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8URAL MATHEMATICAL JOURNAL, Vol. 3, No. 1, 2017
CALIBRATION RELATIONS FOR ANALOGUES OF THE BASIS SPLINES WITH UNIFORM NODES1
Valerii T. Shevaldin
N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences;
Ural Federal University, Ekaterinburg, Russia, Valerii.Shevaldin@imm.uran.ru
Abstract: The paper deals with generalized linear and parabolic B—splines with the uniform nodes constructed by means only one function y(x). For such splines in this paper conditions have been found that guarantee satisfaction of two—scale relations.
Key words: B-spline, uniform nodes, two-scale relations.
Introduction
In contemporary mathematics, various generalizations of the polynomial spline-functions regularly appear. Besides the well-known ^-splines (see, for example [1]), let us note the source-representative splines [2], the Rvachev functions [3], the Leontiev splines [4], the Kvasov iso-geometric splines [5], the Demyanovich B^-splines [6], and so on.
Recently, the Author [7] suggested another generalization of a known construction of the parabolic basis spline (of the B-spline) with the uniform nodes; this spline is constructed by means of only one function p e C 1[-h, h] (h > 0).
In [7], the approximative and form-retaining properties of the local non-interpolating splines were investigated. These ones are linear combinations of shifts of the suggested B-splines. As particular cases, there were considered examples of exponential, elliptic, and hyperbolic local splines with arbitrary collocation of nodes.
It is well known that the polynomial splines have played important role in development of the wavelet theory (see, e.g., [8-10]). Namely, in constructing the wavelet decompositions of the space L2(R), embeddedness of spaces {Vjon refining meshes is used. This embeddedness follows from presence of scaling (multiple-scaled) relations (see [8, § 4.3])) for the basis functions.
But note that not each basis function B(x) satisfies the general scaling equation of the form
B(x) = hjB(2x - jh) (x e R),
jez
and finding such functions B(x) is a complicated problem.
In the present paper, conditions on the function p are given that guarantee implementation of analogues for the two-scaled relations for the generalized parabolic B-splines from work [7] (all necessary definitions are given below). Moreover, an analogous problem is considered for the generalized linear B-splines and corresponding examples are given.
It is worthy to note that we have obtained these results without application of the harmonic analysis techniques.
*The paper was originally published in Trudy Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 2011. Vol. 17, no 3. P. 319-323 (in Russian).
1. Generalized parabolic B—splines
Let h > 0 and C = C[a, b] be the space of continuous functions given on the segment [a, b] with usual definition of the norm
\\f ||c = max \f (x)\.
xE[a,b\
Fix the function p given on the segment [—2h, 2h] and satisfying the following conditions:
p e C—2h, 2h], p(—x) = p(x) (x e [0, 2h]), p(0) = p'(0) = 0. The B-spline corresponding to this function p (see [7]) is described by the formula
p(x), x e [0, h],
(1.1)
Bh,,2(x) = m(h) <
2p(h) - p(x - h) - p(x - 2h), x e [h, 2h],
p(3h - x), 0,
x e [2h, 3h], x e [0,3h].
(1.2)
Here, m = m(h) > 0 is the normalizing multiplier.
In the classic case, the normalized parabolic B-spline with the uniform nodes 0, h, 2h 3h (see [11]) is obtained from this definition if to set p(x) = x2 and m(h) = 1/(2h2).
Note evident properties of the function Bh2(x) that follow from conditions (1.1):
suppBh>2(x) = [0, 3h], B'h 2 e C(R), Bh,2(3h — x) = Bh>2(x),
(i.e., the function Bh,2 is even w.r.t. the middle carrier point x = (3h)/2). If to suppose another condition to be satisfied that the function p(x) does not decrease on [0, h], then the graph of Bh 2(x) will have the form of a symmetric "cap" (w.r.t. the point x = (3h)/2) as a the parabolic B-spline with the uniform nodes.
In [7], for such functions p, the local splines of the following form were investigated
5 (x) = S(f, x) = £ y3 Bh,2( x + y - jh)
jez
where yj = f (jh), f : R ^ R. It has been proved that these splines locally satisfy to the property of retaining the original data yj (of the 1-monotonicity type) in the following sense: if y—\ < yi < yi+i (l e Z), then the spline S(x) does not decrease on the segment [(l — 1/2)h, (l + 1/2)h] (l e Z). Together with the function Bh 2(x), consider the function
B2h,2(x) = m(2h)
p(x),
x e [0,2h],
2p(2h) - p(x - 2h) - p(x - 4h), x e [2h, 4h],
p(6h - x), 0,
x e [4h, 6h], x e [0,6h]
that is obtained from he function Bh,2 by formal substitution of the parameter h by the 2h one. It is evident that, in a general case, the graph of this function can not be obtained from the graph of the function Bh,2 by two-times extension along the horizontal axis as it happened un the classic polynomial case (see [8, § 4.3]). This is since nowhere the demand of homogeneity property of the function p is imposed. But this is the key reasoning in the described constructing. So, in the subsequent investigation of the wavelets on the basis of these basis functions, the embeddedness
of corresponding subspaces {Vjon the refining meshes must be understood in some other sense.
In this paper, we are searching for an answer on the following question. For what functions p satisfying conditions (1.1), there exist real numbers A1, A2, A3, and A4 such that for any x e R the equality holds
B2h(x) = AiBh,2(x) + A2BK2(x - h) + A3Bht2(x - 2h) + ABh>2(x - 3h)? (1.3)
We call this equation the scaling (two-scaled) relation for the generalized B-spline that is determined by formula (1.2). In the subsequent formulas, the expression o/o is supposed to be equal to 1.
