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Integral combinations of Heavisides |
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| Authors: | Paul C Kainen Věra K?rková Andrew Vogt |
| Institution: | 1. Department of Mathematics, Georgetown University, Washington, D.C. 20057‐1233, USA;2. Phone: +1 202 687 2703, Fax: +1 202 687 6067;3. Institute of Computer Science, Acad. of Sci. of the Czech Republic, Pod Vodárenskou vě?í 2, 182 07 Prague 8, Czech Republic;4. Phone: +420 266053231, Fax: +420 286585789 |
| Abstract: | A sufficiently smooth function of d variables that decays fast enough at infinity can be represented pointwise by an integral combination of Heaviside plane waves (i.e., characteristic functions of closed half‐spaces). The weight function in such a representation depends on the derivatives of the represented function. The represen‐tation is proved here by elementary techniques with separate arguments for even and odd d, and unifies and extends various results in the literature (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) |
| Keywords: | Feedforward neural network perceptron Heaviside function plane wave integral formula Radon transform Green's function for iterated Laplacians order of vanishing function of controlled decay |