Embedding of non-polynomial spline spaces

Abstract

Purpose

The aims of the paper are to obtain necessary and sufficient conditions ofexistence and smoothness for non-polynomial spline spaces of fifth order, toestablish the uniqueness of the B _φ -spline spaces in the class C⁴among mentioned spaces (under condition of fixed grid), and toprove the embedding of the B _φ -spline spaces corresponding to embedded grids.

Methods

In the paper, the approximation relations with initial grid and with completechain of vectors are applied to obtain the minimal spline spaces. Usage oflocally orthogonal chain of vectors gives opportunity to construct specialapproximation relations from which the initial space of B _φ splines is constructed.

Results

Deletion of a knot from initial grid gives a new grid, and as result, a newspace of B _φ splines is embedded in the initial space mentioned above.

Conclusions

Consequent deletion of the knots (one by one) generates the sequence of theembedded spaces of B _φ splines. Obtained results are successfully proved. They may be appliedto spline-wavelet decompositions.

Introduction

Wide application of spline approximations in the theory and practise is well known [1–4]. Essential point for spline constructions is the usage of approximationrelations which firstly appeared in the finite element method. Approximationrelations were introduced in [5–7]. Approximation and interpolation properties of spline-waveletdecompositions are based on relations just mentioned [8].

The main problem of spline-wavelet decompositions is the construction of the chain ofembedded spaces. The problem is discussed in the case of the spaces ofnon-polynomial splines for irregular grids [9–12]; there, the chains are constructed for smooth spaces (which are calledspaces of B _φ splines) of the first, second, and third orders. Spline spaces of arbitraryorder were constructed by [8], but the embedding was obtained only for special non-smooth splinespaces.

The aims of this paper are to obtain necessary and sufficient conditions of existenceand smoothness for non-polynomial spline spaces of fifth order, to establish theuniqueness of the B _φ -spline spaces in the class C⁴among mentioned spaces (of fixed grid), and to prove the embedding ofthe B _φ -spline spaces corresponding to embedded grids.

Methods

In the paper, the approximation relations with initial grid and with complete chainof vectors are applied to obtaining of the minimal spline spaces. Usage of locallyorthogonal chain of vectors gives opportunity to construct special approximationrelations from which the initial space of B _φ splines is constructed.

Results and discussion

Spaces of (X,A,φ) splines of fifth order

Consider the grid X={x _j }_j∈Z on interval (α,β),

$$X:\dots <x_{-1}<x_0<x_1<\dots ;$$

where x _j ∈(α,β), $\alpha =\underset{j\to -\infty }{lim}{x}_j$, $\beta =\underset{j\to +\infty }{lim}{x}_j$, $0<c_1\le \frac{x_{j+1}-x_j}{x_j-x_{j-1}}\le c_2$∀j∈Z, c₁, and c₂ are fixed.

Let φ(t) be column vector functionφ(t)∈R⁶,φ∈C⁵[α,β] such that

$$|det(\phi ,\phi \prime ,\phi \mathrm{\prime \prime },\phi \mathrm{\prime \prime \prime },\phi \mathit{\text{IV}},\phi V\left)\right(t\left)\right|\ge c>0\forall t\in [\alpha ,\beta ].$$

(1.1)

A set A={a _j }_j∈Z of column vectors a _j ∈R⁶is called a chain of vectors. Differentenumerations are permitted for the same chain of vectors: two differentenumerations can be distinguished from each other with constant summand anddirection of numbering, for example, $\mathbf{A}={\left\{{\mathbf{a}}_{j^{\prime }}\right\}}_{j^{\prime }\in \mathrm{Z}}$ with j^′=−j + j₀ (where j₀ is an integer constant) is another enumeration of the samechain.

A chain A={a _i }_i∈Z is locally orthogonal to a chain B={b _j }_j∈Zif there exist enumerations such that

$${\mathbf{b}}_j^T{\mathbf{a}}_{j-p}=0\forall j\in \mathrm{Z},p\in I_5,$$

(1.2)

where I₅={1,2,3,4,5}.

Lemma 1

If a chain A is locally orthogonal to a chain B, then B islocally orthogonal to A.

Proof

Using Equation 1.2, we get

$${\mathbf{a}}_j^T{\mathbf{b}}_{j+p}=0\forall j\in \mathrm{Z},p\in I_5.$$

Now, we enumerate the chain {b _j }_j∈Z by definition b_〈j〉=b_{j + 6}.

Then,

$${\mathbf{a}}_j^T{\mathbf{b}}_{〈j-p〉}=0\forall j\in \mathrm{Z},p\in I_5.$$

This concludes the proof. □

By Lemma 1 the local orthogonality is symmetric, and therefore, the discussedchains A and B are called locally orthogonal chains.

A chain A={a _j }_j∈Z is called a non-generated chain ifa _j ≠0∀j∈Z.

Consider a square matrix A _j =(a_j−5,a_j−4,a_j−3,a_j−2,a_j−1,a _j ).

The chain A={a _j }_j∈Z is called a complete chain of vectors ifdetA _j ≠0∀j∈Z. It is clear that the complete chain isnon-generated.

Let $\mathcal{A}$ denote a set of all complete chains.

Lemma 2

Suppose A={a _i }_i∈Zand B={b _j }_j∈Zare non-generated locally orthogonal chains,then the following conditions are equivalent:

• (1) the chain A is a complete chain,

• (2) the chain B is a complete chain.

Proof

Let B be a complete chain. If A is not complete, then thereexists j∈Z such that a _j =c₁a_j−1 + c₂a_j−2 + c₃a_j−3 + c₄a_j−4 + c₅a_j−5, where c₁,c₂,c₃,c₄,c₅∈R¹. Taking into account that A andB are locally orthogonal (see (1.2)), we obtain

$${\mathbf{b}}_j^T{\mathbf{a}}_j=0.$$

(1.3)

From Equation 1.2, we also deduce that

$${\mathbf{b}}_{j+p}^T{\mathbf{a}}_j=0p\in I_5.$$

(1.4)

Combining Equations 1.3 and 1.4 gives a _j =0. This contradiction proves the first part of the lemma. Second partis proved analogously. □

Lemma 3

If A={a _i }_i∈Z and B={b _j }_j∈Z are complete chains and relation (1.2) isfulfilled, then

$${\mathbf{b}}_j^T{\mathbf{a}}_j\ne 0,{\mathbf{b}}_{j+6}^T{\mathbf{a}}_j\ne 0.$$

(1.5)

Proof

The proof is by reductio ad absurdum. If there existsj∈Z such that relation (1.3) holds, then by Equation 1.4, weget a _j =0; this contradicts the completeness of the chain A. The firstrelation in Equation 1.5 is proved. Arguing as above, we find that thesecond relation in Equation 1.5 is true. □

Lemma 4

For an arbitrary complete chain A={a _j }_j∈Z, there exists a non-generated locallyorthogonal chain B={b _j }_j∈Zwhere the directions of vectors b _j are uniquely defined (up to non-zero constant factor).

