**On General Smoothness of Minimal Splines of the Lagrange Type**

YURI K.DEM’YANOVICH Department of Parallel Algorithms

St.-Petersburg State University University str., 7/9, Saint Petersburg

RUSSIA y.demjanovich@spbu.ru

TATJANA O.EVDOKIMOVA Department of Parallel Algorithms

St.-Petersburg State University University str., 7/9, Saint Petersburg

RUSSIA t.evdokimova@spbu.ru

EVELINA V.PROZOROVA Department of Parallel Algorithms

St.-Petersburg State University University str., 7/9, Saint Petersburg

RUSSIA e.prozorova@spbu.ru

*Abstract: In many cases the smoothness of splines is important (for qualitative approximation, for the calculation*
of a number of functionals, etc.). In the case of discontinuity of approximated functions it is difficult to use
ordinary splines. It is desirable to have splines with similar properties of the approximated function. The purpose
of this paper is to introduce the concept of general smoothness with the aid of linear functionals having a definite
location of supports. Splines are often used for processing numerical information flows; a lot of scientific papers
are devoted to these investigations. Sometimes spline treatment implies to the filtration of the mentioned flows or
to their wavelet decomposition. A discrete flow often appears as a result of analog signal sampling, representing
the values of a function, and in this case, the splines of the Lagrange type are used. In all cases, it is highly
desirable that the generalized smoothness of the resulting spline coincides with the generalized smoothness of
the original signal. Here we formulate the necessary and sufficient conditions for general smoothness of splines,
and also a toolkit is being developed to build mentioned splines. The proposed scheme allows us to consider
splines generated by functions from different spaces and to apply the obtained result to sources which can appear
in physics, chemistry, biology, etc.

*Key–Words: splines, general smoothness, chains of vectors, approximate relations*

**1** **Introduction**

The continuity of splines was considered from the mo- ment of their appearance. Often the requirement of a certain continuity (spline or its derivatives) was in- cluded in the definition of a spline (see [1] – [6]).

Basically these considerations concern polynomial splines, the last of which were defined as piecewise polynomial functions. For non-polynomial splines obtained from approximation relations (for the so- called minimal splines), necessary and sufficient con- tinuity conditions have been obtained relatively re- cently (see [17]).

In many cases the continuity of splines is impor- tant (for qualitative approximation, for calculation of functional values, etc.), since the functions approxi- mated by them are, as a rule, continuous. A study of the continuity of splines and their derivatives is de- voted to a lot of work (see also [7] – [18]).

Splines are often used for processing numerical information flows; a lot of scientific papers are de- voted to these investigations. Sometimes spline treat- ment implies to the filtration of the mentioned flows or to their wavelet decomposition. A discrete flow often appears as a result of analog signal sampling, repre- senting the values of a function, and in this case, the

splines of the Lagrange type are used.

In all cases, it is highly desirable that the general- ized smoothness of the resulting spline coincides with the generalized smoothness of the original signal.

When a discontinuous function is approximated, the behavior of the function in a neighborhood of the discontinuity point is often known. Approximation of that function by ordinary splines is difficult. It is de- sirable to have splines with properties similar to prop- erties of the approximated function.

Such properties can be formulated with the help of linear functionals, in particular, the usual continuity of the function can be considered as an equality of the values of two linear functionals applied to this func- tion: one functional is the left limit of the function at this point, and the second is the right limit of it. The purpose of this paper is to introduce the concept of general smoothness with the aid of linear functionals having a definite location of supports. In what follows general smoothness are called pseudo-continuity.

Here we formulate the necessary and sufficient conditions for the pseudo-continuity of splines, and also a toolkit is being developed to build pseudo- continuous splines. The proof use properties of com- plete chains of vectors, local orthogonality mentioned chains, as well as the vector identities.

Thus, in this paper we propose a general scheme
that allows us to consider splines generated by func-
*tions from spaces C*^{l}*, L*_{p}*, W*_{p}* ^{l}*, and handle nonsmooth
data, and sources which can be generated in physics,
chemistry, biology, etc.

**2** **Preliminaries**

*An ordered set A = {a**j**}** _{j∈Z}* of vectors a

*j*

*∈ R*

^{m+1}*is called a chain of vectors. There are different enu-*merations of the same chain; moreover, two enumer- ations can differ by only a constant term and direc-

*tion of enumeration, for example, A = {a*

_{j}

^{0}*| j*

*=*

^{0}*−j + j*_{0}*, j ∈ Z} (where j*_{0} is an integer constant) is
another enumeration of the same chain.

