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# Hilbert's basis theorem

About: Hilbert's basis theorem is a research topic. Over the lifetime, 1019 publications have been published within this topic receiving 22373 citations.

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01 Feb 1966

TL;DR: In this paper, it was shown that a close relationship exists between the notions of majorization, factorization, and range inclusion for operators on a Hilbert space, and that these notions fit together to yield theorems.

Abstract: The purpose of this note is to show that a close relationship exists between the notions of majorization, factorization, and range inclusion for operators on a Hilbert space. Although fragments of these results are to be found scattered throughout the literature (usually buried in proofs), it does not seem to have been noticed how nicely they fit together to yield our theorems. We will also make an attempt at extending our result to the case of unbounded operators in the hope that it might be useful in establishing existence theorems for linear partial differential equations. The author wishes to acknowledge that he discovered these relations in the study of an unpublished manuscript of deBranges and Rovnyak. Also, we acknowledge our indebtedness to P. Halmos for several conversations on this subject and note, in particular, that it was he who first noticed the equivalence of (1) and (3) in Theorem 1. The Hilbert space considered can be either real or complex. We use [1 ] as our basic reference and use the definitions and notation therein with the following exception. For an operator A on the Hilbert space we will denote the range and null space of A by range [A] and null [A ], respectively.

1,041 citations

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TL;DR: In this article, the additive group of R has a direct-sum decomposition R = R + R, +..., where RiRi C R,+j and 1 E R,.

Abstract: Let R be a Noetherian commutative ring with identity, graded by the nonnegative integers N. Thus the additive group of R has a direct-sum decomposition R = R, + R, + ..., where RiRi C R,+j and 1 E R, . I f in addition R, is a field K, so that R is a k-algebra, we will say that R is a G-akebra. The assumption that R is Noetherian implies that a G-algebra is finitely generated (as an algebra over k) and that each R, is a finite-dimensional vector space over k. The Hilbe-rt function of R is defined by

948 citations

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RAND Corporation

^{1}TL;DR: In this article, a mathematical theorem for finding the Hilbert transform of a product of functions in a simplified fashion is presented under certain conditions, and a mathematical application of this theorem for engineering applications is discussed.

Abstract: : A mathematical theorem is presented for finding the Hilbert transform of a product of functions in a simplified fashion, under certain conditions. Engineering applications are discussed. (Author)

537 citations