Proof

We fix integer k; let b _k ∈R⁶be a vector satisfying

$${\mathbf{b}}_k^T{\mathbf{a}}_{k-p}=0,p\in I_5.$$

(1.6)

The conditions (1.6) are the linear system with respect to unknown vectorb _k :

$${({\mathbf{a}}_{k-5},{\mathbf{a}}_{k-4},{\mathbf{a}}_{k-3},{\mathbf{a}}_{k-2},{\mathbf{a}}_{k-1})}^T{\mathbf{b}}_k=0.$$

(1.7)

By assumption, the chain A is complete; hence, vectors a_k−5,a_k−4, a_k−3,a_k−2, and a_k−1are linearly independent. System (1.7) has aunique (up to constant factor) non-trivial solution b _k , which can be defined by the identity

$${\mathbf{b}}_k^T\mathbf{x}\equiv det({\mathbf{a}}_{k-5},{\mathbf{a}}_{k-4},{\mathbf{a}}_{k-3},{\mathbf{a}}_{k-2},{\mathbf{a}}_{k-1},\mathbf{x})\forall \mathbf{x}\in {\mathrm{R}}^6.$$

This concludes the proof. □

Corollary 1

For any complete chain, there exists a locally orthogonal complete chaindefined uniquely up to non-zero constant factors.

Proof

Combining Lemmas 2 and 4, we prove the corollary.

By definition, put

$$M={\cup }_{j\in \mathrm{Z}}(x_j,x_{j+1}),S_j=[x_j,x_{j+6}].$$

For $\mathbf{A}\in \mathcal{A}$ we define the functions ω _j (t), t∈M, and j∈Z, byapproximation relations

$$\begin{array}{l}\sum _{j^{\prime }\in \mathrm{Z}}{\mathbf{a}}_{j^{\prime }}{\omega }_{j\prime }\left(t\right)\equiv \phi \left(t\right)\forall t\in (x_k,x_{k+1})\forall k\in \mathrm{Z};\\ \text{supp}{\omega }_j\subset S_j\forall j\in \mathrm{Z}.\end{array}$$

(1.8)

If t is fixed in (x _k ,x_{k + 1}), then relation (1.8) contains at most six non-zerosummands:

$$\sum _{j=k-5}^k{\mathbf{a}}_j{\omega }_j\left(t\right)\equiv \phi \left(t\right).$$

(1.9)

Discuss relation (1.9) as the system of linear equations with respect tounknown values ω _j (t).

Let j be an arbitrary fixed integer. Consider relations (1.9) withk=j, k=j + 1, k=j+ 2, k=j + 3, k=j + 4, andk=j + 5. By Cramer’s theorem, we obtain

$$\begin{array}{ll}{\omega }_j\left(t\right)& =\frac{det\left({\mathbf{a}}_{j-5},{\mathbf{a}}_{j-4},{\mathbf{a}}_{j-3},{\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1},\phi \left(t\right)\right)}{det\left({\mathbf{a}}_{j-5},{\mathbf{a}}_{j-4},{\mathbf{a}}_{j-3},{\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1},{\mathbf{a}}_j\right)}\\ \text{for}t\in (x_j,x_{j+1}),\end{array}$$

(1.10)

$$\begin{array}{ll}{\omega }_j\left(t\right)& =\frac{det\left({\mathbf{a}}_{j-4},{\mathbf{a}}_{j-3},{\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1},\phi \left(t\right),{\mathbf{a}}_{j+1}\right)}{det\left({\mathbf{a}}_{j-4},{\mathbf{a}}_{j-3},{\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1},{\mathbf{a}}_j,{\mathbf{a}}_{j+1}\right)}\\ \text{for}t\in (x_{j+1},x_{j+2}),\end{array}$$

(1.11)

$$\begin{array}{ll}{\omega }_j\left(t\right)& =\frac{det\left({\mathbf{a}}_{j-3},{\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1},\phi \left(t\right),{\mathbf{a}}_{j+1},{\mathbf{a}}_{j+2}\right)}{det\left({\mathbf{a}}_{j-3},{\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1},{\mathbf{a}}_j,{\mathbf{a}}_{j+1},{\mathbf{a}}_{j+2}\right)}\\ \text{for}t\in (x_{j+2},x_{j+3}),\end{array}$$

(1.12)

$$\begin{array}{ll}{\omega }_j\left(t\right)& =\frac{det\left({\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1},\phi \left(t\right),{\mathbf{a}}_{j+1},{\mathbf{a}}_{j+2},{\mathbf{a}}_{j+3}\right)}{det\left({\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1},{\mathbf{a}}_j,{\mathbf{a}}_{j+1},{\mathbf{a}}_{j+2},{\mathbf{a}}_{j+3}\right)}\\ \text{for}t\in (x_{j+3},x_{j+4}),\end{array}$$

(1.13)

$$\begin{array}{ll}{\omega }_j\left(t\right)& =\frac{det\left({\mathbf{a}}_{j-1},\phi \left(t\right),{\mathbf{a}}_{j+1},{\mathbf{a}}_{j+2},{\mathbf{a}}_{j+3},{\mathbf{a}}_{j+4}\right)}{det\left({\mathbf{a}}_{j-1},{\mathbf{a}}_j,{\mathbf{a}}_{j+1},{\mathbf{a}}_{j+2},{\mathbf{a}}_{j+3},{\mathbf{a}}_{j+4}\right)}\\ \text{for}t\in (x_{j+4},x_{j+5}),\end{array}$$

(1.14)

$$\begin{array}{ll}{\omega }_j\left(t\right)& =\frac{det\left(\phi \left(t\right),{\mathbf{a}}_{j+1},{\mathbf{a}}_{j+2},{\mathbf{a}}_{j+3},{\mathbf{a}}_{j+4},{\mathbf{a}}_{j+5}\right)}{det\left({\mathbf{a}}_j,{\mathbf{a}}_{j+1},{\mathbf{a}}_{j+2},{\mathbf{a}}_{j+3},{\mathbf{a}}_{j+4},{\mathbf{a}}_{j+5}\right)}\\ \text{for}t\in (x_{j+5},x_{j+6}).\end{array}$$

(1.15)

Now, it follows from supp ω _j ⊂[x _j ,x_{j + 6}] (see (1.8)) that the function ω _j (t) is defined for all t∈M.