*Chain A = {a*_{i}*}*_{i∈Z}*is called locally orthogonal*
*to a chain B = {b*_{j}*}** _{j∈Z}*, if there exists an enumera-
tion such that

b^{T}* _{j}*a

_{j−p}*= 0 ∀j ∈ Z,*(1)

*p ∈ I*

_{m}*, I*

_{m}*= {1, 2, . . . , m}.*

The next assertion is obvious.

**Lemma 1. If chain A is locally orthogonal to***chain B, then chain B is locally orthogonal to chain*
*A.*

The local orthogonality is symmetric, and we can
*say that chains A and B are locally orthogonal. Chain*
*A is nondegenerate if it does not contain zero ele-*
ments and degenerate in the opposite case.

*We denote by A** _{j}* a matrix with columns
a

_{j−m}*, a*

_{j−m+1}*, . . . , a*

_{j−1}*, a*

*i.e.,*

_{j}*A** _{j}* =

^{³}a

_{j−m}*, a*

_{j−m+1}*, . . . , a*

_{j−1}*, a*

_{j}^{´}

*.*

*Chain A = {a*

_{i}*}*

_{i∈Z}*is called complete if det A*

_{j}*6= 0*

*for all j ∈ Z. It is clear that a complete chain is non-*degenerate. The set of all complete chains is denoted

*by A .*

**Lemma 2.** *Let A = {a**i**}*_{i∈Z}*and B =*
*{b*_{j}*}*_{j∈Z}*be locally orthogonal and nondegenerate*
**chains. Then chain A is complete if and only if chain****B is complete.**

**Lemma 3. If chains A = {a***i**}*_{i∈Z}*and B =*
*{b*_{j}*}*_{j∈Z}*are complete and (1) holds, then*

b^{T}* _{j}*a

_{j}*6= 0,*b

^{T}*a*

_{j+m+1}

_{j}*6= 0.*(2)

The proof of formula (2) follows by contradiction.

**Lemma 4. For any complete chain of vectors a***nondegenerate locally orthogonal chain exists. The*
*directions of vectors of this chain are uniquely deter-*
*mined.*

**Corollary 1. For any complete chain there exists***a locally orthogonal complete chain that is uniquely*
*determined up to a nonzero constant factor.*

The proof follows from Lemma 2 and Lemma 4.

**Lemma 5.** *Let A = {a*_{k}*} be complete a chain.*

*Suppose a chain B = {b**k**} is obtained by the formula*
b^{T}_{k}*x ≡ det(a*_{k−m}*, a*_{k−m+1}*, . . . , a*_{k−1}*, x)*

*∀x ∈ R*^{m+1}*,*

*and a chain C = {c*_{j}*} is obtained by the relation*
c^{T}_{j}*x ≡ det(b*_{j+1}*, b*_{j+2}*, . . . , b*_{j+m}*, x)* (3)

*∀x ∈ R*^{m+1}*.*

*Then the relation c**j* *= λ** _{j}*a

_{j}*∀j ∈ Z holds, where*

*the constants λ*

_{j}*are not zero.*

**Proof. According to Lemma 2 chain B is com-**
plete. It is obvious that b_{k}*⊥ a** _{k−m}*, b

_{k}*⊥ a*

*,*

_{k−m+1}*. . ., b*

_{k}*⊥ a*

_{k−1}*∀k ∈ Z; the last relations are equiv-*alent to the next ones a

_{j}*⊥ b*

*, a*

_{j+1}

_{j}*⊥ b*

_{j+2}*, . . .,*a

_{j}*⊥ b*

_{j+m}*∀j ∈ Z. Taking into account that the*

*chain C = {c*

*j*

*} is obtained by formula (3), we see*that the relations c

_{j}*⊥ b*

_{j+1}*,*c

_{j}*⊥ b*

_{j+2}*, . . . , c*

_{j}*⊥*b

_{j+m}*∀j ∈ Z are fulfilled. Comparing the last*one with the relation obtained just before we get the needed result.

**3** **Some vector identities**

Let u0*, u*_{1}*, . . . , u*_{m−1}*∈ R*^{m+1} be vector columns.

