Combining interpolation schemes and Lagrange interpolation on the unit sphere in $\mathbb R^{N+1}$
UDC 517.5 We study Lagrange interpolation in $\mathbb R^N$ and on the unit sphere in $\mathbb R^{N+1}$. We show that sequences of unisolvent sets can be combined to get other sequences of unisolvent sets such that the existence of the limits is preserved. Moreover, the limiting operators keep the in...
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| author | Phung, V. M. Nguyen, V. T. Dinh, H. L. Phung, V. M. Nguyen, V. T. Dinh, H. L. |
| author_facet | Phung, V. M. Nguyen, V. T. Dinh, H. L. Phung, V. M. Nguyen, V. T. Dinh, H. L. |
| author_sort | Phung, V. M. |
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| collection | OJS |
| datestamp_date | 2022-07-06T16:22:31Z |
| description | UDC 517.5
We study Lagrange interpolation in $\mathbb R^N$ and on the unit sphere in $\mathbb R^{N+1}$. We show that sequences of unisolvent sets can be combined to get other sequences of unisolvent sets such that the existence of the limits is preserved. Moreover, the limiting operators keep the interpolation conditions under the combining process. |
| doi_str_mv | 10.37863/umzh.v74i4.6512 |
| first_indexed | 2026-03-24T03:28:32Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i4.6512
UDC 517.5
V. M. Phung, V. T. Nguyen (Hanoi Nat. Univ. Education, Vietnam),
H. L. Dinh (Hanoi Dept. Education and Training, Vietnam)
COMBINING INTERPOLATION SCHEMES
AND LAGRANGE INTERPOLATION ON THE UNIT SPHERE IN \BbbR \bfitN +\bfone *
КОМБIНУВАННЯ IНТЕРПОЛЯЦIЙНИХ СХЕМ
ТА IНТЕРПОЛЯЦIЇ ЛАГРАНЖА НА ОДИНИЧНIЙ СФЕРI В \BbbR \bfitN +\bfone
We study Lagrange interpolation in \BbbR N and on the unit sphere in \BbbR N+1. We show that sequences of unisolvent sets can be
combined to get other sequences of unisolvent sets such that the existence of the limits is preserved. Moreover, the limiting
operators keep the interpolation conditions under the combining process.
Вивчається iнтерполяцiя Лагранжа в \BbbR N та на одиничнiй сферi в \BbbR N+1. Доведено, що послiдовностi унiрозв’язних
множин можна скомбiнувати в iншi послiдовностi таким чином, що iснування меж збiжностi буде збережено. I
навiть бiльше, у такому комбiнуваннi граничнi оператори також зберiгають умови iнтерполяцiї.
1. Introduction. Let \scrP d(\BbbR N ) be the vector space of all polynomials of degree at most d in \BbbR N . It
is known that the dimension md(\BbbR N ) of \scrP d(\BbbR N ) equals
\biggl(
N + d
N
\biggr)
. Let E be a nonempty subset of
\BbbR N . Then the polynomials in \scrP d(\BbbR N ), when restricted to E, form a vector space, say \scrP d(E). We
denote by md(E) the dimension of \scrP d(E). If \BbbS N is the unit sphere in \BbbR N+1, then
md(\BbbS N ) =
\biggl(
N + d
N
\biggr)
+
\biggl(
N + d - 1
N
\biggr)
.
More generally, if E is an algebraic variety in \BbbR N , then one can compute md(E) precisely (see
Subsection 2.2). Since \scrP d(E) is a finite dimensional vector space, any two norms on \scrP d(E) are
equivalent. Hence, the convergence on \scrP d(E) can be understood as the convergence under any norm
on \scrP d(E).
A subset X = \{ \bfx 1, . . . ,\bfx md(E)\} of md(E) distinct points of E is said to be unisolvent for \scrP d(E)
if, for every function f defined on X, there exists a unique P \in \scrP d(E) such that f(\bfx ) = P (\bfx )
for all \bfx \in X. This function is called the Lagrange interpolation polynomial of f at X on E
and is denoted by \bfL E [X; f ]. When E = \BbbR N , we write \bfL [X; f ] for \bfL \BbbR N [X; f ]. Given a basis
\scrB = \{ p1, . . . , pmd(E)\} for \scrP d(E), the (generalized) Vandermonde determinant with respect to \scrB and
X is defined by
VDM(\scrB ;X) = \mathrm{d}\mathrm{e}\mathrm{t}[pi(\bfx j)]1\leq i,j\leq md(E).
It is known that X is unisolvent for \scrP d(E) if and only if VDM(\scrB ;X) \not = 0. The Vandermonde
determinant is a polynomial of interpolation points. Hence, it is different from zero for almost
all choices of interpolation points. In other words, a subset A \subset E of md(E) distinct points is
unisolvent for almost all choices of A. On the other hand, given a set of points on E, it is difficult
to check whether it is unisolvent.
* This research was supported by the Vietnam Ministry of Education and Training (grant number B2021-SPH-16).
c\bigcirc V. M. PHUNG, V. T. NGUYEN, H. L. DINH, 2022
542 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
COMBINING INTERPOLATION SCHEMES AND LAGRANGE INTERPOLATION . . . 543
Roughly speaking, a Hermite interpolation problem is more general than a Lagrange interpolation
problem. More precisely, the problem means to find a polynomial which matches, on a set of distinct
points in E, values of a function and its partial derivatives. If the interpolation problem has a unique
solution, then we say that the problem is poised. Unlike the univariate Hermite interpolation, the
multivariate Hermite interpolation on E is not always poised. Moreover, it is difficult to check
whether a particular Hermite problem is poised.
We are concerned with the problem of determining the limits of Lagrange interpolation polyno-
mials, which generalizes the problem considered in [10].
Problem. a) Construct unisolvent sets for \scrP d(E).
b) Let \{ Xn\} be a sequence of unisolvent sets for \scrP d(E). Find conditions such that the sequence
\{ \bfL E [Xn; f ]\} converges for every suitably defined function f and characterize the limiting operator
\bfH (f).
The problem has been solved in some special cases. It is expected that \bfH (f) is a Hermite type
projector on E. Let us consider the problem in \BbbR N . If N = 1 and Xn coalesces to some points, then
Theorem 1.4 in [2] points out that \{ \bfL [Xn; f ]\} converges to the univariate Hermite interpolation at
the limiting points when f is sufficiently smooth. Bloom and Calvi in [1] gave sufficient conditions
to guarantee the convergence of multivariate Lagrange projectors to the Taylor projector. In a recent
work, Phung [10] showed that the limit of the bivariate Lagrange interpolation polynomials at Bos
configurations distributed on straight lines and circles is a Hermite type interpolation polynomial
when the interpolation points coalesce.
The problem is solved in some cases where E is an algebraic hypersurface. When E is a circle in
\BbbR 2, we showed in [11] (Proposition 4.1) that the Lagrange interpolation on E converges to a Taylor
type polynomial when all interpolation points tend to a single point. An extension of this result for
irreducible algebraic curves in \BbbC 2 was given in [7]. It is worth pointing out that analogous result
in [7] also hold when we replace complex curves in \BbbC 2 by real curves in \BbbR 2 (see Example 2.4 for
details). Also in [11], we constructed new Lagrange and Hermite interpolation schemes on 2-sphere
\BbbS 2. The unisolvent sets for \scrP d(\BbbS 2) are located on d + 1 circles on \BbbS 2 in which the kth circles
contains 2k - 1 points. Fortunately, we can write the interpolation polynomials into Newton forms
and use them to prove that Lagrange projectors tend to Hermite type projectors on \BbbS 2 (see Example
2.5). In [10], the first author of this paper gave new Lagrange and Hermite interpolation schemes
on the unit sphere \BbbS 2. More precisely, the unisolvent sets are the images of Bos configurations of
points distributed on straight lines and circles in \BbbR 2 under the trivial parametrizations of the upper
and lower half spheres. Here the special configurations of interpolation points on \BbbS 2 enable us to
reduce the limiting problem on the sphere to a limiting problem in \BbbR 2 which is solvable. As a result,
\bfH (f) is a certain Hermite projector on \BbbS 2. For details, we refer the readers to [10].