Equivalent record of formulas (1.10) to (1.15) is

$$\begin{array}{c}{\omega }_j\left(t\right)=\frac{det\left({\left\{{\mathbf{a}}_{j^{\prime }}\right\}}_{j^{\prime }\in J_k,j^{\prime }\ne j}\left|{|\prime }^j\phi \right(t)\right)}{det\left({\left\{{\mathbf{a}}_{j^{\prime }}\right\}}_{j^{\prime }\in J_k}\right)},t\in (x_k,x_{k+1}),k-j\in I_5;\end{array}$$

(1.16)

here J _k ={k−5,k−4,k−3,k−2,k−1,k},columns of determinants are ordered uniformly, symbol ||^′jφ(t) indicates that column vectorφ(t) has to be placed instead of the columnvector a _j .

By (1.1) the set of functions ω _j ∀j∈Z is a linearly independent system onarbitrary subinterval (a,b) of the interval(α,β). Consider the linear space

$${\mathrm{S}}_5(X,\mathbf{A},\phi )=C{l}_p\mathcal{L}{\left\{{\omega }_j\right\}}_{j\in \mathrm{Z}},$$

where $\mathcal{L}$ is the linear span, and C l _p is the closure in the topology of point convergence. The spaceS₅(X,A,φ) is called a spaceof minimal(X,A,φ)splines of fifthorder. □

Properties of minimal splines

Lemma 5

Suppose A is a complete chain and let k∈Z, t_∗∈[x _k ,x_{k + 1}] be fixed, the following relations areequivalent:

$$\begin{array}{c}\left(1\right)\\ \left(2\right)\end{array}\begin{array}{c}\underset{t\to t_{\ast },t\in (x_k,x_{k+1})}{lim}{\omega }_j\left(t\right)=0;\\ det\left({\left\{{\mathbf{a}}_{j^{\prime }}\right\}}_{j^{\prime }\in J_k,j^{\prime }\ne j}\parallel \prime j\phi \left(t_{\ast }\right)\right)=0.\end{array}$$

Proof

The proof is a consequence of (1.16). □

Lemma 6

Let A be a complete chain. For the validity of the equalities

$$\underset{t\to x_k-0}{lim}{\omega }_{k-6}\left(t\right)=0,\underset{t\to x_k+0}{lim}{\omega }_k\left(t\right)=0,$$

(2.1)

it is necessary (and if a_k−6 and a _k are not collinear, then it is also sufficient) that

$$\begin{array}{c}\underset{t\to x_k-0}{lim}{\omega }_j\left(t\right)=\underset{t\to x_k+0}{lim}{\omega }_j\left(t\right)\forall j\in \{k-5,k-4,k-3,k-2,k-1\}.\end{array}$$

(2.2)

Proof

Necessity. Replacement of k by k−1 in (1.9) gives

$$\sum _{j=k-6}^{k-1}{\mathbf{a}}_j{\omega }_j\left(t\right)\equiv \phi \left(t\right),t\in (x_{k-1},x_k).$$

(2.3)

Hence, by passing to the limits as t→x _k −0 and using the first condition (2.1), we obtain

$$\sum _{j=k-5}^{k-1}{\mathbf{a}}_j{\omega }_j(x_k-0)=\phi \left(x_k\right).$$

(2.4)

Analogously by using (1.9) and the second condition (2.1) ast→x _k + 0, we arrive at

$$\sum _{j=k-5}^{k-1}{\mathbf{a}}_j{\omega }_j(x_k+0)=\phi \left(x_k\right).$$

(2.5)

In view of the linear independence of vectors a_k−5, a_k−4, a_k−3, a_k−2, and a_k−1 by equalities (2.4) to (2.5), we obtainrelations (2.2). These completes the proof of necessity.

Sufficiency. Using (2.3) and (1.9), we have

$$\sum _{j=k-6}^{k-1}{\mathbf{a}}_j{\omega }_j(x_k-0)=\phi \left(x_k\right)$$

(2.6)

and

$$\sum _{j=k-5}^k{\mathbf{a}}_j{\omega }_j(x_k+0)=\phi \left(x_k\right),$$

(2.7)

respectively. It follows from (2.6), (2.7), and (2.2) that a_k−6ω_k−6(x _k −0)=a _k ω _k (x _k + 0); using linear independence of vectors a_k−6and a _k , we obtain equalities (2.1).

This completes the proof of Lemma 6. □

Theorem 1

Let A be a complete chain of vectors; then, the following conditionsare equivalent:

• (1) for any j∈Z functions ω _j (t) are continuous on (α,β);

• (2) limit values at the boundary points of supp ω _j are zero for all j∈Z.

Proof

Sufficiency. Since vector-function φ(t) is continuous,it will suffice to prove the continuity of the function ω _j (t) in the knots of the grid X. If a knot x _k is placed on the boundary of S _j , the continuity of ω _j (t) at x _k follows from the hypotheses of the theorem. If a knot x _k is placed inside S _j , then conditions of Lemma 6 are fulfilled (in the part of necessity),and therefore, the relations (2.2) are true. Sufficiency is proved.

Necessity is obvious.

By definition, put φ _k =φ(x _k ), ${\phi }_k^{\left(i\right)}={\phi }^{\left(i\right)}\left(x_k\right)$. □

Theorem 2

The following conditions are equivalent:

• (1) functions ω _j (t)(∀j∈Z) can be extended to (α,β) continuously;

• (2) relations

  $$det\left({\mathbf{a}}_{j-5},{\mathbf{a}}_{j-4},{\mathbf{a}}_{j-3},{\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1},{\phi }_j\right)=0\forall j\in \mathrm{Z}$$

  (2.8)

are fulfilled.