Consider multivector production

(u_{0}*×u*_{1}*×. . .×u** _{m−1}*)

*x = det(u*

^{T}_{0}

*, u*

_{1}

*, . . . , u*

_{m−1}*, x)*

*∀x ∈ R*^{m+1}*.*
For shortness we denote it by symbol

*m−1*Y

*i=0*

*×u** _{i}* = u

_{0}

*× u*

_{1}

*× . . . × u*

_{m−1}*,*

so that (

*m−1*Y

*i=0*

*×u** _{i}*)

^{T}*x ≡ det(u*

_{0}

*, u*

_{1}

*, . . . , u*

_{m−1}*, x)*(4)

*∀x ∈ R*^{m+1}*.*

**Lemma 6. The next assertions are right:**

*1) a transposition of neighbor factors changes the*
*sign of result,*

*2) two collinear factors result in zero,*

*3) the multivector production has a distributive*
*property; in particular, if i ∈ {1, 2, . . . , m − 2} then*
u_{0}*×u*_{1}*×. . .×u*_{i−1}*×(u*^{0}* _{i}*+u

^{00}

_{i}*)×u*

_{i+1}*×. . .×u*

*=*

_{m−1}= u_{0}*× u*_{1}*× . . . × u*_{i−1}*× u*^{0}_{i}*× u*_{i+1}*× . . . × u** _{m−1}*+
+u

_{0}

*× u*

_{1}

*× . . . × u*

_{i−1}*× u*

^{00}

_{i}*× u*

_{i+1}*× . . . × u*

_{m−1}*,*

*4) if a vector b*

*∈ R*

^{m+1}

*is a factor in*

*multivector production*

^{Q}

^{m−1}

_{i=0}*×u*

_{i}*then the relation*

³Q_{m−1}

*i=0* *×u*_{i}^{´}^{T}*b = 0 is right.*

The proof is followed by formula (4).

* Theorem 1. Let v*0

*, . . . , v*

_{2m−2}*, f ∈ R*

^{m+1}

*be*

*column vectors and let*

*C =*^{³}

*m−1*Y

*i=0*

*×v*_{i}*, . . . ,*

*2m−2*Y

*i=m−1*

*×v*_{i}*, f*^{´}

*be matrix with column f . Then the relation*
*det C = (−1)*^{m}

*m−2*Y

*i=0*

det(v_{i}*, . . . , v*_{i+m}*) · f** ^{T}*v

*(5)*

_{m−1}*is correct.*

**Proof. By definition put**
V* _{j}* =

^{³}(

*m−1*Y

*i=0*

*×v** _{i}*)

*v*

^{T}

_{j}*, (*Y

*m*

*i=1*

*×v** _{i}*)

*v*

^{T}

_{j}*, . . .*

*. . . , (*

*2m−2*Y

*i=m−1*

*×v** _{i}*)

*v*

^{T}

_{j}*, f*

*v*

^{T}

_{j}*,*

^{´}

^{T}*, j = 0, 1, . . . , m.*

Consider the production of matrix *C** ^{T}* and
(v

_{0}

*, . . . , v*

*). We have*

_{m}*C** ^{T}*(v

_{0}

*, . . . , v*

*) = (V*

_{m}_{0}

*, V*

_{1}

*, . . . , V*

_{m}*).*

Now we discuss the vectors
W* _{k}*=

^{³}

*0, 0, . . . , 0*

| {z }

*k+1*

*, (*

*m+j*Y

*i=j+1*

*×v** _{i}*)

*v*

^{T}

_{k}*,*

(

*m+j+1*Y

*i=j+2*

*×v** _{i}*)

*v*

^{T}

_{k}*, . . . , (*

*2m−2*Y

*i=m−1*

*×v** _{i}*)

*v*

^{T}

_{k}*, f*

*v*

^{T}

_{k}^{´}

^{T}*,*

*k = 0, 1, . . . , m − 1,*

and vector
W* _{m}*=

^{³}(

*m−1*Y

*i=0*

*×v** _{i}*)

*v*

^{T}

_{m}*, 0, 0, . . . , 0, f*

*v*

^{T}

_{m}^{´}

^{T}*.*

Taking into account formula (4) and point 4) of Lemma 6, we obtain

V* _{j}* = W

_{j}*∀j ∈ {0, 1, . . . , m};*

therefore we have

*det C** ^{T}*(v

_{0}

*, . . . , v*

*) = det(W*

_{m}_{0}

*, W*

_{1}

*, . . . , W*

_{m}*).*

Calculating the right determinant by the ”delet- ing” of the first row and of the last column, we obtain the determinant of the triangular matrix:

*det C*^{T}*· det(v*_{0}*, . . . , v** _{m}*) =

*= (−1)** ^{m}*(

*m−1*Y

*i=0*

*×v** _{i}*)

*v*

^{T}

_{m}*· (*Y

*m*

*i=1*

*×v** _{i}*)

*v*

^{T}_{0}

*· . . .*

*. . . · (*

*2m−2*Y

*i=m−1*

*×v** _{i}*)

*v*

^{T}

_{m−2}*· f*

*v*

^{T}

_{m−1}*.*Using Lemma 6 and relation (4), we deduce

*det C*^{T}*· det(v*_{0}*, . . . , v** _{m}*) =

*= (−1)** ^{m}*det(v

_{0}

*, . . . , v*

_{m}*) · (−1)*

*det(v*

^{m}_{0}

*, . . . , v*

_{m}*)·*

*·(−1)** ^{m}*det(v

_{1}

*, . . . , v*

_{m+1}*) · . . .*

*. . . · (−1)*

*det(v*

^{m}

_{m−2}*, . . . , v*

_{2m−2}*) · f*

*v*

^{T}

_{m−1}*.*

*Here we have m factors (−1)*

*. Because the relation*

^{m}*(−1)*

^{m}^{2}

*= (−1)*

*holds, we get the equality (5).*

^{m}**Corollary 2. Let v**_{0}*, . . . , v*_{2m−1}*∈ R*^{m+1}*be col-*
*umn vectors. Then the relation*

det^{³}

*m−1*Y

*i=0*

*×v*_{i}*, . . . ,*

*2m−1*Y

*i=m*

*×v*_{i}^{´}=

=

*m−1*Y

*i=0*

det(v_{i}*, . . . , v** _{i+m}*) (6)

*is fulfilled.*

**Proof. If we put f =** ^{Q}^{2m−1}_{i=m}*×v** _{i}* in relation (5)
then we get formula (6).

**4** **Classification of complete chains**

*Consider a complete chain A = {a*_{j}*}** _{j∈Z}* of column
vectors a

*j*

*∈ R*

*.*

^{m+1}It is clear that the multiplication of vectors for a complete chain by nonzero numbers gives a new complete chain. The aforementioned multiplication generates the relation of equivalency between chains.

*Two nondegenerate chains A = {a*_{j}*}** _{j∈Z}* and A

*=*

^{0}*{a*

^{0}

_{j}*}*

*are called equivalent chains, if there are*

_{j∈Z}*nonzero numbers λ*

*j*such a

^{0}

_{j}*= λ*

*a*

_{j}*. The introduced*

_{j}*equivalency is denoted by ∼; it divides the set A on*classes of equivalent chains.

*Consider a complete chain A ∈ A. Let A ⊂ A be*
*the class of equivalent complete chains such that A =*

*{A*^{0}*| A*^{0}*∼ A}. The set of all classes is denoted by*
*K.*

Let us return to the discussion of the local orthog-
*onality of complete chains. Let A = {a**i**}** _{i∈Z}* be a

*complete chain. By definition put B = {b*

*j*

*}*

*, where*

_{j∈Z}b* _{j}* =
Y

*m*

*s=1*

*×a*_{j−s}*∀j ∈ Z.* (7)

We want to prove that chains A and B are locally
*orthogonal. For any p ∈ {1, 2, . . . , m} we have j −*
*p ∈ {j − 1, j − 2, . . . , j − m}, so that the vector a** _{j−p}*
locates among the factors of multivector production
in formula (7). Let us verify the relation (1). Using
Lemma 6 (see its point 4), we get relation (1). This
completes the proof.

