Intereting Posts

Hatcher Problem 2.2.36
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find maximum area

Wikipedia says:

the normalized

cardinal B-splinestend to the Gaussian function

and writes them as “**B**_{k}“. Meanwhile, cnx.org Signal Reconstruction says:

- What is the maximum overshoot of interpolating splines in $d$ dimensions?
- estimating a particular analytic function on a bounded sector.
- Why the quadrature formula is exact one not an approximation?
- How to create a computationally cheap function passing through given points?
- 2D array downsampling and upsampling using bilinear interpolation
- Interpolation inequality

The

basis splinesB_{n}are shown … as the order increases, the functions approach the Gaussian function, which is exactlyB_{∞}.

but then says

as the order increases, the

cardinal basis splinesapproximate the sinc function, which is exactlyη_{∞}.

Likewise, Signal Reconstruction with Cardinal Splines uses similar notation of *η*^{n} for “cardinal spline”.

So which is it? Does a “cardinal basis spline” approximate a Gaussian or a sinc? “B-spline” and “basis spline” are the same thing, right? Is there any relationship to this cardinal spline?

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- Function generation by input $y$ and $x$ values
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- Looking for a Calculus Textbook

Roughly speaking …

If you fix certain quantities (degree, knots, orders of continuity) then the set of all splines forms a vector space, which, of course, has several different bases.

Two common bases are the cardinal splines, and the b-splines.

It’s true that “b-spline” is an abbreviation of “basis spline”, but the vector space of splines has other bases, besides the b-splines. Confusing, I guess.

I wasn’t aware of the limits as degree goes to infinity, but the statements sound plausible. Assuming you use the right knot sequences, I can see how the b-spline basis functions might tend to the Gaussian function, and the cardinal basis functions might tend to the sinc function.

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