Proof

Let us show that the condition (2) in Theorem 1 is equivalent to relations(2.8). Indeed, by (1.10), we see that equalities (2.8) are equivalent to

$${\omega }_j(x_j+0)=0\forall j\in \mathrm{Z}.$$

(2.9)

Changing j to j + 6 in (2.8), we get

$$det\left({\mathbf{a}}_{j+1},{\mathbf{a}}_{j+2},{\mathbf{a}}_{j+3},{\mathbf{a}}_{j+4},{\mathbf{a}}_{j+5},{\phi }_{j+6}\right)=0\forall j\in \mathrm{Z};$$

(2.10)

now using (1.15), we obtain

$${\omega }_j(x_{j+6}-0)=0\forall j\in \mathrm{Z}.$$

(2.11)

Thus, the equalities (2.8) are equivalent to the equalities (2.9) and (2.11);the last ones indicate that (2) is fulfilled. To conclude the proof, itremains to use Theorem 1. □

Theorem 3

Suppose φ^(S)∈C(α,β),where S is a positive integer; then, the following conditions areequivalent:

• (1) the functions ${\omega }_j^{\left(S\right)}\left(t\right)(\forall j\in \mathrm{Z})$ can be extended to (α,β) continuously,

• (2) relations

  $$\begin{array}{l}det\left({\mathbf{a}}_{j-5},{\mathbf{a}}_{j-4},{\mathbf{a}}_{j-3},{\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1},{\phi }_j^{\left(S\right)}\right)=0\forall j\in \mathrm{Z}\\ \left(2.8\right)\end{array}$$

are fulfilled.

Proof

By differentiating Equation 1.9 and using the arguments applied for proofs ofTheorems 1 and 2, we arrive at the required result. □

B _φ splines of fifth order

Discuss a special case of relations (1.8) choosing a specific chain {a _j }_j∈Z.

First, we define vectors d _k ∈R⁶by identities

$${\mathbf{d}}_k^T\mathbf{x}\equiv det({\phi }_k,\phi k\prime ,\phi k\mathrm{\prime \prime },\phi k\mathrm{\prime \prime \prime },\phi k\mathit{\text{IV}},\mathbf{x})\forall \mathbf{x}\in {\mathrm{R}}^6,k\in \mathrm{Z},$$

(3.1)

and introduce vectors ${\mathbf{a}}_j^{\star }$ with symbolic determinant

$$\begin{array}{c}{\mathbf{a}}_j^{\star }=det\left(\begin{array}{lllll}{\phi }_{j+1}& \phi j+1\prime & \phi j+1\mathrm{\prime \prime }& \phi j+1\mathrm{\prime \prime \prime }& \phi j+1\mathit{\text{IV}}\\ \underset{j+2}{\overset{T}{\mathbf{d}}}{\phi }_{j+1}& \underset{j+2}{\overset{T}{\mathbf{d}}}\phi j+1\prime & \underset{j+2}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime }& \underset{j+2}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime \prime }& \underset{j+2}{\overset{T}{\mathbf{d}}}\phi j+1\mathit{\text{IV}}\\ \underset{j+3}{\overset{T}{\mathbf{d}}}{\phi }_{j+1}& \underset{j+3}{\overset{T}{\mathbf{d}}}\phi j+1\prime & \underset{j+3}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime }& \underset{j+3}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime \prime }& \underset{j+3}{\overset{T}{\mathbf{d}}}\phi j+1\mathit{\text{IV}}\\ \underset{j+4}{\overset{T}{\mathbf{d}}}{\phi }_{j+1}& \underset{j+4}{\overset{T}{\mathbf{d}}}\phi j+1\prime & \underset{j+4}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime }& \underset{j+4}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime \prime }& \underset{j+4}{\overset{T}{\mathbf{d}}}\phi j+1\mathit{\text{IV}}\\ \underset{j+5}{\overset{T}{\mathbf{d}}}{\phi }_{j+1}& \underset{j+5}{\overset{T}{\mathbf{d}}}\phi j+1\prime & \underset{j+5}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime }& \underset{j+5}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime \prime }& \underset{j+5}{\overset{T}{\mathbf{d}}}\phi j+1\mathit{\text{IV}}\end{array}\right).\end{array}$$

(3.2)

$${h_X}^{\text{def}}={\text{max}}_{j\in Z}\left(x_{j+1}-x_j\right).$$

Theorem 4

Suppose that (1.1) is valid. Then, there exists δ>0 suchthat for h _X <δ, the chain ${\left\{{\mathbf{a}}_j^{\star }\right\}}_{j\in \mathrm{Z}}$ is complete.

Proof

Using (1.1), we apply Taylor’s expansion for x=x _j elements of determinant (3.2); as a result, we get asymptotic behaviorof ${\mathbf{a}}_j^{\star }$ (as h _X →0) uniformly with respect tox∈(α,β). Using theasymptotics, the reader will easily prove the theorem. □

Theorem 5

The chains of vectors ${\left\{{\mathbf{d}}_j^T\right\}}_{i\in \mathrm{Z}}$ and ${\left\{{\mathbf{a}}_i^{\star }\right\}}_{i\in \mathrm{Z}}$ are locally orthogonal:

$${\mathbf{d}}_{j+p}^T{\mathbf{a}}_j^{\star }=0\forall j\in \mathrm{Z},p\in I_5.$$

(3.3)

