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Generalizations of Shannon Sampling Theorem
Published in Ahmed I. Zayed, Advances in Shannon’s Sampling Theory, 2018
An earlier attempt to derive such a series expansion was made by I. Kluvánek [48], who derived a general form of the WSK sampling theorem in the setting of harmonic analysis on locally compact abelian groups. By a suitable restriction to ℜN, we can obtain Theorem 3.6 as a special case of Kluvánek’s result. Nevertheless, Kluvánek’s result does not seem to easily yield any explicit sampling series expansion for a function that is band-limited to a general region in ℜN. More recent results have been obtained by H. Feichtinger [22], H. Feichtinger and K. Gröchenig [23–25], and K. Gröchenig [30]. Their approach to sampling theorems appears to be new and promising, but a bit abstract. They have also extended their results to locally compact abelian groups using harmonic analysis techniques. This connection between the WSK sampling theorem and harmonic analysis on locally compact abelian groups is not surprising since the Poisson summation formula is known to play a fundamental role in both areas; see [88] and [89].
Locally Starplus-Compactness in L-Topological Spaces
Published in Fuzzy Information and Engineering, 2018
The class of locally compact spaces is far more wider than the class of compact spaces. The locally compact spaces often arise in topology and applications of topology to geometry, analysis and algebra. For example, the study of locally compact abelian group forms the foundation of harmonic analysis. It is well known that every compact space is locally compact but the converse need not be true. For example, the Euclidean space is locally compact but not compact. Topological manifolds share the local properties of Euclidean space and hence are locally compact. A locally compact space can be imbedded in a compact space, which is its compactification. One of the simplest compapctification of a space is the one point compactification, wherein one simply adjoins one new point to the space. The classical example of one point compactification is the embedding of the Gaussian plane of complex numbers into the Riemann sphere. The category of locally compact spaces has been applied in almost every subdiscipline of mathematics and hence it is important to formulate an appropriate version of local compactness in the L-fuzzy setting.