For convenience, we will say that a sequence of unisolvent sets \{ Xn\} \subset E is normal (resp.,
regular) if E is an algebraic variety in \BbbR N (resp., E = \BbbR N ) and the sequence \{ \bfL E [Xn; f ]\} converges
for every suitably defined function f. Precise examples of such sequences are presented in Subsection
2.3. In this paper, we first want to find methods to combine regular and normal sequences to create
similar sequences. In this direction, we prove in Theorem 3.1 that a regular sequence in \BbbR N can
combine with a normal sequence on an algebraic variety in \BbbR N to form a regular sequence. Moreover,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
544 V. M. PHUNG, V. T. NGUYEN, H. L. DINH
the interpolation conditions of the limiting operators preserve under the combining process. In
particular, the union of suitable normal sequences on algebraic varieties in \BbbR N is a regular sequence
in which the limiting operator inherits the interpolation conditions from the limiting operators on
algebraic varieties.
We also investigate the problem on the unit sphere \BbbS N in \BbbR N+1. We first construct unisolvent
sets on \BbbS N . They are of the form
X0 = R+(A0) \cup R - (A0) \cup R+(B0),
where R\pm is the trivial parametrizations of the half spheres
R+(\bfx ) =
\Bigl(
\bfx ,
\sqrt{}
1 - \| \bfx \| 2
\Bigr)
, R - (\bfx ) =
\Bigl(
\bfx , -
\sqrt{}
1 - \| \bfx \| 2
\Bigr)
, \| \bfx \| \leq 1,
and A0, B0 are unisolvent for \scrP d - 1(\BbbR N ) and \scrP d(E), respectively. Here E is a hyperplane in \BbbR N .
We show in Theorem 4.2 that if \{ An\} is a regular sequence in the unit ball in \BbbR N and \{ Bn\} is a
normal sequence on a hyperplane, then
Xn = R+(An) \cup R - (An) \cup R+(Bn)
is a normal sequence on \BbbS N . Furthermore, the limiting operators composed with R\pm also preserve
the interpolation conditions. Our new theorems generalize results in [10].
Notations and conventions. The points in \BbbR N are denoted by bold letters. The Euclidean norms
of \bfx \in \BbbR N is denoted by \| \bfx \| . The symbol \BbbS N stand for the unit sphere in \BbbR N+1. For \bfa \in \BbbR N and
r > 0, we denote by \BbbB N (\bfa , r) the Euclidean ball of centre \bfa and radius r. We write \BbbB N = \BbbB N (0, 1),
the unit ball. Throughout this paper, we denote by \scrF ,\scrA the \BbbR -algebras of functions defined in \BbbR N
that contain the space of all polynomials \scrP (\BbbR N ). For a closed subset K of \BbbR N , we write Cm(K)
for the space of all continuously differentiable functions in neighborhoods of K. We always assume
that d is a positive integer.
2. Regular and normal sequences of unisolvent sets. 2.1. Regular sequences of unisolvent
sets in \BbbR \bfitN .
Definition 2.1. Let \{ An\} be a sequence of unisolvent sets for \scrP d(\BbbR N ) and \scrF be an algebra of
functions defined in \BbbR N . We say the \{ An\} is \scrF -regular if, for any f \in \scrF , the sequence \{ \bfL [An; f ]\}
is convergent in \scrP d(\BbbR N ).
Let \{ An\} be a \scrF -regular sequence for \scrP d(\BbbR N ). We can define
\Lambda (f) := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An; f ], f \in \scrF .
Then \Lambda : \scrF \rightarrow \scrP d(\BbbR N ) is linear map. It can be regarded as a Hermite type interpolation operator
in \BbbR N .
Lemma 2.1. If \{ An\} be a \scrF -regular sequence of unisolvent sets for \scrP d(\BbbR N ) and \Lambda is the limit
of the sequence of Lagrange projectors \{ \bfL [An; \cdot ]\} , then
\Lambda (f\Lambda (g)) = \Lambda (fg), f, g \in \scrF .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
COMBINING INTERPOLATION SCHEMES AND LAGRANGE INTERPOLATION . . . 545
Proof. We first claim that \bfL [An; f\bfL [An; g]] = \bfL [An; fg]. Both sides are polynomials of degree
at most d and agree at any points of An,
\bfL [An; f\bfL [An; g]](\bfa ) = f(\bfa )\bfL [An; g](\bfa ) = f(\bfa )g(\bfa ) = \bfL [An; fg](\bfa ), \bfa \in An.
The desired relation follows from the uniqueness of Lagrange interpolation. Let \{ p1, . . . , pm\} be a
basis for \scrP d(\BbbR N ) with m = md(\BbbR N ). We write \Lambda (g) =
\sum m
i=1
cipi and \bfL [An; g] =
\sum m
i=1
cni pi.
By the hypothesis, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty cni = ci, i = 1, . . . ,m. We have
\Lambda (f\Lambda (g)) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An; f\Lambda (g)] = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
m\sum
i=1
ci\bfL [An; fpi] =
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
m\sum
i=1
cni \bfL [An; fpi] = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An; f\bfL [An; g]] =
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An; fg] = \Lambda (fg).
Here, in the third equation, we use the fact that \{ \bfL [An; fpi]\} is bounded since it converges to \Lambda (fpi),
i = 1, . . . ,m.
Lemma 2.2. Let \{ An\} be a \scrF -regular sequence for \scrP d(\BbbR N ). If \{ Pn\} \subset \scrP k(\BbbR N ) converges to
P \in \scrP k(\BbbR N ), then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An; fPn] = \Lambda (fP ), f \in \scrF .
Proof. Let \{ p1, . . . , pm\} be a basis for \scrP k(\BbbR N ) with m = mk(\BbbR N ). We write P =
\sum m
i=1
aipi
and Pn =
\sum m
i=1
ani pi. By the hypothesis, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty ani = ai, i = 1, . . . ,m. It follows that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An; fPn] = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
m\sum
i=1
ani \bfL [An; fpi] =
m\sum
i=1
ai\Lambda (fpi) = \Lambda (fP ),
where, in the second equation, we use the fact that \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \bfL [An; fpi] = \Lambda (fpi), i = 1, . . . ,m.
2.2. Normal sequences of unisolvent sets on algebraic varieties. Throughout this subsection
we always assume that E is a real algebraic variety in \BbbR N such that its ideal
\scrI (E) = \{ p \in \scrP (\BbbR N ) : p is identically zero on E\}
is principal, i.e., generated by a single element q \in \scrP (\BbbR N ) with \mathrm{d}\mathrm{e}\mathrm{g} q \geq 1.
We recall some arguments in [3]. Let \Phi : \scrP d(\BbbR N ) \rightarrow \scrP d(E) be the continuous surjective linear
map defined by \Phi (Q) = Q| E . Then \mathrm{k}\mathrm{e}\mathrm{r}\Phi = \scrI (E) \cap \scrP d(\BbbR N ). For each Q \in \mathrm{k}\mathrm{e}\mathrm{r}\Phi , our assumption
implies that q divides Q. This enables us to find Q1 \in \scrP d - deg q(\BbbR N ) such that Q = qQ1. Hence
\mathrm{k}\mathrm{e}\mathrm{r}\Phi \subset q\scrP d - deg q(\BbbR N ). The converse inclusion is trivial. It follows that \mathrm{k}\mathrm{e}\mathrm{r}\Phi = q\scrP d - deg q(\BbbR N ).
Consequently,
md(E) = \mathrm{d}\mathrm{i}\mathrm{m}\scrP d(\BbbR N ) - \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{k}\mathrm{e}\mathrm{r}\Phi = md(\BbbR N ) - md - deg q(\BbbR N ).
Here we make the convention that md - deg q(\BbbR N ) = 0 when \mathrm{d}\mathrm{e}\mathrm{g} q > d.
