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| We are going to consider various ''operators'' on these functions. Here, an operator <math>\operatorname{F}</math> is a function that takes one function <math>f\!</math> into another function <math>\operatorname{F}f.</math> | | We are going to consider various ''operators'' on these functions. Here, an operator <math>\operatorname{F}</math> is a function that takes one function <math>f\!</math> into another function <math>\operatorname{F}f.</math> |
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− | <pre>
| + | The first couple of operators that we need to consider are logical analogues of those that occur in the classical ''finite difference calculus'', namely: |
− | The first couple of operators that we need to consider are logical analogues | + | |
− | of those that occur in the classical "finite difference calculus", namely: | + | {| align="center" cellpadding="6" width="90%" |
| + | | The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math> |
| + | |- |
| + | | The ''enlargement" operator'' <math>\Epsilon,\!</math> written here as <math>\operatorname{E}.</math> |
| + | |} |
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− | 1. The "difference" operator [capital Delta], written here as D.
| + | These days, <math>\operatorname{E}</math> is more often called the ''shift operator''. |
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− | 2. The "enlargement" operator [capital Epsilon], written here as E.
| + | In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse, passing from the space <math>U = X \times Y</math> to its ''differential extension'', <math>\operatorname{E}U,</math> that has the following description: |
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− | These days, E is more often called the "shift" operator.
| + | {| align="center" cellpadding="6" width="90%" |
| + | | <math>\operatorname{E}U ~=~ U \times \operatorname{d}U ~=~ X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math> |
| + | |- |
| + | | with |
| + | |- |
| + | | <math>\operatorname{d}X = \{ \texttt{(} \operatorname{d}x \texttt{)}, \operatorname{d}x \}</math> and <math>\operatorname{d}Y = \{ \texttt{(} \operatorname{d}y \texttt{)}, \operatorname{d}y \}.</math> |
| + | |} |
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− | In order to describe the universe in which these operators operate,
| + | <pre> |
− | it will be necessary to enlarge our original universe of discourse.
| |
− | We mount up from the space U = X x Y to its "differential extension",
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− | EU = U x dU = X x Y x dX x dY, with dX = {(dx), dx} and dY = {(dy), dy}.
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| The interpretations of these new symbols can be diverse, but the easiest | | The interpretations of these new symbols can be diverse, but the easiest |
| for now is just to say that dx means "change x" and dy means "change y". | | for now is just to say that dx means "change x" and dy means "change y". |