*Suppose that chain B ∈ A, B = {b**j**}** _{j∈Z}*, is

*local orthogonal to chain A ∈ A, A = {a*

*i*

*}*

*,*

_{i∈Z}*i.e. relation (1) is fulfilled. Consider classes A =*

*{A*

^{0}*| A*

^{0}*∼ A} and B = {B*

^{0}*| B*

^{0}*∼ A}.*

Discuss chains A^{0}*∈ A, A*^{0}*= {a*^{0}_{j}*}** _{i∈Z}*, and
B

^{0}*∈ B, B*

^{0}*= {b*

^{0}

_{j}*}*

*. According to the definition of equivalence we have a*

_{j∈Z}

^{0}

_{i}*= λ*

*a*

_{i}*, b*

_{i}

^{0}

_{j}*= µ*

*b*

_{j}*, where*

_{j}*λ*

_{i}*, µ*

*are nonzero numbers. Multiplying relation (1)*

_{j}*by µ*

*j*

*λ*

*, we obtain a relation*

_{j−p}(b^{0}* _{j}*)

*a*

^{T}

^{0}

_{j−p}*= 0 ∀j ∈ Z,*

*p ∈ I*

_{m}*.*

The last one is equivalence to the local orthogonality
of chains A* ^{0}*and B

*.*

^{0}*Thus, a chain of class A is local orthogonal to*
*each chain of class B, and otherwise, a chain of the*
*class B is local orthogonal to each chain of class A.*

*In the discussed case the classes A and B are called*
*local orthogonal classes. Applying Lemma 5, we see*
that each class has the unique local orthogonal class.

Using formula (5), it is possible to make Lemma 5 more precise: the next assertion is right.

**Lemma 7.** *Let A = {a*_{k}*} be complete chain.*

*Suppose chain B = {b*_{k}*} is obtained by formula*
b* _{k}*= a

_{k−m}*× a*

_{k−m+1}*× . . . × a*

_{k−1}*,*(8)

*and chain C = {c*

_{j}*} is obtained by formula*

c* _{j}* = b

_{j+1}*× b*

_{j+2}*× . . . × b*

_{j+m}*.*(9)

*Then the relation*

c_{j}*= (−1)*^{m}

*m−2*Y

*i=0*

det(a_{i−m+j+1}*, . . . , a** _{i+j+1}*)a

*(10)*

_{j}*is right.*

**Proof. By formula (9) we have**

c^{T}* _{j}*x = det(b

_{j+1}*, . . . , b*

_{j+m)}*, x) ∀x ∈ R*

^{m+1}

*.*

Taking into account formula (8), we obtain
c^{T}* _{j}*x = det

^{³}

Y*j*
*i=j+1−m*

*×a*_{i}*,*

*j+1*Y

*i=j+2−m*

*×a*_{i}*, . . .*

*j+m−1*Y

*i=j*

*×a*_{i}*, x*^{´}*.*

*By substitution i*^{0}*= i + m − j − 1 we have*
c^{T}* _{j}*x = det

^{³}

*m−1*Y

*i** ^{0}*=0

*×a*_{i}^{0}_{−m+j+1}*,*
Y*m*
*i** ^{0}*=1

*×a*_{i}^{0}_{−m+j+1}*, . . .*

*. . . ,*

*2m−2*Y

*i*^{0}*=m−1*

*×a*_{i}^{0}_{−m+j+1}*, x*^{´}*.*

Using (5) for v_{i}* ^{0}* = a

_{i}

^{0}

_{−m+j+1}*, f = x, ∀x ∈ R*

^{m+1}, we deduce relation (10).

*The passage from class A to the orthogonal class*
*B is denoted by* * ^{?}*(star), so that for discussed classes

*we have B = A*

^{?}*, A = B*

^{?}*. It is evident that A =*

*(A*

*)*

^{?}

^{?}*. Thus, the passage from class A to the local*

*orthogonal class is involution in the set K.*

**5** **Space of minimal splines of the La-** **grange type**

*On a finite or infinite interval (α, β) ⊂ R*^{1} we con-
sider a grid

*X : . . . < x*_{−1}*< x*_{0}*< x*_{1} *< . . . ;* (11)
*here α = lim**j→−∞**x*_{j}*, β = lim*_{j→+∞}*x*_{j}*.*

*Let A = {a**j**}** _{j∈Z}*be a complete chain of column
vectors a

_{j}*∈ R*

*.*

^{m+1}Introduce some notation

*G = ∪*_{j∈Z}*(x*_{j}*, x*_{j+1}*),* *S*_{j}*= [x*_{j}*, x*_{j+m+1}*],*
*J*_{k}*= {k − m, k − m + 1, . . . , k − 1, k} ∀k, j ∈ Z.*

*Let U = U (G) be a linear space of functions de-*
*fined on the set G; the last one depends on grid (11).*