Next we give an analog of the notion of regular sequences presented in the above subsection.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
546 V. M. PHUNG, V. T. NGUYEN, H. L. DINH
Definition 2.2. Let E be an algebraic variety in \BbbR N such that \scrI (E) is generated by a non-
constant polynomial q. Let \{ Bn\} \subset E be a sequence of unisolvent sets for \scrP d(E) and \scrA be an
algebra of functions defined on E. The sequence \{ Bn\} is said to be \scrA -normal if \{ \bfL E [Bn; f ]\} is
convergent in \scrP d(E) for every f \in \scrA .
Let \Phi : \scrP d(\BbbR N ) \rightarrow \scrP d(E) be the continuous surjective linear map defined above. Note that
\mathrm{k}\mathrm{e}\mathrm{r}\Phi is a subspace of the finite-dimensional vector space \scrP d(\BbbR N ). We now denote by \scrQ the
supplementary space of \mathrm{k}\mathrm{e}\mathrm{r}\Phi in \scrP d(\BbbR N ), i.e.,
\scrP d(\BbbR N ) = \scrQ \oplus \mathrm{k}\mathrm{e}\mathrm{r}\Phi .
Then \Phi restricted on \scrQ is a bijective from \scrQ onto \scrP d(E). Furthermore, the two maps
\Phi | \scrQ : \scrQ \rightarrow \scrP d(E) and (\Phi | \scrQ ) - 1 : \scrP d(E) \rightarrow \scrQ
are continuous maps between two normed spaces. If P \in \scrP d(E), then (\Phi | \scrQ ) - 1(P ) \in \scrP d(\BbbR N ) and
(\Phi | \scrQ ) - 1(P )| E = P.
If B0 be a unisolvent set for \scrP d(E), then we define
\bfI [B0; f ] := (\Phi | \scrQ ) - 1(\bfL E [B0; f ]), f \in \scrA .
It is easy to see that \bfI [B0; \cdot ] : \scrA \rightarrow \scrQ is a linear map.
Lemma 2.3. The operator \bfI [B0; \cdot ] has the following properties:
a) for every f \in \scrA , \bfI [B0; f ] interpolates f at B0, i.e., \bfI [B0; f ](\bfb ) = f(\bfb ), \bfb \in B0;
b) if f \in \scrA , f = 0 on B0, then \bfI [B0; f ] = 0.
Proof. By definition we have
\bfI [B0; f ](\bfb ) = \bfL E [B0; f ](\bfb ) = f(\bfb ) \forall \bfb \in B0.
If f = 0 on B0, then \bfL E [B0; f ] = 0. It follows that \bfI [B0; f ] = (\Phi | \scrQ ) - 1(\bfL E [B0; f ]) = 0.
The following result gives a connection between the above operator and a normal sequence.
Lemma 2.4. Let E be an algebraic variety in \BbbR N such that \scrI (E) is generated by a non-
constant polynomial q. Let Bn \subset E be a unisolvent set for \scrP d(E) and \scrA be an algebra of functions.
Then the sequence \{ Bn\} is \scrA -normal for \scrP d(E) if and only if the sequence \{ \bfI [Bn; f ]\} is convergent
in \scrP d(\BbbR N ) for every f \in \scrA , where \bfI [Bn; \cdot ] : \scrA \rightarrow \scrQ defined by
\bfI [Bn; f ] := \Phi - 1(\bfL E [Bn; f ]), f \in \scrA .
Proof. We first assume that \{ Bn\} is \scrA -normal for \scrP d(E). Since \Phi - 1 is continuous and
\{ \bfL E [Bn; f ]\} is convergent, \bfI [Bn; f ] is also convergent.
Conversely, the sequence \{ \bfL E [Bn; f ]\} is convergent, because \{ \bfI [Bn; f ]\} is convergent and \Phi is
continuous.
We set
\Pi (f) := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfI [Bn; f ], f \in \scrA .
Then \Pi : \scrA \rightarrow \scrP d(\BbbR N ) is a linear map which can be viewed as a Hermite type interpolation operator
on E. Next we investigate some properties of \Pi .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
COMBINING INTERPOLATION SCHEMES AND LAGRANGE INTERPOLATION . . . 547
Lemma 2.5. Let E be an algebraic variety in \BbbR N such that \scrI (E) is generated by a non-
constant polynomial q. Let \{ Bn\} \subset E be an \scrA -normal sequence for \scrP d(E). Then, for any f, g \in \scrA ,
we have:
a) \Pi (qf) = 0;
b) \Pi (f\Pi (g)) = \Pi (fg); in particular, \Pi (\Pi (g)) = \Pi (g).
Proof. a) Since Bn \subset E, the function qf vanishes on Bn. Hence \bfI [Bn; qf ] = 0 for all n \geq 1.
It follows that \Pi (qf) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \bfI [Bn; qf ](qf) = 0.
b) We have f\bfI [Bn; g] - fg = 0 on Bn. Hence, by Lemma 2.3, \bfI [Bn; f\bfI [Bn; g] - fg] = 0. It
follows that
\bfI [Bn; f\bfI [Bn; g]] = \bfI [Bn; fg].
We now apply the arguments given in the proof of Lemma 2.1, with \bfL [An; \cdot ] and \Lambda replaced by
\bfI [Bn; \cdot ] and \Pi , respectively, to obtain the desired relation.
Lemma 2.6. Let E be an algebraic variety in \BbbR N such that \scrI (E) is generated by a non-
constant polynomial q. Let \{ Bn\} be an \scrA -normal sequence for \scrP d(E). If \{ Pn\} \subset \scrP k(\BbbR N ) con-
verges to P \in \scrP k(\BbbR N ), then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfI [Bn; fPn] = \Pi (fP ), f \in \scrA .
Proof. The proof is similar to that of Lemma 2.2. Let \{ p1, . . . , pm\} be a basis for \scrP k(\BbbR N ) with
m = mk(\BbbR N ). We write P =
\sum m
i=1 aipi and Pn =
\sum m
i=1 a
n
i pi. By the hypothesis, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty ani = ai,
i = 1, . . . ,m. It follows that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfI [Bn; fPn] = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
m\sum
i=1
ani \bfI [Bn; fpi] =
m\sum
i=1
ai\Pi (fpi) = \Pi (fP ),
where, in the second equation, we use the fact that \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \bfI [Bn; fpi] = \Pi (fpi), i = 1, . . . ,m.
2.3. Examples.
Example 2.1. Let \{ An\} be a sequence of unisolvent sets for \scrP d(\BbbR N ) such that, for every multi-
index \alpha with | \alpha | = d+ 1,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An;\bfx
\alpha ] = 0.
Bloom and Calvi proved in [1] that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}\{ \| \bfx \| : \bfx \in An\} = 0
and
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An; f ] = \bfT d
0(f) \forall f \in Cmd(\BbbR N ) - 1(\{ 0\} ),
where \bfT d
0(f) stands for the Taylor expansion of f at 0 to the order d. In other words, \{ An\} is
Cmd(\BbbR N ) - 1(\{ 0\} )-regular. In [1], the authors also gave some examples of \{ An\} satisfying the above
assumption.
Example 2.2. A set of N hyperplanes H = \{ h1, . . . , hN\} in \BbbR N is said to be in general position
if the intersection of the N hyperplanes is a singleton, that is,
N\bigcap
j=1
hj = \{ \bfa H\} .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
548 V. M. PHUNG, V. T. NGUYEN, H. L. DINH
More generally, a collection \scrH of d \geq N hyperplanes in \BbbR N is said to be in general position if
(1) every H \in
\bigl( \scrH
N
\bigr)
, a subset of N hyperplanes of \scrH , is in general position;
(2) the map
H \in
\biggl(
\scrH
N
\biggr)
\mapsto - \rightarrow \bfa H =
N\bigcap
j=1
hj
is one-to-one.
Let us set
\Theta \scrH =
\Bigl\{
\bfa H : H \in
\biggl(
\scrH
N
\biggr) \Bigr\}
,
which is called a natural lattice of degree d - N. It was proved in [8] that \Theta \scrH is unisolvent for
\scrP d - N (\BbbR N ). Moreover, the corresponding Lagrange interpolation polynomial has a simple formula.