Proof

Using (3.2), we get

$$\begin{array}{l}{\mathbf{d}}_{j+p}^T{\mathbf{a}}_j^{\star }\\ =det\left(\begin{array}{lllll}\underset{j+p}{\overset{T}{\mathbf{d}}}{\phi }_{j+1}& \underset{j+p}{\overset{T}{\mathbf{d}}}\phi j+1\prime & \underset{j+p}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime }& \underset{j+p}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime \prime }& \underset{j+p}{\overset{T}{\mathbf{d}}}\phi j+1\mathit{\text{IV}}\\ \underset{j+2}{\overset{T}{\mathbf{d}}}{\phi }_{j+1}& \underset{j+2}{\overset{T}{\mathbf{d}}}\phi j+1\prime & \underset{j+2}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime }& \underset{j+2}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime \prime }& \underset{j+2}{\overset{T}{\mathbf{d}}}\phi j+1\mathit{\text{IV}}\\ \underset{j+3}{\overset{T}{\mathbf{d}}}{\phi }_{j+1}& \underset{j+3}{\overset{T}{\mathbf{d}}}\phi j+1\prime & \underset{j+3}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime }& \underset{j+3}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime \prime }& \underset{j+3}{\overset{T}{\mathbf{d}}}\phi j+1\mathit{\text{IV}}\\ \underset{j+4}{\overset{T}{\mathbf{d}}}{\phi }_{j+1}& \underset{j+4}{\overset{T}{\mathbf{d}}}\phi j+1\prime & \underset{j+4}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime }& \underset{j+4}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime \prime }& \underset{j+4}{\overset{T}{\mathbf{d}}}\phi j+1\mathit{\text{IV}}\\ \underset{j+5}{\overset{T}{\mathbf{d}}}{\phi }_{j+1}& \underset{j+5}{\overset{T}{\mathbf{d}}}\phi j+1\prime & \underset{j+5}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime }& \underset{j+5}{\overset{T}{\mathbf{d}}}\phi j+1\mathrm{\prime \prime \prime }& \underset{j+5}{\overset{T}{\mathbf{d}}}\phi j+1\mathit{\text{IV}}\end{array}\right).\end{array}$$

If p=1, then (3.1) implies that the first row of the determinant iszero; if p=2,3,4,5, then there are two equal rows in the discusseddeterminant. This completes the proof of Theorem 5.

Suppose the chain ${\mathbf{A}}^{\star }={\left\{{\mathbf{a}}_j^{\star }\right\}}_{j\in \mathrm{Z}}$ is complete. We replace a _j by ${\mathbf{a}}_j^{\star }$ in (1.8). Then, we obtain specific minimal splines, whichare denoted by ${\omega }_j^{\star }$. □

Theorem 6

The relations ${\omega }_j^{\star }\in C^4(\alpha ,\beta )\forall j\in \mathrm{Z}$ are true.

Proof

Relations (3.3) can be rewritten as ${\mathbf{d}}_j\perp {\mathbf{a}}_{j-i}^{\star },i\in I_5$ or, equivalently,

$${\mathbf{d}}_j\perp \mathcal{L}({\mathbf{a}}_{j-5}^{\star },{\mathbf{a}}_{j-4}^{\star },{\mathbf{a}}_{j-3}^{\star },{\mathbf{a}}_{j-2}^{\star },{\mathbf{a}}_{j-1}^{\star }).$$

(3.4)

Using (3.1), we have

$${\mathbf{d}}_j\perp \mathcal{L}({\phi }_j,\phi j\prime ,\phi j\mathrm{\prime \prime },\phi j\mathrm{\prime \prime \prime },\phi j\mathit{\text{IV}}).$$

(3.5)

It follows from (3.4) and (3.5) that

$$\mathcal{L}({\mathbf{a}}_{j-5}^{\star },{\mathbf{a}}_{j-4}^{\star },{\mathbf{a}}_{j-3}^{\star },{\mathbf{a}}_{j-2}^{\star },{\mathbf{a}}_{j-1}^{\star })=\mathcal{L}({\phi }_j,\phi j\prime ,\phi j\mathrm{\prime \prime },\phi j\mathrm{\prime \prime \prime },\phi j\mathit{\text{IV}});$$

therefore, ${\phi }_k^{\left(S\right)}\in \mathcal{L}({\mathbf{a}}_{j-5}^{\star },{\mathbf{a}}_{j-4}^{\star },{\mathbf{a}}_{j-3}^{\star },{\mathbf{a}}_{j-2}^{\star },{\mathbf{a}}_{j-1}^{\star })$ for S=0,1,2,3,4. Thus,

$$det(\underset{j-5}{\overset{\star }{\mathbf{a}}},{\mathbf{a}}_{j-4}^{\star },{\mathbf{a}}_{j-3}^{\star },{\mathbf{a}}_{j-2}^{\star },{\mathbf{a}}_{j-1}^{\star },{\phi }_k^{\left(S\right)})=0.$$

To conclude the proof, we refer to Theorem 3. □

By definition, put

$${\mathrm{S}}_5^{\star }(X,\phi )=C{l}_p\mathcal{L}{\left\{{\omega }_j^{\star }\right\}}_{j\in \mathrm{Z}}.$$

According to Theorem 6, we obtain

$${\mathrm{S}}_5^{\star }(X,\phi )\subset C4(\alpha ,\beta )$$

by S₅(X,φ) denoting the set of all spaces of thesplines of fifth order (where grid X and function φ arefixed):

$${\mathbf{S}}_5(X,\phi )=\left\{{\mathrm{S}}_5\right(X,\mathbf{A},\phi \left)\right|\forall \mathbf{A}\in \mathcal{A}\}.$$

Theorem 7

There is a unique space ${\mathrm{S}}_5^{\star }(X,\phi )$ in S₅that is contained in C⁴(α,β).

$$S_5^{\ast }(X,\phi )={\mathbf{S}}_5(X,\phi )\cap C^4(\alpha ,\beta ).$$

Proof

By Theorem 3, the condition ω _j ∈C⁴(α,β) ∀j∈Z isequivalent to relations (2.12) for all S∈{0,1,2,3,4}. Supposea chain {a _j }_j∈Zsatisfies

$$\begin{array}{l}det\left({\mathbf{a}}_{j-5},{\mathbf{a}}_{j-4},{\mathbf{a}}_{j-3},{\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1},\phi j\left(S\right)\right)=0\\ \forall j\in \mathrm{Z},S=0,1,2,3,4.\end{array}$$

(3.6)

We wish to prove that vector a _j differs from ${\mathbf{a}}_j^{\star }$ with nonzero constant factors.

By Lemma 4, there exists a non-generated chain B={b _j }_j∈Zlocally orthogonal to chain {a _j }_j∈Z; directions of vectors b _j are defined uniquely (up to constant factor):

$${\mathbf{b}}_j\perp \mathcal{L}\{{\mathbf{a}}_{j-5},{\mathbf{a}}_{j-4},{\mathbf{a}}_{j-3},{\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1}\}.$$

Using condition (3.6), we have

$${\phi }_j^{\left(S\right)}\in \mathcal{L}\{{\mathbf{a}}_{j-5},{\mathbf{a}}_{j-4},{\mathbf{a}}_{j-3},{\mathbf{a}}_{j-2},{\mathbf{a}}_{j-1}\},S=0,1,2,3,4,$$

hence, b _j ⊥φ j(S), S=0,1,2,3,4.

