximations from the individual subspaces which make up the intersection. The method of subspace corrections is an iterative algorithm for finding the solution of a linear equation in a Hilbert space by approximately solving equations restricted on a number of closed subspaces which make up the entire space. The new identity given in the paper provides a sharpest possible estimate for the rate of convergence of these algorithms. It is also proved in the paper that the method of alternating projections is essentially equivalent to the method of subspace corrections. Some simple examples of multigrid and domain decomposition methods are given to illustrate the application of the new identity.

     

Energy method in the partial Fourier space and application to stability problems in the half space

The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space Rn. In this paper, we study half space problems in and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x^′∈Rn−1. For the variable x₁∈R₊ in the normal direction, we use L² space or weighted L² space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t→∞. The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13].     

Convolution of operators and applications

Supported by the Grant C.A.I.C.Y.T. n 85-0338     

Broyden's method in Hilbert space

Broyden's method is formulated for the solution of nonlinear operator equations in Hilbert spaces. The algorithm is proven to be well defined and a linear rate of convergence is shown. Under an additional assumption on the initial approximation for the derivative we prove the superlinear rate of convergence.     

Convolution quadrature time-domain boundary element method for 2-D fluid-saturated porous media

For large-scale wave analyses of fluid-saturated porous media, a conventional time-domain boundary element method (BEM) cannot be applied because of the following reasons: (1) no time-domain fundamental solutions are known for some problems, (2) the method sometimes suffers from instability, and (3) the analyses require large amounts of computational time and memory. In this study, an innovative time-domain BEM is developed for a fluid-saturated porous medium. The formulation presented herein overcomes the above disadvantages using a convolution quadrature method (CQM), first proposed by Lubich, and hybrid-parallelization with both MPI and OpenMP. Problems involving the scattering of an incident plane wave by cavities in a 2-D poroelastic medium are solved as a means of validating the proposed method.     

Remarks on the Voronoi method in Paley-Wiener space

In Voronoi method, the reconstruction functions χn are not well-behaved, i.e., not continuous, which affect the convergence rate of this algorithm. In this paper, first, we obtain well-behaved reconstruction functions in place of χn; second, when the sampling set is denser, we obtain a faster convergent reconstruction iterative algorithm using analog to the mth order B-spline reconstruction functions φn(f)(x).     

Computational schemes of the Davidon-Fletcher-Powell method in infinite-dimensional space

The disadvantage of the extension of the Davidon-Fletcher-Powell method to infinite-dimensional space is that the information to be stored in the computer increases with the number of iterations. In this paper, a computational scheme is proposed to remove this disadvantage and make the extension method more practicable. The linear operator which determines the direction of one-dimensional search in the method is formulated by integral kernels to derive the scheme. Furthermore, polynomial interpolation methods are proposed to save computer storage. The computational scheme which is presented here and the polynomial interpolation method are successfully applied to an optimal control problem.The authors would like to thank Dr. N. Adachi for valuable discussion. Computations were carried out at the computing centers of Osaka University, Kyoto University, and Tokyo University.     

Heat Equation and Convolution Inequalities

It is known that many classical inequalities linked to convolutions can be obtained by looking at the monotonicity in time of convolutions of powers of solutions to the heat equation, provided that both the exponents and the coefficients of diffusions are suitably chosen and related. This idea can be applied to give an alternative proof of the sharp form of the classical Young’s inequality and its converse, to Brascamp–Lieb type inequalities, Babenko’s inequality and Prékopa–Leindler inequality as well as the Shannon’s entropy power inequality. This note aims in presenting new proofs of these results, in the spirit of the original arguments introduced by Stam [35] to prove the entropy power inequality.     

Yetter-Drinfel''''d模与卷积Hopf模

In this paper, we first give a sufficient and necessary condition for a Hopf algebra to be a Yetter-Drinfel'd module, and prove that the finite dual of a Yetter-Drinfel'd module is still a Yetter-Drinfel'd module. Finally, we introduce a concept of convolution module.     

Singular Measures and Convolution Operators

We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type （1, 1） inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated with parallelotopes do not decrease with the dimension.     

Convolution semigroups of states

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C ₀-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.     

Convolution of convex valuations

We show that the natural “convolution” on the space of smooth, even, translation-invariant convex valuations on a euclidean space V, obtained by intertwining the product and the duality transform of S. Alesker J. Differential Geom. 63: 63–95, 2003; Geom.Funct. Anal. 14:1–26, 2004 may be expressed in terms of Minkowski sum. Furthermore the resulting product extends naturally to odd valuations as well. Based on this technical result we give an application to integral geometry, generalizing Hadwiger’s additive kinematic formula for SO(V) Convex Geometry, North Holland, 1993 to general compact groups acting transitively on the sphere: it turns out that these formulas are in a natural sense dual to the usual (intersection) kinematic formulas.      

Convolution Structures and Arithmetic Cohomology

Convolution structures are group-like objects that were extensively studied by harmonic analysts. We use them to define H ⁰ and H ¹ for Arakelov divisors over number fields. We prove the analogs of the Riemann–Roch and Serre duality theorems. This brings more structure to the works of Tate and van der Geer and Schoof. The H ¹ is defined by a procedure very similar to the usual ech cohomology. Serres duality becomes Pontryagin duality of convolution structures. The whole theory is parallel to the geometric case.