the normalized cardinal B-splines tend to the Gaussian function
and writes them as “Bk“. Meanwhile, cnx.org Signal Reconstruction says:
The basis splines Bn are shown … as the order increases, the functions approach the Gaussian function, which is exactly B∞.
but then says
as the order increases, the cardinal basis splines approximate the sinc function, which is exactly η∞.
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?
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.