Thus,

$${\mathbf{b}}_j\perp \mathcal{L}\{{\phi }_j,\phi j\prime ,\phi j\mathrm{\prime \prime },\phi j\mathrm{\prime \prime \prime },\phi j\mathit{\text{IV}}\}.$$

By assumption (1.1), vectors φ _j , φ j′, φ j ′′, φ j ′′′, and φ j IV are linearly independent; it follows that b _j =c _j d _j (see (3.1)), where constants c _j are non-zero (for all j∈Z). The chains {a _j }_j∈Z and ${\left\{{\mathbf{a}}_j^{\star }\right\}}_{j\in \mathrm{Z}}$, which are locally orthogonal to the chains {b _j }_j∈Z and {d _j }_j∈Z respectively, also differ by non-zero constantfactors c j⋆: ${\mathbf{a}}_j=c_j^{\star }{\mathbf{a}}_j^{\star }$. Taking into account Equations 1.10 to 1.15, we have ${\omega }_j={\omega }_j^{\star }/c_j^{\star }$; this completes the proof. □

Consider the relations

$$\sum _{i=k-5}^k{\mathbf{a}}_i^{\star }{\omega }_i^{\star }\left(t\right)=\phi \left(t\right),$$

(3.7)

where t∈(x _k ,x_{k + 1}). Multiply both sides (3.7) on the left by ${\mathbf{d}}_j^T$, wherej=k−5,k−4,k−3,k−2,k−1,k.By local orthogonality of the chains ${\left\{{\mathbf{d}}_j^T\right\}}_{i\in \mathrm{Z}}$ and ${\left\{{\mathbf{a}}_i^{\star }\right\}}_{i\in \mathrm{Z}}$, we obtain six scalar relations

$$\begin{array}{l}\left(\begin{array}{llllll}{\mathbf{d}}_{k-5}^T{\mathbf{a}}_{k-5}^{\star }& {\mathbf{d}}_{k-5}^T{\mathbf{a}}_{k-4}^{\star }& {\mathbf{d}}_{k-5}^T{\mathbf{a}}_{k-3}^{\star }& {\mathbf{d}}_{k-5}^T{\mathbf{a}}_{k-2}^{\star }& {\mathbf{d}}_{k-5}^T{\mathbf{a}}_{k-1}^{\star }& {\mathbf{d}}_{k-5}^T{\mathbf{a}}_k^{\star }\\ 0& {\mathbf{d}}_{k-4}^T{\mathbf{a}}_{k-4}^{\star }& {\mathbf{d}}_{k-4}^T{\mathbf{a}}_{k-3}^{\star }& {\mathbf{d}}_{k-4}^T{\mathbf{a}}_{k-2}^{\star }& {\mathbf{d}}_{k-4}^T{\mathbf{a}}_{k-1}^{\star }& {\mathbf{d}}_{k-4}^T{\mathbf{a}}_k^{\star }\\ 0& 0& {\mathbf{d}}_{k-3}^T{\mathbf{a}}_{k-3}^{\star }& {\mathbf{d}}_{k-3}^T{\mathbf{a}}_{k-2}^{\star }& {\mathbf{d}}_{k-3}^T{\mathbf{a}}_{k-1}^{\star }& {\mathbf{d}}_{k-3}^T{\mathbf{a}}_k^{\star }\\ 0& 0& 0& {\mathbf{d}}_{k-2}^T{\mathbf{a}}_{k-2}^{\star }& {\mathbf{d}}_{k-2}^T{\mathbf{a}}_{k-1}^{\star }& {\mathbf{d}}_{k-2}^T{\mathbf{a}}_k^{\star }\\ 0& 0& 0& 0& {\mathbf{d}}_{k-1}^T{\mathbf{a}}_{k-1}^{\star }& {\mathbf{d}}_{k-1}^T{\mathbf{a}}_k^{\star }\\ 0& 0& 0& 0& 0& {\mathbf{d}}_k^T{\mathbf{a}}_k^{\star }\end{array}\right)\\ \times \left(\begin{array}{l}{\omega }_{k-5}^{\star }\left(t\right)\\ {\omega }_{k-4}^{\star }\left(t\right)\\ {\omega }_{k-3}^{\star }\left(t\right)\\ {\omega }_{k-2}^{\star }\left(t\right)\\ {\omega }_{k-1}^{\star }\left(t\right)\\ {\omega }_k^{\star }\left(t\right)\end{array}\right)\end{array}$$

$$\begin{array}{l}=\left({\mathbf{d}}_{k-5}^T\phi \left(t\right),{\mathbf{d}}_{k-4}^T\phi \left(t\right),{\mathbf{d}}_{k-3}^T\phi \left(t\right),{\mathbf{d}}_{k-2}^T\phi \left(t\right),{\mathbf{d}}_{k-1}^T\phi \left(t\right),\right.\\ {\mathbf{d}}_k^T\phi \left(t\right){)}^T\end{array}$$

(3.8)

In the same way, multiplying both sides of (3.7) by ${\mathbf{d}}_j^T$ for j=k + 1,k + 2,k +3,k + 4,k + 5,k + 6 and using the localorthogonality of the chains ${\left\{{\mathbf{d}}_j^T\right\}}_{i\in \mathrm{Z}}$ and ${\left\{{\mathbf{a}}_i^{\star }\right\}}_{i\in \mathrm{Z}}$, we get the next system of relations

