Definition:Convolution of Real Sequences
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This page is about Convolution of Real Sequences. For other uses, see Convolution.
This page has been identified as a candidate for refactoring of advanced complexity. In particular: These are not equivalent definitions, so we cannot implement this as a simple case of "Definition 1" and "Definition 2". The concepts might need to be separated for $\N$-sequences and $\Z$-sequences Until this has been finished, please leave {{Refactor}} in the code.
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Definition 1
Let $\sequence f$ and $\sequence g$ be real sequences.
The convolution of $f$ and $g$ is defined as:
- $\ds \sequence {f_i} * \sequence {g_i} := \sum_{j \mathop = 0}^i f_j g_{i - j}$
Definition 2
Let $f: \Z \to \R$ and $g: \Z \to \R$ be mappings from the integers to the real numbers.
The convolution of $f$ and $g$ is defined as:
- $\ds \map f i * \map g i := \sum_{j \mathop \in \Z} f_j g_{i - j}$
Sources
This page may be the result of a refactoring operation. As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering. In particular: These all apply specifically to Definition 1 If you have access to any of these works, then you are invited to review this list, and make any necessary corrections. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{SourceReview}} from the code. |
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Frontispiece
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Chapter $4$: Notation for some useful Functions: Summary of special symbols: Table $4.1$ Special symbols
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Inside Back Cover