Fourier Transform Geometry

How is the Fourier Transform related to Sound Wave Geometry?

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In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the number line by a point denoted .

The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line. More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded.

The projectively extended real line may be identified with the projective line over the reals in which three points have been assigned specific values (e.g. 0, 1 and ). The projectively extended real line must not be confused with the extended real number line, in which +∞ and −∞ are distinct.

Wheels are a type of algebra where division is always defined. In particular, division by zero is meaningful. The real numbers can be extended to a wheel, as can any commutative ring.

The function y = 1/x.

As x approaches 0 from the right, y approaches infinity.

As x approaches 0 from the left, y approaches negative infinity.

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The indeterminate forms typically considered in the literature are denoted 0/0, ∞/∞, 0 × ∞, ∞ − ∞, 00, 1 and ∞0.





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