*Consider m + 1-component vector function ϕ(t)*
*with components in the space U (G). Discuss the next*
supposition

*(L) The component of vector function ϕ(t) are the*
*linear independent system on the set G ∩ (c, d) for any*
*interval (c, d) ⊂ (α, β).*

*We define functions ω**j**(t), t ∈ G, j ∈ Z, by*
approximate relations

X

*j*^{0}

a_{j}^{0}*ω*_{j}^{0}*(t) ≡ ϕ(t) ∀t ∈ G,* (12)

*ω*_{j}*(t*^{0}*) ≡ 0 ∀t*^{0}*∈ G\S*_{j}*.* (13)
According to (12) – (13) we obtain

X*k*
*j*^{0}*=k−m*

a_{j}^{0}*ω*_{j}^{0}*(t) ≡ ϕ(t) ∀t ∈ (x*_{k}*, x*_{k+1}*) ∀k ∈ Z.*

(14)
*By (14) we have supp ω*_{j}*⊂ S*_{j}*, ω*_{j}*∈ U (G)*

*∀j ∈ Z, and*

*ω*_{j}*(t) =* det^{³}*{a*_{j}^{0}*}*_{j}^{0}_{∈J}_{k}_{,j}^{0}_{6=j}*k* ^{0j}*ϕ(t)*^{´}

det^{³}*{a*_{j}^{0}*}*_{j}^{0}_{∈J}_{k}^{´} (15)

*∀t ∈ (x*_{k}*, x*_{k+1}*) ∀j ∈ J*_{k}*,*

*where symbol k* * ^{0j}* denotes that the determined of
the numerator is deduced from the denominator by the
replacement of the column a

_{j}*with column ϕ(t) (with*preservation of the previous order of columns). Thus, the linear space

*S = {*e *u |*e *u(t) =*e ^{X}

*j∈Z*

*c*_{j}*ω*_{j}*(t) ∀t ∈ G, ∀c*_{j}*∈ R*^{1}*},*
(16)
*is contained in the space U (G).*

By (12) – (16) we see that the set of elements of
the space S does not change if vectors a^{e} * _{j}* are mul-

*tiplied with the nonzero coefficients λ*

_{j}*. Let A be a*class of equivalent chains, which contains the chain

*A = {a*

_{j}*}*

*. It is clear that space S does not depend*

_{j∈Z}*on the choice of a chain in class A, but that it depends*

*on the choice of class A. Therefore we will denote the*space (16) byS

^{e}

_{(X,A,ϕ)}*. This space is called the space*

*of minimal splines of the Lagrange type.*

**6** **On representation of minimal** **splines**

*Let {d**j**}** _{j∈Z}* be complete chain. In formulas (12),
(15) – (16) we put

a* _{j}* =

*j+m*Y

*s=j+1*

*×d*_{s}*.* (17)

*Using (15) for j = k, we have*

*ω*_{k}*(t) =* det(a_{k−m}*, a*_{k−m+1}*, . . . , a*_{k−1}*, ϕ(t))*
det(a_{k−m}*, a*_{k−m+1}*, . . . , a*_{k−1}*, a** _{k}*) (18)

*∀t ∈ (x*_{k}*, x*_{k+1}*).*

By (17) equality (18) can be rewritten:

*ω*_{k}*(t) =* *N*
*D,*

where

*N = det*^{³}
Y*k*
*s=k−m+1*

*×d*_{s}*,*

*k+1*Y

*s=k−m+2*

*×d*_{s}*, . . .*

*. . . ,*

*k+m−1*Y

*s=k*

*×d*_{s}*, ϕ(t)*^{´}*,* (19)

*D = det*^{³}
Y*k*
*s=k−m+1*

*×d*_{s}*,*

*k+1*Y

*s=k−m+2*

*×d*_{s}*, . . .*

*. . . ,*

*k+m−1*Y

*s=k*

*×d*_{s}*,*

*k+m*Y

*s=k+1*

*×d*_{s}^{´} (20)

*∀t ∈ (x*_{k}*, x*_{k+1}*).*

Using Theorem 1 for v*i* = d_{i+k−m+1}*, f = ϕ(t)*
we transform (19) – (20). As a result we obtain

