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Wikipedia says:

the normalized cardinal B-splines tend to the Gaussian function

and writes them as "Bk". Meanwhile, Signal Reconstruction says:

The basis splines Bn are shown ... as the order increases, the functions approach the Gaussian function, which is exactly B.

basis splines

but then says

as the order increases, the cardinal basis splines approximate the sinc function, which is exactly η.

cardinal basis splines

Likewise, Signal Reconstruction with Cardinal Splines uses similar notation of ηn 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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up vote 1 down vote accepted

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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Ok, so "cardinal B-spline" and "cardinal basis spline" are not the same thing? – endolith Sep 11 '12 at 14:22
The word "cardinal" is used in two different senses (unfortunately). I just discovered that some people use the term "cardinal b-spline" to refer to a b-spline whose knots are successive integers. Cardinal basis splines are the things I mentioned in my answer -- they are the splines shown in your second set of pictures. So, not the same thing. – bubba Sep 16 '12 at 6:25

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