$$\begin{array}{l}\left(\begin{array}{llllll}{\mathbf{d}}_{k+1}^T{\mathbf{a}}_{k-5}^{\star }& 0& 0& 0& 0& 0\\ {\mathbf{d}}_{k+2}^T{\mathbf{a}}_{k-5}^{\star }& {\mathbf{d}}_{k+2}^T{\mathbf{a}}_{k-4}^{\star }& 0& 0& 0& 0\\ {\mathbf{d}}_{k+3}^T{\mathbf{a}}_{k-5}^{\star }& {\mathbf{d}}_{k+3}^T{\mathbf{a}}_{k-4}^{\star }& {\mathbf{d}}_{k+3}^T{\mathbf{a}}_{k-3}^{\star }& 0& 0& 0\\ {\mathbf{d}}_{k+4}^T{\mathbf{a}}_{k-5}^{\star }& {\mathbf{d}}_{k+4}^T{\mathbf{a}}_{k-4}^{\star }& {\mathbf{d}}_{k+4}^T{\mathbf{a}}_{k-3}^{\star }& {\mathbf{d}}_{k+4}^T{\mathbf{a}}_{k-2}^{\star }& 0& 0\\ {\mathbf{d}}_{k+5}^T{\mathbf{a}}_{k-5}^{\star }& {\mathbf{d}}_{k+5}^T{\mathbf{a}}_{k-4}^{\star }& {\mathbf{d}}_{k+5}^T{\mathbf{a}}_{k-3}^{\star }& {\mathbf{d}}_{k+5}^T{\mathbf{a}}_{k-2}^{\star }& {\mathbf{d}}_{k+5}^T{\mathbf{a}}_{k-1}^{\star }& 0\\ {\mathbf{d}}_{k+6}^T{\mathbf{a}}_{k-5}^{\star }& {\mathbf{d}}_{k+6}^T{\mathbf{a}}_{k-4}^{\star }& {\mathbf{d}}_{k+6}^T{\mathbf{a}}_{k-3}^{\star }& {\mathbf{d}}_{k+6}^T{\mathbf{a}}_{k-2}^{\star }& {\mathbf{d}}_{k+6}^T{\mathbf{a}}_{k-1}^{\star }& {\mathbf{d}}_{k+6}^T{\mathbf{a}}_k^{\star }\end{array}\right)\\ \times \left(\begin{array}{l}{\omega }_{k-5}^{\star }\left(t\right)\\ {\omega }_{k-4}^{\star }\left(t\right)\\ {\omega }_{k-3}^{\star }\left(t\right)\\ {\omega }_{k-2}^{\star }\left(t\right)\\ {\omega }_{k-1}^{\star }\left(t\right)\\ {\omega }_k^{\star }\left(t\right)\end{array}\right)\end{array}$$

$$\begin{array}{l}=\left({\mathbf{d}}_{k+1}^T\phi \left(t\right),{\mathbf{d}}_{k+2}^T\phi \left(t\right),{\mathbf{d}}_{k+3}^T\phi \left(t\right),{\mathbf{d}}_{k+4}^T\phi \left(t\right),{\mathbf{d}}_{k+5}^T\phi \left(t\right),\right.\\ {\left.{\mathbf{d}}_{k+6}^T\phi \left(t\right)\right)}^T\end{array}$$

(3.9)

Discuss relations (3.8) and (3.9) as the systems of linear equations with respectto unknown values ω _j (t), j∈J _k . Triangular matrices of these systems are non-singular since theirdiagonal elements are nonzero (see Lemma 3).

Note. Indeed, the solutions of the systems (3.8) and (3.9) areidentical.

Let I₀={0,1,2,3,4,5}, $I_p^1=\{-p,-p+1,\dots ,5\}$, $I_p^2=\{-4,-3,\dots ,6-p\}$, where p∈I₀.

Lemma 7

The formulas (3.8) define the functions ${\omega }_{k-p}^{\star }\left(t\right)\forall p\in I_0$ on the interval t∈(x _k ,x_{k + 1}) by values of the vector-functionφ(t) on this interval complemented with values ofmentioned vector function and its derivatives up to the fourth order in theknots $x_{k+i}\forall i\in I_p^1$.

Proof

From (3.1), it follows that vector d _j is defined by vector-function φ(t) and itsderivatives up to the fourth order in the knot x _j . Using (3.2), we see that the vector ${\mathbf{a}}_s^{\star }$ is defined by vector-function φ(t)and its derivatives up to the fourth order in the knots x_{s + i}∀i∈I₅. To conclude the proof, one should use the formula (3.8).□

Lemma 8

According to (3.9) the functions ${\omega }_{k-p}^{\star }\left(t\right)\forall p\in I_0$ for t∈(x _k ,x_{k + 1}) are defined by the vector-functionφ(t) on this interval complemented with values ofthe mentioned vector function and its derivatives up to the fourth order inthe knots $x_{k+i}\forall i\in I_p^2$.

Proof

We again recall (3.1) and conclude that d _j is defined by φ _j and ${\phi }_j^{\left(q\right)}$ for q∈{1,2,3,4}; besides, according toformula (3.2), the vector a _s is defined by φ_{s + i}and ${\phi }_{s+i}^{\left(q\right)}$, where q∈{1,2,3,4},i∈I₅. Now, the result follows from (3.9). □

Lemma 9

The functions ${\omega }_{k-p}^{\star }\left(t\right)$ for p∈I₀and t∈(x _k ,x_{k + 1}) are defined by $\phi \left(t\right)\left|t\in (x_k,x_{k+1})\right.$ complemented with φ_{k + i}and ${\phi }_{k+i}^{\left(q\right)}$ for q∈{1,2,3,4}, $i\in I_p^1\bigcap \underset{p}{\overset{2}{I}}$.

Proof

Since the values of vector function and its derivatives in the knots areunrestricted, the desired result follows from Lemmas 7 and 8. □

Theorem 8

The function ${\omega }_j^{\star }\left(t\right)$ is completely defined by $\phi \left(t\right)\left|t\in [x_j,x_{j+6}]\right.$.

Proof

Using Lemma 9, we put k=j,j + 1,j +2,j + 3,j + 4,j + 5. This completes the proofof Theorem 8. □

Calibration relations

We enlarge the origin grid X by deletion of knot x_{k + 1}, i.e., we put

$${\tilde{x}}_j=x_j\text{for}j\le k,{\tilde{x}}_j=x_{j+1}\text{for}j\ge k+1,$$

and discuss a new grid $\tilde{X}={\left\{{\tilde{x}}_j\right\}}_{j\in \mathrm{Z}}$,

$$\tilde{X}:\dots <{\tilde{x}}_{-1}<{\tilde{x}}_0<{\tilde{x}}_1<\dots .$$

We put ${\tilde{\phi }}_j={\phi }^{\prime }\left({\tilde{x}}_j\right)$, $\tilde{\phi }j\prime =\phi \prime \left({\tilde{x}}_j\right)$ and introduce the vectors ${\tilde{\mathbf{d}}}_k\in {\mathrm{R}}^6$ by identities

$${\tilde{\mathbf{d}}}_k^T\mathbf{x}\equiv det({\tilde{\phi }}_k,\tilde{\phi }k\prime ,\tilde{\phi }k\mathrm{\prime \prime },\tilde{\phi }k\mathrm{\prime \prime \prime },\tilde{\phi }k\mathit{\text{IV}},\mathbf{x})\forall \mathbf{x}\in {\mathrm{R}}^6,k\in \mathrm{Z}.$$

