منابع مشابه
A Representation Theorem for Schauder Bases in Hilbert Space
A sequence of vectors {f1, f2, f3, . . . } in a separable Hilbert space H is said to be a Schauder basis for H if every element f ∈ H has a unique norm-convergent expansion f = ∑ cnfn. If, in addition, there exist positive constants A and B such that A ∑ |cn| ≤ ∥∥∥∑ cnfn∥∥∥2 ≤ B∑ |cn|, then we call {f1, f2, f3, . . . } a Riesz basis. In the first half of this paper, we show that every Schauder ...
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A unified approach to sampling theorems for (wide sense) stat ionary random processes rests upon Hilbert space concepts. New results in sampling theory are obtained along the following lines: recovery of the process x(t) from nonperiodic samples, or when any finite number of samples are deleted; conditions for obtaining x (t) when only the past is sampled; a criterion for restoring x(t) from a ...
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Let X be a set of vectors in R m. X is said to be a Hilbert base if every vector in R m which can be written both as a linear combination of members of X with nonnegative coeecients and as a linear combination with integer coeecients can also be written as a linear combination with nonnegative integer coeecients. Denote by H the collection of the graphs whose family of cuts is a Hilbert base. I...
متن کاملLocal Theory of Frames and Schauder Bases for Hilbert Space
We develope a local theory for frames on finite dimensional Hilbert spaces. We show that for every frame (fi) m i=1 for an n-dimensional Hilbert space, and for every ǫ > 0, there is a subset I ⊂ {1, 2, . . . ,m} with |I| ≥ (1 − ǫ)n so that (fi)i∈I is a Riesz basis for its span with Riesz basis constant a function of ǫ, the frame bounds, and (‖fi‖) m i=1 , but independent of m and n. We also con...
متن کاملOperator-valued bases on Hilbert spaces
In this paper we develop a natural generalization of Schauder basis theory, we term operator-valued basis or simply ov-basis theory, using operator-algebraic methods. We prove several results for ov-basis concerning duality, orthogonality, biorthogonality and minimality. We prove that the operators of a dual ov-basis are continuous. We also dene the concepts of Bessel, Hilbert ov-basis and obta...
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ژورنال
عنوان ژورنال: Pacific Journal of Mathematics
سال: 1968
ISSN: 0030-8730,0030-8730
DOI: 10.2140/pjm.1968.26.441