Let d \geq N and let \Theta (s) be a sequence of natural lattices of degree d - N in \BbbR N . We assume
that \Theta (s) is the lattice generated by the family of hyperplanes
\scrH (s) = \{ h(s)1 , . . . , h
(s)
d \} with h
(s)
j (\bfx ) = \langle \bfn (s)
j ,\bfx \rangle - c
(s)
j , \| \bfn (s)
j \| = 1, j = 1, . . . , d,
where \langle \cdot , \cdot \rangle is the scalar product in \BbbR N . Consider the following two conditions:
(C1) all points of the lattices tend to the origin as s \rightarrow \infty , that is, \mathrm{m}\mathrm{a}\mathrm{x}\{ \| \bfa \| : \bfa \in \Theta (s)\} \rightarrow 0 as
s \rightarrow \infty ;
(C2) the volumes
vol
\Bigl(
\bfn
(s)
j1
, . . . ,\bfn
(s)
jN
\Bigr)
, 1 \leq j1 < j2 < . . . < jN \leq d,
of the parallelotope spanned by the vectors \bfn
(s)
j1
, . . . ,\bfn
(s)
jN
are bounded from below, always from 0,
uniformly in s.
We proved in [6] (Theorem 3.1) that if the above two conditions hold, then
\mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow \infty
\bfL [\Theta (s); f ] = \bfT d - N
0 (f), f \in Cd - N+1(\{ 0\} ).
Hence, we can say that \{ \Theta (s)\} is Cd - N+1(\{ 0\} )-regular.
Example 2.3. Let \delta > 0 and d \geq 2 be a positive integer. We define m =
\biggl[
d
2
\biggr]
+ 1 and
S = \{ s1, . . . , sm\} with sk = d - 2k+2 for k = 1, . . . ,m. Let A = \{ \bfa 1, . . . ,\bfa m\} be m distinct points
in \BbbR 2. Each point \bfa k is associated to a sequence of circles \{ Cn
k \} with Cn
k = \{ \bfx : \| \bfx - \bfa k\| = rk,n\}
such that \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty rk,n = 0. For 1 \leq k \leq m and n \geq 1, let Xn
k be a \delta -separate set of 2sk+1 points
on Cn
k , i.e.,
\| \bfb - \bfc \| \geq \delta rk,n, \bfb , \bfc \in Xn
k , \bfb \not = \bfc .
Then Xn := \cup m
k=1X
n
k is a unisolvent set for \scrP d(\BbbR 2) and is called a Bos configuration on circles.
Let \scrF be the class of all bivariate functions of class Csk in neighborhoods of the \bfa k ’s. It was proved
in [9] that, for any f \in \scrF ,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [Xn; f ] = \bfH [(A,S); f ],
where \bfH [(A,S); f ] is a Hermite type projector satisfying the relations
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COMBINING INTERPOLATION SCHEMES AND LAGRANGE INTERPOLATION . . . 549\biggl(
\partial
\partial x1
- i
\partial
\partial x2
\biggr) k
\bfH [(A,S); f ](\bfa k) =
\biggl(
\partial
\partial x1
- i
\partial
\partial x2
\biggr) k
f(\bfa k), 1 \leq k \leq m, 0 \leq j \leq sk,
where i is the imaginary unit and \bfx = (x1, x2). It follows that \{ Xn\} is \scrF -regular. A generalization
of the above result can be found in [10].
Example 2.4. Normal sets can be constructed on irreducible algebraic curves in \BbbR 2. Indeed, let
A(n) = \{ \bfa (n)0 , . . . ,\bfa
(n)
2d \} be distinct points on the circle C(0, \rho ) and \bfb \in C(0, \rho ) such that \bfa (n)j \rightarrow \bfb
as n \rightarrow \infty . Let g be a real-valued functions in C2d(\Omega ), where \Omega is a neighborhood of \bfb in C(0, \rho ).
Proposition 4.1 in [11] asserts that the following limit exists:
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL C(0,\rho )[A
(n), g]. (2.1)
Moreover, the limit depends only on \bfb and g and is denoted by \bfT d
\bfb (g) which satisfies the relations\Bigl(
\bfT d
\bfb (g)(\rho \mathrm{c}\mathrm{o}\mathrm{s}\alpha , \rho \mathrm{s}\mathrm{i}\mathrm{n}\alpha )
\Bigr) (k) \bigm| \bigm| \bigm|
\alpha =\alpha \ast
= (g(\rho \mathrm{c}\mathrm{o}\mathrm{s}\alpha , \rho \mathrm{s}\mathrm{i}\mathrm{n}\alpha ))(k)
\bigm| \bigm| \bigm|
\alpha =\alpha \ast
, k = 0, . . . , 2d,
where \bfb = (\rho \mathrm{c}\mathrm{o}\mathrm{s}\alpha \ast , \rho \mathrm{s}\mathrm{i}\mathrm{n}\alpha \ast ). In other words, \{ A(n)\} is \scrA -normal on C(0, \rho ), where \scrA is the
algebra of all functions of class C2d in neighborhoods of \bfb on C(0, \rho ).
More general result also holds when we replace the circle by irreducible algebraic curves in \BbbR 2.
In [7], we studied the polynomial interpolation on irreducible algebraic curves in \BbbC 2. However, with
simple adaptions, every result remains true in the real settings. The passage to the real case can be
found in [4].
Let q be an irreducible polynomial in \BbbR 2 such that V := \{ \bfx \in \BbbR 2 : q(\bfx ) = 0\} contains at least
one regular point. In [5], the authors defined the notion of d-Taylorian points on V. Note that all but
finitely many points on V are d-Taylorian (see [5], Theorem 4.10). Bos and Calvi proved in [4] that,
for every function f of class Cmd(V ) - 1 on a neighbourhood of a d-Taylorian point \bfa \in V, there
exists a unique polynomial P \in \scrP d(V ) such that, for every local parametrization \scrL = (0, U,R) of
V at \bfa with R(0) = \bfa , we have
(P \circ R)(i)(0) = (f \circ R)(i)(0), i = 0, . . . ,md(V ) - 1.
The above interpolation polynomial is called the d-Taylor polynomial of f at \bfa and is denoted by
\bfT d
\bfa (f). We proved in [7] that it is the limit of Lagrange interpolation on V. More precisely, we
proved in Theorems 4.1 and 4.2 that if An \subset V, n \in \BbbN is a sequence of unisolvent sets for \scrP d(V )
whose points tend to a d-Taylorian point \bfa , i.e., \mathrm{m}\mathrm{a}\mathrm{x}\{ \| \bfx - \bfa \| : \bfx \in An\} \rightarrow 0 as n \rightarrow \infty , then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL V [An ; f ] = \bfT d
\bfa (f)
for every function f of class Cmd(V ) - 1 on a neighborhood of \bfa in V. Hence every sequence of
unisolvent sets \{ An\} \subset V tending to a d-Taylorian point \bfa is normal.
Example 2.5. In [11], we give some normal sequences on the unit sphere in \BbbR 3. Associated with
each point \bfa = (\mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{c}\mathrm{o}\mathrm{s}\varphi , \mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{s}\mathrm{i}\mathrm{n}\varphi , \mathrm{c}\mathrm{o}\mathrm{s} \theta ) \in \BbbS 2, we denote R\bfa by the local parametrization of \BbbS
at \bfa ,
R\bfa = (R1
\bfa , R
2
\bfa , R
3
\bfa ),
where
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550 V. M. PHUNG, V. T. NGUYEN, H. L. DINH
R1
\bfa (u, v) = (\mathrm{c}\mathrm{o}\mathrm{s} \theta \mathrm{c}\mathrm{o}\mathrm{s}\varphi )u - (\mathrm{s}\mathrm{i}\mathrm{n}\varphi )v + (\mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{c}\mathrm{o}\mathrm{s}\varphi )
\sqrt{}
1 - u2 - v2,
R2
\bfa (u, v) = (\mathrm{c}\mathrm{o}\mathrm{s} \theta \mathrm{s}\mathrm{i}\mathrm{n}\varphi )u+ (\mathrm{c}\mathrm{o}\mathrm{s}\varphi )v + (\mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{s}\mathrm{i}\mathrm{n}\varphi )
\sqrt{}
1 - u2 - v2,
R3
\bfa (u, v) = - (\mathrm{s}\mathrm{i}\mathrm{n} \theta )u+ (\mathrm{c}\mathrm{o}\mathrm{s} \theta )
\sqrt{}
1 - u2 - v2.