*N =*

*= (−1)*^{m}

*m−2*Y

*i=0*

det(d_{i+k−m+1}*, . . . , d*_{i+k+1}*)·ϕ*^{T}*(t)d*_{k}*,*

*D =*

*m−1*Y

*i=0*

det(d_{i+k−m+1}*, . . . , d*_{i+k+1}*).*

Thus, we have

*ω*_{k}*(t) = (−1)*^{m}*·* d^{T}_{k}*ϕ(t)*

det(d_{k}*, . . . , d** _{k+m}*)

*.*(21)

**7** **Pseudo-continuity** **of minimal** **splines**

*We assume that for integer k there exists a pair of lin-*
*ear functionals F*_{k}^{−}*and F*_{k}^{+} *in the adjoint space U*^{∗}*with supports in segments [x*_{k−1}*, x*_{k}*] and [x*_{k}*, x** _{k+1}*]
accordingly. We assume that the action of a functional
on a vector function is understood as componentwise,
so that we obtain a constant vector.

*A function u ∈ U is called pseudo-continuous in*
*x*_{k}*if F*_{k}^{−}*u = F*_{k}^{+}*u. We denote by C*_{U}*(x** _{k}*),

*C*_{U}*(x*_{k}*) = {u | u ∈ U, F*_{k}^{−}*u = F*_{k}^{+}*u}.*

*It is obvious that C*_{U}*(x** _{k}*) is a linear space and

*C*

_{U}*(x*

_{k}*) ⊂ U . The linear space of all (m + 1)-*dimensional vector-valued functions with components

*in C*

*U*

*(x*

*) is denoted by C*

_{k}

_{U}*(x*

*).*

_{k}*We consider the conditions under which ω**j**, j ∈*
*Z, belong to C*_{U}*(x** _{k}*).

**Lemma 8. Assume that j, k ∈ Z are such that***k − j ∈ J*_{m}*and the functional F ∈ U*^{∗}*has support in*
*[x*_{k}*, x*_{k+1}*]. Then F ω*_{j}*= 0, if and only if*

det^{³}*{a*_{j}^{0}*}*_{j}^{0}_{∈J}_{k}_{,j}^{0}_{6=j}*k* ^{0}^{j}*F ϕ*^{´}*= 0.*

The proof follows from (15).

**Lemma 9. Let ϕ ∈ C***U**(x*_{k}*) and A ∈ A. If*
*F*_{k}^{−}*ω*_{k−m−1}*= 0,* *F*_{k}^{+}*ω*_{k}*= 0,* (22)
*then*

*F*_{k}^{−}*ω*_{j}*= F*_{k}^{+}*ω** _{j}* (23)

*If a*

_{k−m−1}*and a*

_{k}*are not collinear, then (23) is a*

*sufficient condition for the validity of (22).*

* Proof. Necessity. Replacing k with k − 1 in (14),*
we get

*k−1*X

*j=k−m−1*

a_{j}*ω*_{j}*(t) ≡ ϕ(t),* *t ∈ (x*_{k−1}*, x*_{k}*), (24)*

which implies

*k−1*X

*j=k−m*

a_{j}*F*_{k}^{−}*ω*_{j}*= F*_{k}^{−}*ϕ* (25)

in view of the first relation in (22).

Similarly, by (14) and the second relation in (22),

*k−1*X

*j=k−m*

a_{j}*F*_{k}^{+}*ω*_{j}*= F*_{k}^{+}*ϕ.* (26)

Since the vectors a*k−m**, a*_{k−m+1}*, . . . , a** _{k−1}* are

*linearly independent and F*

_{k}

^{−}*ϕ = F*

_{k}^{+}

*ϕ by assump-*tion, from (25) and (26) we obtain (23). Necessity is proved.

Sufficiency. Using (24), we find

*k−1*X

*j=k−m−1*

a_{j}*F*_{k}^{−}*ω*_{j}*= F*_{k}^{−}*ϕ,* (27)

and (14) implies
X*k*
*j=k−m*

a_{j}*F*_{k}^{+}*ω*_{j}*= F*_{k}^{+}*ϕ.* (28)

Subtracting (28) from (27) and taking into ac-
*count (23) and the condition ϕ ∈ C*_{U}*(x** _{k}*), we find
a