Theorem 1. Let the function p satisfy conditions (1.1). Then equality (1.3) holds iff there exists such a number X e R, for which the following equalities hold:
= p(t + h) - 2p(h) + p(t - h) + p(t) = p(t - 2h) - 2p(h) + p(t - h) + p(t) = = W) = t(t - h) =
2p(2h) - p(t - 2h) - p(t) - p(t - h) 2p(2h) - p(t - h) - p(t + h) - p(t)
2p(h) - p(t - h) 2p(h) - p(t)
^ t<h.
(1.4)
Proof. By virtue of symmetry of the generalized B-spline w.r.t. the middle of the segment carrier, it is possible to think that A1 = A4 and A2 = A3.
Consider equality (1.3) as an equation w.r.t. the coefficients A1, A2, A3 and A4 on each segment [0, h], [h, 2h],..., [5h, 6h]. We obtain that
A1 = A4 = m{2K)
Ao = A3 =
m(h) '
m(2h) [p(t + h) - 2p(h) + p(t - h) + ^(t)j
m(h) p(t)
_ m(2h) [^(2h - t) - 2p(h) + p(t - h) + ^(t)j _
m(h) p(t - h)
_ m(2h) [2^(2h) - p(t - 2h) - p(t) - p(t - h)] _ _ m(h) 2p(h) - p(t - h) _
_ m(2h) [2^(2h) - p(t - h) - p(t + h) - p(t)] _ m(h) 2p(h) - p(t) '
□
Examples. Give examples of three functions p that satisfy equalities (1.4). In the sequel for simplicity, we put m(2h) _ m(h).
Example 1. Let p(x) _ x2 (the parabolic splines). Then Ai _ A4 _ 1, Ao _ A3 _ 3 are the binomial coefficients from [8, formula 4.3.4].
Example 2. Let p(x) _ cosh(^x) - 1 (0 > 0) (the exponential splines corresponding to the linear differential operator of the third order with the form L3 _ L3(D) _ D(D2 - ¡32), where D is the differentiation symbol). Then A1 _ A4 _ 1, A2 _ A3 _ 1 + 2cosh¡h.
Example 3. Let p(x) = 1 — cos ax (a > 0), i.e., be the trigonometric splines corresponding to the linear differential operator of the third order with the form L3 = L3(D) = D(D2 + a2). Then Ai = A4 = 1, A2 = A3 = 1 + 2 cos ah.
In connection with the latter two examples, note the Author's work [12]. There the scaling relations are constructed for the B-L-splines (of an arbitrary order) in more generalized form than in (1.3).
2. Generalized linear B—splines
The scheme suggested for obtaining the two-scaled relations can be expanded onto the generalized linear B-splines.
Let the function p be given on the segment [0,2h] and satisfy the following conditions:
p e C[0,2h], p(0) = 0. (2.1)
The generalized linear B-spline is described by the formula
{p(x), x e [0,h], p(2h — x), x e [h,2h], (2.2)
0, x e [0,2h].
Here, m(h) > 0 is the normalizing multiplier. If to put p(x) = x and m(h) = 1/h, then formula (2.2) defines the normalized linear B-spline (see, for example [11]).
It is evident that suppBhi = [0, 2h], Bhi e C(R), Bhi(2h — x) = Bhi(x) (x e [0,h]). Also, consider the function
{p(x), x e [0,2h], p(4h — x), x e [2h, 4h], 0, x e [0,4h]
that was obtained by formal substitution of the parameter h by 2h one in the function Bh,i. We are interested in the question: for what p the equality holds
B2h,i(x) = CiBh,i(x) + C2Bh,i(x — h) + C3Bh,i(x — 2h) (x e R), (2.3)
where Ci, C2, and C3 are some real numbers?
Theorem 2. Let the function p satisfy conditions (2.1). Then equality (2.3) holds iff there exists such a number X e R for which the following equalities hold:
X = p(* + h) — p(h — t) = p(2h — t) - p(t) (0 < t < h). (2.4)
p(t) p(h — t) V - - ; v ;
Proof. It is similar to one of Theorem 1. Under this,
m(2h)
C\ = C3 =
m(2h) [p(t + h
C2 =
m(h)
m(2h) [p(t + h) - p(h - t)] _ m(2h) [p(2h - t) - p(t)]
m(h) p(t) m(h) p(h -1)
□
Examples. As in the previous paragraph, it is possible to give examples of three functions p satisfying equalities (2.4). Again for simplicity, we put m(2h) = m(h).
Example 4. Let p(x) = x (the linear splines). Then Ci = C3 = 1, C2 = 2 are the binomial coefficients from [8, formula 4.3.4].
Example 5. Let p(x) = sinh/3x (0 > 0), i.e., be the exponential splines of the second order corresponding to the linear differential operator of the form L2 = L2(D) = D2 — 02. Then Ci = C3 = 1, C2 = 2 cosh ph.
Example 6. Let p(x) = sinh ax (a > 0), i.e., be the trigonometric splines of the second order corresponding to the linear differential operator of the form L2 = L2(D) = D2 + a2. Then Ci = C3 = 1, C2 = 2 cos ah.
3. Conclusion
It would be interesting to construct examples of other functions p satisfying relations (1.4) or (2.4). But it is not clear, how using only one function p, it is possible to construct analogues of the polynomial B-splines of more high degrees, i.e., to derive formules of the type (1.2).
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