For \rho \in (0, 1], we consider the linear polynomial
q\bfa ,\rho (\bfx ) = (\mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{c}\mathrm{o}\mathrm{s}\varphi )x1 + (\mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{s}\mathrm{i}\mathrm{n}\varphi )x2 + (\mathrm{c}\mathrm{o}\mathrm{s} \theta )x3 -
\sqrt{}
1 - \rho 2.
The plane \{ q\bfa ,\rho = 0\} cuts a small spherical cap off the sphere in which the circle of the base denoted
by \scrC (\bfa , \rho ) is of radius \rho and the peak point is \bfa .
Let \delta > 0. Let \bfa 0, . . . ,\bfa d be d + 1 distinct points on \BbbS 2. Each point \bfa j is associated with a
sequence of circles \scrC (\bfa j , \rho (n)j ) where \rho
(n)
j \in (0, 1) and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \rho
(n)
j = 0. For each n \geq 1, let
A
(n)
j be a set of 2(d - j) + 1 distinct points on \scrC (\bfa j , \rho (n)j ) such that A(n)
j is \delta -separate. Let us set
A(n) =
\bigcup d
j=0A
(n)
j . Let \scrA be the algebra of functions of class C(d - j) in a neighborhood of \bfa j in \BbbS 2,
j = 0, . . . , d. For each f \in \scrA , we have
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL \BbbS 2 [A
(n); f ] = \Pi (1)(f).
Here the right-hand side is a Hermite type interpolation defined in [11] (Theorem 3.4) which satisfies
the relation \biggl(
\partial
\partial u
- i
\partial
\partial v
\biggr) k
(\Pi (1)(f) \circ R\bfa j )(0, 0) =
\biggl(
\partial
\partial u
- i
\partial
\partial v
\biggr) k
(f \circ R\bfa j )(0, 0),
j = 0, . . . , d, k = 0, 1 . . . , d - j.
The above assertion is proved in [11] (Theorem 3.7) and gives an \scrA -normal sequence.
Another type of normal sequence is constructed in [11] (Theorem 4.4).
Let \bfa 0, . . . ,\bfa d be d + 1 not necessarily distinct points on \BbbS 2 and \rho 0, . . . , \rho d \in (0, 1]. On each
circle \scrC (\bfa j , \rho j), we take a point \bfb j which does not lie on \scrC (\bfa k, \rho k) for j > k. For j = 0, . . . , d
and n \geq 1, let A
(n)
j be a set of 2(d - j) + 1 distinct points on \scrC (\bfa j , \rho j) such that A
(n)
j \rightarrow \bfb j .
Set A(n) =
\bigcup d
j=0A
(n)
j . Then, for each function f of class C2(d - k) in a neighborhood of \bfb k on \BbbS 2,
k = 0, . . . , d, we have
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL \BbbS 2 [A
(n); f ] = \Pi (2)(f), (2.2)
where A
(n)
j \rightarrow \bfb j means that all points in A
(n)
j tend to \bfb j as n \rightarrow \infty . Here the right-hand side is a
Hermite type interpolation defined in [11] (Theorem 4.3) which satisfies the relation\Bigl(
(\Pi (2)(f) \circ R\bfa j )(\rho j \mathrm{c}\mathrm{o}\mathrm{s}\alpha , \rho j \mathrm{s}\mathrm{i}\mathrm{n}\alpha )
\Bigr) (k) \bigm| \bigm| \bigm|
\alpha =\alpha j
=
\bigl(
(f \circ R\bfa j )(\rho j \mathrm{c}\mathrm{o}\mathrm{s}\alpha , \rho j \mathrm{s}\mathrm{i}\mathrm{n}\alpha )
\bigr) (k) \bigm| \bigm| \bigm|
\alpha =\alpha j
for all j = 0, . . . , d and k = 0, . . . , 2(d - j).
Example 2.6. Let q be a linear polynomial in \BbbR N and V = \{ \bfx \in \BbbR N : q(\bfx ) = 0\} . Then V
can be viewed as a (N - 1)-dimensional space. Then any \scrF -regular sequence in V is a \scrF -normal
sequence on the subset V of \BbbR N . Hence Examples 2.1 – 2.3 give normal sequences on V. This type
of normal sequence is used in Theorem 4.2 below.
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COMBINING INTERPOLATION SCHEMES AND LAGRANGE INTERPOLATION . . . 551
3. Combining interpolation schemes. The aim of this section is to study the behavior of
Lagrange projectors in \BbbR N when the interpolation points are collected from regular sequences and
normal sequences. The method is inspired from [3].
Theorem 3.1. Let d, k be non-negative integer with k < d. Let E be an algebraic variety in
\BbbR N such that \scrI (E) is generated by a non-constant polynomial q with \mathrm{d}\mathrm{e}\mathrm{g} q = d - k. Let \{ Bn\} \subset E
be an \scrA -normal sequence for \scrP d(\BbbR N ). Let \{ An\} be a \scrF -regular sequence for \scrP k(\BbbR N ) such that
A \cap \{ q = 0\} = \varnothing with A =
\bigcup \infty
n=1An. Assume that 1/q \in \scrF . Then \{ An \cup Bn\} is (\scrA \cap \scrF )-regular
for \scrP d(\BbbR N ). Moreover, if \Lambda and \Pi are define by
\Lambda (f) := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An; f ], \Pi (f) := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfI [Bn; f ], f \in \scrA \cap \scrF ,
then the operator \bfH defined by
\bfH (f) := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An \cup Bn; f ]
satisfies the relations
\Lambda (\bfH (f)) = \Lambda (f), \Pi (\bfH (f)) = \Pi (f).
Proof. We first prove that Xn := An \cup Bn is unisolvent for \scrP d(\BbbR N ). Since An \cap Bn = \varnothing , we
have
\sharp Xn = \sharp An+\sharp Bn = \mathrm{d}\mathrm{i}\mathrm{m}\scrP k(\BbbR N )+\mathrm{d}\mathrm{i}\mathrm{m}\scrP d(E) = mk(\BbbR N )+md(\BbbR N ) - md - deg q(\BbbR N ) = md(\BbbR N ).
Hence, it suffices to show that P \in \scrP d(\BbbR N ) that vanishes on Xn is identically zero. Since P = 0
on Bn and Bn is unisolvent for \scrP d(E), P = 0 on E. This enables us to find P1 \in \scrP d - deg q(\BbbR N )
such that P = qP1. Since \{ q = 0\} \cap An = \varnothing , P1 must vanish on An. It follows that P1 = 0 since
it belongs to \scrP k(\BbbR N ). Hence, P = 0.
We next prove a Newton form formula for interpolation polynomials
\bfL [Xn; f ] = \bfI [Bn; f ] + q\bfL
\biggl[
An;
f - \bfI [Bn; f ]
q
\biggr]
. (3.1)
Indeed, the right-hand side of (3.1) denoted by Q is a polynomial of degree at most d in \BbbR N . For
each \bfb \in Bn we have q(\bfb ) = 0. Hence Q(\bfb ) = \bfI [Bn; f ](\bfb ) = f(\bfb ). On the other hand, for each
\bfa \in An, we have
Q(\bfa ) = \bfI [Bn; f ](\bfa ) + q(\bfa )\bfL
\biggl[
An;
f - \bfI [Bn; f ]
q
\biggr]
(\bfa ) =
= \bfI [Bn; f ](\bfa ) + q(\bfa )
f(\bfa ) - \bfI [Bn; f ](\bfa )
q(\bfa )
= f(\bfa ).
From what has already been proved, we conclude that Q interpolates f at Xn. Therefore, Q =
= \bfL [Xn; f ].