(4.1)

Now, we define vectors ${\tilde{\mathbf{a}}}_j^{\star }$ by symbolic determinant:

$$\begin{array}{c}{\tilde{\mathbf{a}}}_j^{\star }=det\left(\begin{array}{lllll}{\tilde{\phi }}_{j+1}& \tilde{\phi }j+1\prime & \tilde{\phi }j+1\mathrm{\prime \prime }& \tilde{\phi }j+1\mathrm{\prime \prime \prime }& \tilde{\phi }j+1\mathit{\text{IV}}\\ \underset{j+2}{\overset{T}{\tilde{\mathbf{d}}}}{\tilde{\phi }}_{j+1}& \underset{j+2}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\prime & \underset{j+2}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathrm{\prime \prime }& \underset{j+2}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathrm{\prime \prime \prime }& \underset{j+2}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathit{\text{IV}}\\ \underset{j+3}{\overset{T}{\tilde{\mathbf{d}}}}{\tilde{\phi }}_{j+1}& \underset{j+3}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\prime & \underset{j+3}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathrm{\prime \prime }& \underset{j+3}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathrm{\prime \prime \prime }& \underset{j+3}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathit{\text{IV}}\\ \underset{j+4}{\overset{T}{\tilde{\mathbf{d}}}}{\tilde{\phi }}_{j+1}& \underset{j+4}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\prime & \underset{j+4}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathrm{\prime \prime }& \underset{j+4}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathrm{\prime \prime \prime }& \underset{j+4}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathit{\text{IV}}\\ \underset{j+5}{\overset{T}{\tilde{\mathbf{d}}}}{\tilde{\phi }}_{j+1}& \underset{j+5}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\prime & \underset{j+5}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathrm{\prime \prime }& \underset{j+5}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathrm{\prime \prime \prime }& \underset{j+5}{\overset{T}{\tilde{\mathbf{d}}}}\tilde{\phi }j+1\mathit{\text{IV}}\end{array}\right).\end{array}$$

(4.2)

As before, we discuss approximation relations

$$\begin{array}{l}\sum _{j^{\prime }\in \mathrm{Z}}{\tilde{\mathbf{a}}}_{j^{\prime }}^{\star }{\tilde{\omega }}_{j^{\prime }}^{\star }\left(t\right)\equiv \phi \left(t\right)\forall t\in ({\tilde{x}}_k,{\tilde{x}}_{k+1})\forall k\in Z;\\ \text{supp}{\tilde{\omega }}_j=[{\tilde{x}}_j,{\tilde{x}}_{j+6}]\forall j\in \mathrm{Z}\end{array}$$

(4.3)

and put

$${\mathrm{S}}_5^{\star }(\tilde{X},\phi )=C{l}_p\mathcal{L}{\left\{{\tilde{\omega }}_j^{\star }\right\}}_{j\in \mathrm{Z}}.$$

Theorem 9

If the grid $\tilde{X}$ is so fine that the chain ${\tilde{\mathbf{A}}}^{\star }={\left\{{\tilde{\mathbf{a}}}_j^{\star }\right\}}_{j\in \mathrm{Z}}$ is complete, then

$${\mathrm{S}}_5^{\star }(\tilde{X},\phi )\subset {\mathrm{S}}_5^{\star }(X,\phi ).$$

(4.4)

Proof

From the definition of the grid $\tilde{X}$ and (4.2) to (4.3), we deduce that

$${\tilde{\omega }}_j^{\star }\left(t\right)\equiv {\omega }_j^{\star }\left(t\right),{\tilde{\mathbf{a}}}_j^{\star }={\mathbf{a}}_j^{\star }\text{for}j\le k-6,$$

(4.5)

$${\tilde{\omega }}_j^{\star }\left(t\right)\equiv {\omega }_{j+1}^{\star }\left(t\right),{\tilde{\mathbf{a}}}_j^{\star }={\mathbf{a}}_{j+1}^{\star }\text{for}j\ge k+1.$$

(4.6)

With Equations 1.8 and 4.3, we have

$$\sum _{j^{\prime }\in \mathrm{Z}}{\tilde{\mathbf{a}}}_{j^{\prime }}^{\star }{\tilde{\omega }}_{j^{\prime }}^{\star }\left(t\right)\equiv \sum _{j\in \mathrm{Z}}{\mathbf{a}}_j^{\star }{\omega }_j^{\star }\left(t\right)\forall t\in (\alpha ,\beta ),$$

and (after annihilation of identical terms, defined by relations (4.5) to(4.6)) we obtain

$$\sum _{j=k-5}^k{\tilde{\mathbf{a}}}_j^{\ast }{\tilde{\omega }}_j^{\ast }\left(t\right)\equiv \sum _{j\prime =k-5}^{k+1}{\mathbf{a}}_{j\prime }^{\ast }{\omega }_{j\prime }^{\ast }\left(t\right)\forall t\in (\alpha ,\beta ).$$

(4.7)

Discuss relations (4.7) as a system of linear equations with respect tounknown values $\left\{{\tilde{\omega }}_{k-i}^{\star }\right(t\left)\right\}$, i∈I₀. Because of the completeness of the chain ${\tilde{\mathbf{A}}}^{\star }$, the matrix of the system is nonsingular. Now, we obtainmentioned functions $\left\{{\tilde{\omega }}_{k-i}^{\star }\right(t\left)\right\}$, i∈I₀, as linear combinations of basic functions ${\omega }_j^{\star }\left(t\right)$ of the space ${\mathrm{S}}_5^{\star }(X,\phi )$; these representations are calibration relations indiscussed case [8]. The inclusion (4.4) is proved. □

Conclusions

Consequent deletion of the knots (one by one) generates the sequence of the embeddedspaces of B _φ splines. Obtained results are successfully proved. They may be applied tospline-wavelet decompositions.

Author’s information

YKD is the Head of the Department of Parallel Algorithms. He is also a professor anda doctor of science.