Journal•ISSN: 0002-9947

# Transactions of the American Mathematical Society

About: Transactions of the American Mathematical Society is an academic journal. The journal publishes majorly in the area(s): Bounded function & Abelian group. It has an ISSN identifier of 0002-9947. Over the lifetime, 18156 publication(s) have been published receiving 589880 citation(s). The journal is also known as: American Mathematical Society. Transactions.

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5,312 citations

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TL;DR: In this paper, a short historical introduction is given to indicate the different manners in which these kernels have been used by various investigators and discuss the more important trends of the application of these kernels without attempting, however, a complete bibliography of the subject matter.

Abstract: : The present paper may be considered as a sequel to our previous paper in the Proceedings of the Cambridge Philosophical Society, Theorie generale de noyaux reproduisants-Premiere partie (vol 39 (1944)) which was written in 1942-1943 In the introduction to this paper we outlined the plan of papers which were to follow In the meantime, however, the general theory has been developed in many directions, and our original plans have had to be changed Due to wartime conditions we were not able, at the time of writing the first paper, to take into account all the earlier investigations which, although sometimes of quite a different character, were, nevertheless, related to our subject Our investigation is concerned with kernels of a special type which have been used under different names and in different ways in many domains of mathematical research We shall therefore begin our present paper with a short historical introduction in which we shall attempt to indicate the different manners in which these kernels have been used by various investigators, and to clarify the terminology We shall also discuss the more important trends of the application of these kernels without attempting, however, a complete bibliography of the subject matter (KAR) P 2

5,299 citations

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TL;DR: In this article, the authors examined viscosity solutions of Hamilton-Jacobi equations, and proved the existence assertions by expanding on the arguments in the introduction concerning the relationship of the vanishing-viscosity method and the notion of viscoity solutions.

Abstract: Publisher Summary This chapter examines viscosity solutions of Hamilton–Jacobi equations. The ability to formulate an existence and uniqueness result for generality requires the ability to discuss non differential solutions of the equation, and this has not been possible before. However, the existence assertions can be proved by expanding on the arguments in the introduction concerning the relationship of the vanishing viscosity method and the notion of viscosity solutions, so users can adapt known methods here. The uniqueness is then the main new point.

2,300 citations

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TL;DR: In this paper, the authors study the properties of multiresolution approximation and prove that it is characterized by a 2π periodic function, which is further described in terms of wavelet orthonormal bases.

Abstract: A multiresolution approximation is a sequence of embedded vector spaces V j jmember Z for approximating L 2 (R) functions. We study the properties of a multiresolution approximation and prove that it is characterized by a 2π periodic function which is further described. From any multiresolution approximation, we can derive a function ψ(x) called a wavelet such that √ 2 j ψ(2 j x −k) (k ,j)member Z 2 is an orthonormal basis of L 2 (R). This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space H s .

2,218 citations

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2,146 citations