Next we find the limit of polynomials in (3.1). Note that \{ \bfI [Bn; f ]\} \subset \scrP d(\BbbR N ) converges to
\Pi (f). Hence, we can use Lemma 2.2, Lemma 2.6 and (3.1) to get
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [Xn; f ] = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\biggl(
\bfI [Bn; f ] + q\bfL
\biggl[
An;
f
q
\biggr]
- q\bfL
\biggl[
An;
\bfI [Bn; f ]
q
\biggr] \biggr)
=
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552 V. M. PHUNG, V. T. NGUYEN, H. L. DINH
= \Pi (f) + q
\biggl(
\Lambda
\biggl(
f
q
\biggr)
- \Lambda
\biggl(
\Pi (f)
q
\biggr) \biggr)
= \Pi (f) + q\Lambda
\biggl(
f - \Pi (f)
q
\biggr)
.
It follows that \{ Xn\} is (\scrF \cup \scrA )-regular and
\bfH (f) = \Pi (f) + q\Lambda
\biggl(
f - \Pi (f)
q
\biggr)
, f \in \scrA \cap \scrF .
By using Lemma 2.5, we obtain
\Pi (\bfH (f)) = \Pi (\Pi (f)) + \Pi
\biggl(
q\Lambda
\biggl(
f - \Pi (f)
q
\biggr) \biggr)
= \Pi (f).
To prove the last relation, we use Lemma 2.1 to get
\Lambda (\bfH (f)) = \Lambda (\Pi (f)) + \Lambda
\biggl(
q\Lambda
\biggl(
f - \Pi (f)
q
\biggr) \biggr)
= \Lambda (\Pi (f)) + \Lambda
\biggl(
q
f - \Pi (f)
q
\biggr)
= \Lambda (f).
Theorem is proved.
Theorem 3.2. Let m \geq 2 be a positive integer. For each 1 \leq j \leq m, let Ej be an algebraic
variety in \BbbR N such that \scrI (Ej) is generated by a non-constant polynomial qj with \mathrm{d}\mathrm{e}\mathrm{g} qj = rj . Let
d \in \BbbN be such that
r1 + . . .+ rm - 1 < d \leq r1 + . . .+ rm.
We define s1, s2, . . . , sm by the relation
s1 = d, sj = d - r1 - . . . - rj - 1, j = 2, . . . ,m.
Let \{ Bj,n\} \subset Ej be a \scrA j -normal sequence for \scrP sj (Ej) such that \{ qk = 0\} \cap (
\bigcup \infty
n=1Bj,n) = \varnothing for
j > k and 1/qi \in \scrA i+1 for 1 \leq i \leq m - 1. Set Xn =
\bigcup m
j=1Bj,n for n \geq 1.
a) In the case d < r1 + . . . + rm, the sequence \{ Xn\} is
\Bigl( \bigcap m
j=1\scrA j
\Bigr)
-regular. Moreover, the
limiting operator defined
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [Xn; f ] = \bfH (f), f \in
m\bigcap
j=1
\scrA j
satisfies the relation
\Pi j(\bfH (f)) = \Pi j(f), 1 \leq j \leq m, (3.2)
where
\Pi j(f) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfI [Bj,n; f ].
b) In the case d = r1 + . . .+ rm, if \{ \bfa m+1,n\} is a sequence in \BbbR N lying outside
\bigcup m
j=1\{ qj = 0\}
and converging to \bfa /\in
\bigcup m
j=1\{ qj = 0\} , then the sequence \{ Xn \cup \{ \bfa m+1,n\} \} is (C0(\{ \bfa \} )\cap \cap m
j=1\scrA j)-
regular. Moreover, the limit operator defined
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [Xn; f ] = \bfH (f), f \in C0(\{ \bfa \} ) \cap
m\bigcap
j=1
\scrA j
satisfies (3.2) along with the additional relation \bfH (f)(\bfa ) = f(\bfa ).
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COMBINING INTERPOLATION SCHEMES AND LAGRANGE INTERPOLATION . . . 553
Proof. We only prove the statement corresponding to the case d < r1 + . . . + rm. Since Bj,n
is unisolvent for \scrP sj (Ej), Theorem 3.3 in [3] asserts that
\bigcup m
j=k Bj,n is unisolvent for \scrP sk(\BbbR N ) for
1 \leq k \leq m and n \geq 1. In particular, Xn is unisolvent for \scrP d(\BbbR N ). The proof is now by induction
in m.
We first assume that m = 2. Then Xn = B1,n \cup B2,n. Since B2,n is unisolvent for \scrP d - r1(\BbbR N ),
\bfL [B2,n; f ] = \bfI [B2,n; f ], and hence \{ B2,n\} is \scrA 2-regular,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [B2,n; f ] = \Pi 2(f), f \in \scrA 2.
By using Theorem 3.1, we get the regularity of \{ Xn\} . Moreover,
\Pi j(\bfH (f)) = \Pi j(f), j = 1, 2.
Assume that the assertion holds up to m - 1 \geq 2; we will prove it for m. We set \widetilde Xn =
\bigcup m
j=2Bj,n.
Then \widetilde Xn is unisolvent for \scrP s2(\BbbR N ) and, by the induction hypothesis, \{ \widetilde Xn\} is
\Bigl( \bigcap m
j=2\scrA j
\Bigr)
-regular.
Furthermore, the limiting operator
\widetilde \bfH (f) := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [ \widetilde Xn; f ], f \in
m\bigcap
j=2
\scrA j ,
satisfies the relation
\Pi j( \widetilde \bfH (f)) = \Pi j(f), 2 \leq j \leq m. (3.3)
Applying Theorem 3.1 for \{ B1,n\} and \{ \widetilde Xn\} , we conclude that \{ Xn\} is
\Bigl( \bigcap m
j=1\scrA j
\Bigr)
-regular and
\widetilde \bfH (\bfH (f)) = \widetilde \bfH (f), \Pi 1(\bfH (f)) = \Pi 1(f),
where
\bfH (f) := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [Xn; f ].
Combining the above relation with (3.3), we obtain, for 2 \leq j \leq m,
\Pi j(f) = \Pi j( \widetilde \bfH (f)) = \Pi j
\Bigl( \widetilde \bfH (\bfH (f))
\Bigr)
= \Pi j(\bfH (f)).
Here, in the third relation we use (3.3) for \bfH (f) in the place of f. The assertion holds for m.
Theorem is proved.
Remark 3.1. From Theorems 3.1 and 3.2, we can use examples in Subsection 2.3 to build new
regular sequences. The details are left to the readers.
4. Polynomial interpolation on the unit sphere. In this section, we construct unisolvent sets
for \scrP d(\BbbS N ) from unisolvent sets lying in unit ball \BbbB N . We also investigate the limit of the Lagrange
interpolation on \BbbS N corresponding a regular sequence and a normal sequence in \BbbB N . We recall the
trivial parametrizations of the half spheres
R+(\bfx ) =
\Bigl(
\bfx ,
\sqrt{}
1 - \| \bfx \| 2
\Bigr)
, R - (\bfx ) =
\Bigl(
\bfx , -
\sqrt{}
1 - \| \bfx \| 2
\Bigr)
, \bfx \in \BbbB N .
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554 V. M. PHUNG, V. T. NGUYEN, H. L. DINH
Theorem 4.1. Let A0 \subset \BbbB N be unisolvent for \scrP d - 1(\BbbR N ). Let E be a hyperplane in \BbbR N such
that A0 \cap E = \varnothing . Let B0 \subset E \cap \BbbB N be unisolvent for \scrP d(E). Then the set
X0 = R+(A0) \cup R - (A0) \cup R+(B0)
is unisolvent for \scrP d(\BbbS N ). Moreover, if f is a function defined on \BbbS N , then
\bfL \BbbS N [X0; f ] \circ R\pm (\bfx ) =
\pm
\sqrt{}
1 - \| \bfx \| 2\bfL [A0; f1](\bfx ) + \bfL [(A0; f2), (B0; f3)](\bfx )
2
, \bfx \in \BbbB N , (4.1)
where
f1(\bfx ) =
f \circ R+(\bfx ) - f \circ R - (\bfx )\sqrt{}
1 - \| \bfx \| 2
, f2(\bfx ) = f \circ R+(\bfx ) + f \circ R - (\bfx ), \bfx \in \BbbB N ,
and
f3(\bfx ) = 2f \circ R+(\bfx ) -
\sqrt{}
1 - \| \bfx \| 2\bfL [A0; f1](\bfx ), \bfx \in \BbbB N .