_{k−m−1}*F*

_{k}

^{−}*ω*

*= a*

_{k−m−1}

_{k}*F*

_{k}^{+}

*ω*

*. Since a*

_{k}*k−m−1*and a

*are linearly independent, we get (22).*

_{k}**Theorem 2. Let ϕ ∈ C**_{U}*(x*_{k}*) and A ∈ A. Then*
*the functions ω**j**(t) (∀j ∈ Z), in the point x*_{k}*are*
*pseudo-continuous if and only if (22) holds.*

**Proof. Sufficiency. If (22) holds, then it is obvi-**
ous that

*F*_{k}^{−}*ω*_{k−m−1}*= F*_{k}^{+}*ω*_{k−m−1}*,* *F*_{k}^{+}*ω*_{k}*= F*_{k}^{−}*ω*_{k}*,*
since the right-hand sides of the last identities vanish
because of the location of supports of the function-
*als. If k is such that x** _{k}* lies outside the support of

*ω*

_{j}*, then F*

_{k}

^{−}*ω*

_{j}*= F*

_{k}^{+}

*ω*

*since the right-hand and left- hand sides of these identities vanish because the sup- ports of functionals do not intersect the supports of functions to which these functionals are applied. By the relations (23), which are valid in view of Lemma 9, sufficiency is established. Necessity is obvious.*

_{j}*For the equal (by assumption) F*_{k}^{−}*ϕ and F*_{k}^{+}*ϕ we*
set Φ*k*: Φ*k* *= F*_{k}^{−}*ϕ = F*_{k}^{+}*ϕ.*

**Theorem 3. Let ϕ ∈ C***U**(x*_{k}*), k ∈ Z, and let*
*A ∈ A. Then ω*_{j}*(t) (∀j ∈ Z) are pseudo-continuous*
*on the grid X if and only if*

d^{T}* _{j}*Φ

_{j}*= 0 ∀j ∈ Z.*(29)

**Proof. By (21) equalities (29) are equivalent to**relations

*F*_{j}^{+}*ω*_{j}*= 0,* *∀j ∈ Z.* (30)
Consider a chaind^{e}* _{j}* =

^{Q}

^{j−1}

_{s=j−m}*×a*

*. According*

_{s}*to Lemma 7, the chain is equivalent to {d*

*j*

*}. There-*fore formula (29) can be written in equivalent form

det^{³}a_{j−m}*, a*_{j−m+1}*, . . . , a*_{j−1}*, Φ*_{j}^{´}= 0 *∀j ∈ Z.*

*Replacing j with j + m + 1 in the last formula,*
we find

det^{³}a_{j+1}*, a*_{j+2}*, . . . , a*_{j+m}*, Φ*_{j+m+1}^{´}*= 0 ∀j ∈ Z,*
*which implies F*_{j+m+1}^{−}*ω*_{j}*= 0 ∀j ∈ Z, in view of*
*(15). Replacing k = j + m + 1, we arrive at the*
relation

*F*_{k}^{−}*ω** _{k−m−1}*= 0

*∀k ∈ Z.*(31) Thus, the identities (29) are equivalent to (30) and (31). It remains to use Lemma 9.

**8** **Conclusion**

This paper discusses continuity of a function as a co- incidence of values of two linear functionals on the function where mentioned functionals have their sup- ports in adjacent segments. It gives the opportunity to discuss different sorts of continuity.

*For example, for adjacent segments (x*_{k−1}*, x** _{k}*)

*and (x*

_{k}*, x*

*) we put*

_{k+1}*F*_{k}^{−}*u = lim*

*τ →−0*

Z _{0}

*τ* *ψ(ξ)u(x*_{k}*+ ξ)dξ,*
*F*_{k}^{+}*u = lim*

*τ →+0*

Z _{τ}

0 *ψ(ξ)u(x*_{k}*+ ξ)dξ,*

*where ψ(τ ) is a weight function. In that case the*
*equality F*_{k}^{−}*u = F*_{k}^{+}*u is ”mean weighted continuity”.*

Consider another example:

*F*_{k}^{−}*u = lim*

*τ →−0**ψ(τ )u*^{0}*(x*_{k}*+ τ ),*
*F*_{k}^{+}*u = lim*

*τ →+0**ψ(τ )u*^{0}*(x*_{k}*+ τ ).*

*Now the equality F*_{k}^{−}*u = F*_{k}^{+}*u illustrates ”weighted*
continuity of derivative”.

**Acknowledgements: The research was partially sup-**
ported by RFBR (grant No. 15-01-08847).

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