login
A290826
The sequence b(n,m) of the m polynomial coefficients of the n-th order B-spline wavelet scaled by n!2^n, read by rows, with n in {0,1,2,...} and m in {1,2,3,...,2(n+1)^2}.
1
1, -1, 2, 0, -6, 4, 6, -8, -2, 4, 4, 0, 0, -20, 24, -6, 40, -96, 54, -40, 144, -126, 20, -96, 114, -4, 24, -36, 8, 0, 0, 0, -56, 96, -48, 8, 168, -576, 624, -216, -280, 1440, -2400, 1296, 280, -1920, 4320, -3184, -168, 1440, -4080, 3816, 56, -576, 1968, -2232, -8, 96, -384, 512
OFFSET
0,3
COMMENTS
The n-th order B-spline wavelet function B_n(x) may be calculated with the expression B_n(x) = (1/(2^n)) Sum_{k=0..n+1} (-1)^k binomial(n+1,k) N_n(2x-k)
where
* N_n(x) is the n-th order B-spline and as such (see A289358)
N_n(x) = (1/n!)Sum_{m=0..n+1} (-1)^m binomial(n+1,m)(x-m)^n step(x-m),
* step(x) is the step function defined as step(x) = {1 for x >= 0
{0 otherwise.
From these definitions, it is apparent that the coefficients of the polynomials induced by n!2^n*B_n(x) are integers and can be "flattened" (as in the Pascal triangle A007318) to form an integer sequence, part of which is listed above.
REFERENCES
Daniel J. Greenhoe, "Wavelet Structure and Design" (2013), isbn13: 9780983801139, page 318, Theorem H.1.
Ole Christensen, "An Introduction to Frames and Riesz Bases" (2003), isbn13: 9780817642952, pages 335--338, equation (14.41) and Figure 14.4.
Ole Christensen, "Frames and bases: An Introductory Course" (2008), isbn13: 9780817646776, pages 283--284.
LINKS
Daniel J. Greenhoe, Technical report for this sequence.
Daniel J. Greenhoe, Maxima script supporting this sequence.
Daniel J. Greenhoe, B-splines and B-spline wavelets, Technical Report [version 0.20], July 2017.
Ole Christensen, equation (14.41), "An Introduction to Frames and Riesz Bases" (2003), ISBN 9780817646783.
Ole Christensen, equation (11.63), "Frames and bases: An Introductory Course" (2008), ISBN 9780817646783.
Daniel J. Greenhoe, Wavelet Structure and Design, [version 1.20], January 2017, "Mathematical Structure and Design" series, volume 3, Theorem H.1, pages 267--268.
Daniel J. Greenhoe, Structure and Analysis of Mathematical Spaces, [version 0.30], January 2017, "Mathematical Structure and Design" series, volume 2, Proposition 11.4 (Power Sums), pages 186--187.
FORMULA
The n-th order B-spline B_n(x) may be expressed as a set of polynomials as described in the COMMENTS section.
EXAMPLE
The m=2(n+1)^2 coefficients for the n-th order B-spline wavelet B_n(x) begin at the sequence index value p=2Sum_{k=0..n}k^2=2(1/6)n(n+1)(2n+1) and end at index value p+2(n+1)^2-1.
Each set of m=2(n+1)^2 coefficients for n=0,1,2,... can be written in the form of an (2n+2)X(n+1) matrix A_n as...
A_0 = [ 1] for n=0 (index values 0 to 1)
[-1]
[ 2 0]
A_1 = [-6 4] for n=1 (index values 2 to 9)
[ 6 -8]
[-2 4]
[ 4 0 0]
[-20 24 -6]
A_2 = [ 40 -96 54] for n=2 (index values 10 to 27)
[-40 144 -126]
[ 20 -96 114]
[ -4 24 -36]
That is, the sequence of integers induces a sequence of (2n+2)X(n+1) matrices (A_0, A_1, A_2, ...).
Taking the n=2 case for example, the coefficients for B_2(x) begin at index value p=2(0+1+4)=10, end at index value p+2(3^2)-1=27, and are scaled by 2!2^2=8. Using this data yields the following expression for B_2(x):
[ 4 0 0 : for 0 <= x < 0.5] [x^2]
[-20 24 -6 : for 0.5 <= x < 1.0] [ x ]
8B_2(x) = [ 40 -96 54 : for 1.0 <= x < 1.5] [ 1 ]
[-40 144 -126 : for 1.5 <= x < 2.0]
[ 20 -96 114 : for 2.0 <= x < 2.5]
[ -4 24 -36 : for 2.5 <= x < 3.0]
[ 0 0 0 : otherwise ]
{ 4x^3 : for 0 <= x < 0.5}
{-20x^3 +24x^2 -6 : for 0.5 <= x < 1.0}
= { 40x^3 -96x^2 +54 : for 1.0 <= x < 1.5}
{-40x^3 +144x^2 -126 : for 1.5 <= x < 2.0}
{ 20x^3 -96x^2 114 : for 2.0 <= x < 2.5}
{ -4x^3 +24x^2 -36 : for 2.5 <= x < 3.0}
{ 0 : otherwise }
Note: Sum_{k=1..n}k^2 is called a "power sum".
For proof that p=Sum_{k=0..n}k^2=(1/6)n(n+1)(2n+1) (as indicated above),
see the "Structure and Analysis of Mathematical Spaces" link.
PROG
(Maxima)
reset();
kill(all);
load(orthopoly);
n:2;
Bnx:(1/n!)*(1/2^n)*sum((-1)^k*binomial(n+1, k)
*sum((-1)^m*binomial(n+1, m)*(2*x-m-k)^n*unit_step(2*x-m-k), m, 0, n+1), k, 0, n+1);
assume(x<=0); print(n!*2^n, "B(x)= ", expand(n!*2^n*Bnx), " for x<=0"); forget(x<=0);
for i:0 thru (n+0.5) step 0.5 do(
assume(x>i, x<(i+0.5)),
print(n!*2^n, "B(x)= ", expand(n!*2^n*Bnx), " for ", i, "<x<", i+0.5),
forget(x>i, x<(i+0.5))
);
assume(x>(n+1)); print(n!*2^n, "B(x)= ", expand(n!*2^n*Bnx), " for x>", n+1); forget(x>(n+1));
CROSSREFS
Cf. A289358.
Sequence in context: A242561 A372767 A131595 * A124228 A115879 A115880
KEYWORD
sign,tabf
AUTHOR
Daniel J. Greenhoe, Aug 11 2017
STATUS
approved