Proof. We first prove that X0 is unisolvent for \scrP d(\BbbS N ). The proof is motivated by [12] (Theorem
2.1) and is similar to [10] (Theorem 3.1). We take a non-zero affine polynomial q in \BbbR N such that
E = \{ \bfx \in \BbbR N : q(\bfx ) = 0\} . We have \sharp B0 = \mathrm{d}\mathrm{i}\mathrm{m}\scrP d(E) = md(\BbbR N ) - md - 1(\BbbR N ). Hence
\sharp X0 = 2md - 1(\BbbR N ) +md(\BbbR N ) - md - 1(\BbbR N ) = md(\BbbR N ) +md - 1(\BbbR N ) = \mathrm{d}\mathrm{i}\mathrm{m}\scrP d(\BbbS N ).
To prove the theorem, it suffices to verify that if P \in \scrP d(\BbbS N ) that vanishes on X0, i.e.,
P \circ R+(\bfa ) = P \circ R - (\bfa ) = P \circ R+(\bfb ) = 0, \forall \bfa \in A0 \bfb \in B0, (4.2)
then P is identically zero. Let us set
P1(\bfx ) :=
P \circ R+(\bfx ) - P \circ R - (\bfx )\sqrt{}
1 - \| \bfx \| 2
. (4.3)
Then P1 belongs to \scrP d - 1(\BbbR N ). Relation (4.2) implies that P1(\bfa ) = 0 for all \bfa \in A0. Since A0
is unisolvent for \scrP d - 1(\BbbR N ), P1 must be identically zero. This enables us to write P \circ R+(\bfx ) =
= P \circ R - (\bfx ) for every \bfx \in \BbbB N . Likewise, consider the polynomial P2 \in \scrP d(\BbbR N ) defined by
P2(\bfx ) := P \circ R+(\bfx ) + P \circ R - (\bfx ). (4.4)
Then P2(\bfx ) = 2P \circ R+(\bfx ) for every \bfx \in \BbbB . From (4.2) we see that
P2(\bfa ) = P2(\bfb ) = 0 \forall \bfa \in A0 \forall \bfb \in B0.
This forces P2 = 0, because A0 \cup B0 is unisolvent for \scrP d(\BbbR N ) due to Theorem 3.1. From what has
already been proved, we conclude that
P \circ R+(\bfx ) = P \circ R - (\bfx ) = 0 \forall \bfx \in \BbbB N .
It follows that P = 0 on \BbbS N , and the proof of the first assertion is complete.
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COMBINING INTERPOLATION SCHEMES AND LAGRANGE INTERPOLATION . . . 555
It remains to prove the formulas. For convenience, we set P = \bfL \BbbS N [X; f ]. Let P1, P2 be defined
as in (4.3) and (4.4). By the interpolation condition, we have
P1(\bfa ) =
P \circ R+(\bfa ) - P \circ R - (\bfa )\sqrt{}
1 - \| \bfa \| 2
=
f \circ R+(\bfa ) - f \circ R - (\bfa )\sqrt{}
1 - \| \bfa \| 2
= f1(\bfa ), \bfa \in A0.
It follows that P1 = \bfL [A0, f1]. Combining this with the setting in (4.3), we get
P \circ R - (\bfx ) = P \circ R+(\bfx ) -
\sqrt{}
1 - \| \bfx \| 2\bfL [A0, f1](\bfx ), \bfx \in \BbbB N . (4.5)
From (4.4), it follows that
P2(\bfa ) = f \circ R+(\bfa ) + f \circ R - (\bfa ) = f2(\bfa ), \bfa \in A0. (4.6)
On the other hand, for all \bfb \in B0, by using (4.5), we obtain
P2(\bfb ) = P \circ R+(\bfb ) + P \circ R - (\bfb ) = 2f \circ R+(\bfb ) -
\sqrt{}
1 - \| \bfb \| 2\bfL [A0, f1](\bfb ) = f3(\bfb ). (4.7)
We conclude from (4.6) and (4.7) that P2 interpolates f2 at A0 and f3 at B0. Hence
P2 = \bfL [(A0; f2), (B0; f3)].
Consequently,
P \circ R+(\bfx ) - P \circ R - (\bfx ) =
\sqrt{}
1 - \| \bfx \| 2\bfL [A0, f1](\bfx ),
P \circ R+(\bfx ) + P \circ R - (\bfx ) = \bfL [(A0; f2), (B0; f3)](\bfx ),
\bfx \in \BbbB N .
Combining the last two relations, we obtain the desired equations in \BbbB N . By continuity, we get the
equations in \BbbB N .
Theorem is proved.
Theorem 4.2. Let \rho \in (0, 1). Let E be a hyperplane in \BbbR N generated by an affine polynomial
q. Let \{ Bn\} \subset E \cap \BbbB N (0, \rho ) be an \scrA -normal sequence for \scrP d(E) corresponding to the sequence
of linear maps \{ \bfI [Bn; \cdot ]\} . Let \{ An\} \subset \BbbB N (0, \rho ) be a \scrF -regular sequence for \scrP d - 1(\BbbR N ) with
An \cap E = \varnothing , n \geq 1. We define
Xn = R+(An) \cup R - (An) \cup R+(Bn).
Assume that (1 - \| \bfx \| 2)\pm 1/2, 1/q \in \scrF and (1 - \| \bfx \| 2)1/2 \in \scrA . We set
\scrC =
\Bigl\{
f : \BbbS N \rightarrow \BbbR
\bigm| \bigm| f \circ R\pm \in \scrF , f \circ R+ \in \scrA
\Bigr\}
.
Then, for any function f \in \scrC , the following limit exists:
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL \BbbS N [Xn; f ].
Moreover, the limit denoted by \bfH \BbbS N satisfies the relations
\Lambda (\bfH \BbbS N \circ R+) = \Lambda (f \circ R+), \Lambda (\bfH \BbbS N \circ R - ) = \Lambda (f \circ R - ), \Pi (\bfH \BbbS N \circ R+) = \Pi (f \circ R+),
where \Lambda and \Pi are defined by
\Lambda (g) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An; g], g \in \scrF , \Pi (h) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfI [Bn;h], h \in \scrA .
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556 V. M. PHUNG, V. T. NGUYEN, H. L. DINH
Proof. We first find the limit of the sequence of Lagrange interpolation polynomials. By Theorem
4.1, Xn is unisolvent for \scrP d(\BbbS N ). For convenience, we set Pn := \bfL \BbbS N [Xn; f ]. Then Theorem 4.1
gives
Pn \circ R\pm (\bfx ) =
\pm
\sqrt{}
1 - \| \bfx \| 2\bfL [An; f1](\bfx ) + \bfL [(An; f2), (Bn; f3,n)](\bfx )
2
, \bfx \in \BbbB N , (4.8)
where
f1(\bfx ) =
f \circ R+(\bfx ) - f \circ R - (\bfx )\sqrt{}
1 - \| \bfx \| 2
, f2(\bfx ) = f \circ R+(\bfx ) + f \circ R - (\bfx ), \bfx \in \BbbB N ,
and
f3,n(\bfx ) = 2f \circ R+(\bfx ) -
\sqrt{}
1 - \| \bfx \| 2\bfL [An; f1](\bfx ), \bfx \in \BbbB N .
