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Block pulse functions, the most fundamental of all piecewise constant basis functions. (English) Zbl 0834.42016
Summary: It is established that block pulse functions (BPFs) are superior to the delayed unit step function (DUSF) proposed by Hwang (1983). The superiority is mainly due to the most elementary nature of BPFs in comparison to any other PCBF function. It is also proved that the operational matrix for integration in the BPF domain is connected to the integration operational matrix in the DUSF domain by simple linear transformation involving invertible Toeplitz matrices. The transformation appears to be transparent because the integration operational matrices are found to match exactly. The reason for such transparency is explained mathematically. Finally, Hwang claimed superiority of DUSFs compared to Walsh functions in obtaining the solution of functional differential equations using a stretch matrix in the DUSF domain. It is shown that the stretch matrices of Walsh and DUSF domains are also related by linear transformation and use of any of these two matrices leads to exactly the same result. This is supported by an example. It may be noted that for any analysis approach involving any of the PCBFs, the accuracy of the final result is always the same. This is because all the PCBFs are connected to one another by linear transformations.
MSC:
42C10Fourier series in special orthogonal functions