Showing papers on "Biorthogonal system published in 1969"
TL;DR: In this article, it was shown that a function whose coefficients in the series are all zero must itself be zero, and conditions under which the series, or a subsequence of its partial sums, converges to the corresponding function.
Abstract: This article is concerned with the representation of functions in domains of the complex plain by series in the systems , , .In § 1 we construct systems biorthogonal to the systems , , and find the asymptotic behaviour of functions of these systems.In § 2 we determine in a natural way the coefficients of the series in the systems in question by means of the biorthogonal systems. We also find the asymptotic behaviour of the coefficients for large indices. We obtain formulae for the remainder, that is, the difference between the function and a partial sum of the corresponding series.In § 3 we prove that a function whose coefficients in the series are all zero must itself be zero. This result makes it possible, in principle, to reconstruct a function from the coefficients of its series.In § 4 we give conditions under which the series, or a subsequence of its partial sums, converges to the corresponding function.
52 citations
17 citations
TL;DR: This paper presents an alternate approach to this problem based on the direct product of matrices, easily understood by anyone familiar with matrix theory, and it yields results in a form convenient for implementation and generalization.
Abstract: Posner (1968) has recently discussed a decoding scheme for certain orthogonal and biorthogonal codes which is based on the fast Fourier transform on a finite abelian group. In this paper, we present an alternate approach to this problem based on the direct product of matrices. This approach is easily understood by anyone familiar with matrix theory, and it yields results in a form convenient for implementation and generalization.
14 citations
TL;DR: In this paper, an extension of the Pade method is made introducing a two-step approximant as suggested by the biorthogonal algorithm considered in a previous work.
Abstract: An extension of the Pade method is made introducing a two-step approximant as suggested by the biorthogonal algorithm considered in a previous work. In this way we have a new tool for the analytic continuation of power series. The two-step approximant is particularly useful in the cases where the usual Pade method does not work, for example, the total T-matrix and the half off-shell partial-wave amplitude. The full off-shell extension of the two-body amplitude is considered and a process is developed to obtain separable forms.
2 citations
TL;DR: The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space as discussed by the authors, which is interesting for lp -spaces because of the paucity of standard bases in these spaces.
Abstract: Our aim in this paper is to generalize certain ideas and results of Bary (1) on biorthogonal systems in separable Hilbert spaces to their counterparts in separable lp -spaces, 1 < p.The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space. That of this paper is to characterize the concept of a Bary basis which is a generalization of the idea of standard basis of an lp -space. The result is interesting for lp -spaces because of the paucity of standard bases in these spaces. Before summarizing our results, we shall introduce some notation and recall a few pertinent definitions and facts. The symbols and denote mutually conjugate lp -spaces, where is the space lt and the space ls with 1 < r <2 and 2 < s = r/(r – 1).
2 citations
01 Jan 1969
TL;DR: A generalization of the basis concept for Hilbert and Banach spaces is discussed in this article, where the starting point is a family of elements of X and a biorthogonal family of continuous linear functionals on X.
Abstract: Having introduced the basis concept for Hilbert and Banach spaces, certain generalizations appear to be at least as important. First of all one discards of almost all requirements used for the definition of a basis. Beginning with the absolute minimum one takes a linear topological space X, does not use the concepts of totalness and countability and avoids all mention of series expansions. Thus, the starting point will be a family {xλ} of elements of X and a biorthogonal family {fλ} of continuous linear (coefficient) functionals on X. The biorthogonal system {x λ ,f λ } is said to be maximal if there is no biorthogonal system in which it is properly contained. A biorthogonal system with respect to X is called a generalized basis for X if, in addition, x ∊ X and f λ (x) = 0 for all X implies x = 0. A generalized basis is always a maximal biorthogonal system. If the set of basis elements {x λ } of a biorthogonal system {x λ ,f λ } in X is total in X, then {x λ ,f λ } is called a dual generalized basis for X. Moreover, if such a basis is also a generalized basis for X, it is called a Markushevich basis for X if the set {x λ } is countable, and an extended Markushevich basis for X if {x λ } is not countable. Finally, introducing again the concept of a series expansion for elements of X, then a Markushevich basis for X becomes a Schauder basis for X.