B-Splines and orthonormal sets in Paley- Wiener space

Abstract

It is known that the cardinal B-splines can be used to generate a multiresolution analysis, and hence an orthonormal wavelet basis for \({{L}^{2}}(\mathbb{R}) \). We show that they can also be used to generate orthonormal sets, as well as, frames in the Paley-Wiener space \(B_{\sigma }^{2} \), which is also known as the space of bandlimited functions with bandwidth σ. A new orthonormal set of functions in \(B_{\sigma }^{2} \), obtained from one single function by translating it by integer multiples of 2π/σ is given explicitly in terms of Young’s function that was introduced by W. H. Young in 1912.KeywordsEntire FunctionCosine FunctionSingle FunctionInteger MultipleMultiresolution AnalysisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.