Exercise 4.2.2

Answers

\begin{array}{l} (\sum_{m = 0}^{2} w^{km} c_{m}) (\sum_{n = 0}^{2} w^{kn} d_{n}) & = (c_{0} + w^{k} c_{1} + w^{2 k} c_{2}) (d_{0} + w^{k} d_{1} + w^{2 k} d_{2}) \\ = c_{0} d_{0} + w^{k} c_{0} d_{1} + w^{2 k} c_{0} d_{2} + w^{k} c_{1} d_{0} + w^{2 k} c_{1} d_{1} + w^{3 k} c_{1} d_{2} + w^{2 k} c_{2} d_{0} + w^{3 k} c_{2} d_{1} + w^{4 k} c_{2} d_{2} \\ = c_{0} d_{0} + w^{k} c_{0} d_{1} + w^{2 k} c_{0} d_{2} + w^{k} c_{1} d_{0} + w^{2 k} c_{1} d_{1} + c_{1} d_{2} + w^{2 k} c_{2} d_{0} + c_{2} d_{1} + w^{k} c_{2} d_{2} \\ = (c_{0} d_{0} + c_{1} d_{2} + c_{2} d_{1}) + w^{k} (c_{0} d_{1} + c_{1} d_{0} + c_{2} d_{2}) + w^{2 k} (c_{0} d_{2} + c_{1} d_{1} + c_{2} d_{0}) \\ = {(c ⊛ d)}_{0} + w^{k} {(c ⊛ d)}_{1} + w^{2 k} {(c ⊛ d)}_{2} \\ = \sum_{p = 0}^{2} w^{kp} {(c ⊛ d)}_{p} \end{array}

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2020-03-20 00:00