Analysis similar to that in the proof of Theorem 3.1 shows that
\bfL [(An; f2), (Bn; f3,n)] = \bfI [Bn; f3,n] + q\bfL
\biggl[
An;
f2 - \bfI [Bn; f3,n]
q
\biggr]
=
= \bfI [Bn; f3,n] + q
\biggl(
\bfL
\biggl[
An;
f2
q
\biggr]
- \bfL
\biggl[
An;
\bfI [Bn; f3,n]
q
\biggr] \biggr)
. (4.9)
We will find the limit of each term in (4.9). By hypothesis we have
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [An; f1] = \Lambda (f1), \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL
\biggl[
An;
f2
q
\biggr]
= \Lambda
\biggl(
f2
q
\biggr)
. (4.10)
Hence Lemma 2.6 shows that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfI [Bn; f3,n] = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\Bigl(
\bfI [Bn; 2f \circ R+] - \bfI
\Bigl[
Bn;
\sqrt{}
1 - \| \bfx \| 2\bfL [An; f1](\bfx )
\Bigr] \Bigr)
=
= \Pi (2f \circ R+) - \Pi
\Bigl( \sqrt{}
1 - \| \bfx \| 2\Lambda (f1)(\bfx )
\Bigr)
=
= \Pi
\Bigl(
2f \circ R+(\bfx ) -
\sqrt{}
1 - \| \bfx \| 2\Lambda (f1)(\bfx )
\Bigr)
=: \Phi (f).
By using Lemma 2.2, we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL
\biggl[
An;
\bfI [Bn; f3,n]
q
\biggr]
= \Lambda
\biggl(
\Phi (f)
q
\biggr)
.
From (4.9), we see that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfL [(An; f2), (Bn; f3,n)] = \Phi (f) + q
\biggl(
\Lambda
\biggl(
f2
q
\biggr)
- \Lambda
\biggl(
\Phi (f)
q
\biggr) \biggr)
=: \Psi (f). (4.11)
Since \scrP d(\BbbR N ) is a finite dimensional vector space, the convergence on \scrP d(\BbbR N ) can be understood
as the convergence under any norm on \scrP d(\BbbR N ). Hence the convergence in (4.10) and (4.11) can be
regarded as the uniform convergence on \BbbB N , because the relation
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COMBINING INTERPOLATION SCHEMES AND LAGRANGE INTERPOLATION . . . 557
p \mapsto - \rightarrow \mathrm{s}\mathrm{u}\mathrm{p}
\bfx \in \BbbB N
| p(\bfx )| , p \in \scrP d(\BbbR N )
define a norm on \scrP d(\BbbR N ). It follows from (4.8) that
Pn \circ R\pm (\bfx ) - \rightarrow
\pm
\sqrt{}
1 - \| \bfx \| 2\Lambda (f1)(\bfx ) + \Psi (f)(\bfx )
2
uniformly on \BbbB N . (4.12)
Note that \Lambda (f1) \in \scrP d - 1(\BbbR N ) and \Psi (f) \in \scrP d(\BbbR N ). We define
\bfH \BbbS N (f)(\bfx , xN+1) =
xN+1\Lambda (f1)(\bfx ) + \Psi (f)(\bfx )
2
. (4.13)
Evidently, \bfH \BbbS N (f) \in \scrP d(\BbbR N+1) and
\bfH \BbbS N (f) \circ R\pm (\bfx ) =
\pm
\sqrt{}
1 - \| \bfx \| 2\Lambda (f1)(\bfx ) + \Psi (f)(\bfx )
2
, \bfx \in \BbbB N . (4.14)
Combining (4.12) and (4.14), we deduce that \{ Pn\} converges to \bfH \BbbS N uniformly on \BbbS N .
It remains to prove the desired properties of \bfH \BbbS N (f). From (4.14) we have
\bfH \BbbS N (f) \circ R+(\bfx ) - \bfH \BbbS N (f) \circ R - (\bfx )\sqrt{}
1 - \| \bfx \| 2
= \Lambda (f1)(\bfx ), \bfx \in \BbbB N , (4.15)
and
\bfH \BbbS N (f) \circ R+(\bfx ) +\bfH \BbbS N (f) \circ R - (\bfx ) = \Psi (f)(\bfx ), \bfx \in \BbbB N . (4.16)
It follows from (4.15) that
\Lambda
\bigl(
\bfH \BbbS N (f) \circ R+(\bfx ) - \bfH \BbbS N (f) \circ R - (\bfx )
\bigr)
= \Lambda
\Bigl( \sqrt{}
1 - \| \bfx \| 2\Lambda (f1)(\bfx )
\Bigr)
=
= \Lambda
\Bigl( \sqrt{}
1 - \| \bfx \| 2f1(\bfx )
\Bigr)
= \Lambda
\bigl(
f \circ R+(\bfx ) - f \circ R - (\bfx )
\bigr)
, (4.17)
where we use Lemma 2.1 in the second relation. From (4.16) we see that
\Lambda
\bigl(
\bfH \BbbS N (f) \circ R+(\bfx ) +\bfH \BbbS N (f) \circ R - (\bfx )
\bigr)
= \Lambda (\Psi (f)(\bfx )) .
The next goal is to determine \Lambda (\Psi (f)) . Using Lemma 2.1 again, we conclude from the definition
of \Psi (f) that
\Lambda (\Psi (f)) = \Lambda (\Phi (f)) + \Lambda
\biggl(
q\Lambda
\biggl(
f2
q
\biggr) \biggr)
- \Lambda
\biggl(
q\Lambda
\biggl(
\Phi (f)
q
\biggr) \biggr)
=
= \Lambda (\Phi (f)) + \Lambda
\biggl(
q
f2
q
\biggr)
- \Lambda
\biggl(
q
\Phi (f)
q
\biggr)
= \Lambda (f2).
Hence
\Lambda
\bigl(
\bfH \BbbS N (f) \circ R+(\bfx ) +\bfH \BbbS N (f) \circ R - (\bfx )
\bigr)
= \Lambda
\bigl(
f \circ R+(\bfx ) + f \circ R - (\bfx )
\bigr)
. (4.18)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
558 V. M. PHUNG, V. T. NGUYEN, H. L. DINH
Combining (4.17) and (4.18), we obtain
\Lambda (\bfH \BbbS N \circ R+) = \Lambda (f \circ R+), \Lambda (\bfH \BbbS N \circ R - ) = \Lambda (f \circ R - ).
From (4.16) we conclude that
\Pi
\bigl(
\bfH \BbbS N (f) \circ R+(\bfx ) +\bfH \BbbS N (f) \circ R - (\bfx )
\bigr)
= \Pi (\Psi (f)(\bfx )) .
On the other hand,
\Pi (\Psi (f)) = \Pi (\Phi (f)) + \Pi
\biggl(
q\Lambda
\biggl(
f2
q
\biggr) \biggr)
- \Pi
\biggl(
q\Lambda
\biggl(
\Phi (f)
q
\biggr) \biggr)
= \Pi (\Phi (f)) =
= \Pi
\Bigl(
\Pi
\Bigl(
2f \circ R+(\bfx ) -
\sqrt{}
1 - \| \bfx \| 2\Lambda (f1)(\bfx )
\Bigr) \Bigr)
=
= \Pi
\Bigl(
2f \circ R+(\bfx ) -
\sqrt{}
1 - \| \bfx \| 2\Lambda (f1)(\bfx )
\Bigr)
=
= \Pi
\bigl(
2f \circ R+(\bfx ) - \bfH \BbbS N (f) \circ R+(\bfx ) +\bfH \BbbS N (f) \circ R - (\bfx )
\bigr)
,
where we use Lemma 2.5 in the second and forth equations. It follows that
\Pi
\bigl(
\bfH \BbbS N (f) \circ R+(\bfx ) +\bfH \BbbS N (f) \circ R - (\bfx )
\bigr)
=
= \Pi
\bigl(
2f \circ R+(\bfx ) - \bfH \BbbS N (f) \circ R+(\bfx ) +\bfH \BbbS N (f) \circ R - (\bfx )
\bigr)
.
The last relation finally gives
\Pi (\bfH \BbbS N \circ R+) = \Pi (f \circ R+).
Theorem is proved.
Corollary 4.1. Under the assumptions of Theorem 4.2, we have
\bfH \BbbS N (f)(\bfx , xN+1) =
xN+1\Lambda (f1)(\bfx ) + \Phi (f)(\bfx )
2
,
where
\Psi (f) = \Phi (f) + q
\biggl(
\Lambda
\biggl(
f2
q
\biggr)
- \Lambda
\biggl(
\Phi (f)
q
\biggr) \biggr)
with
\Phi (f) = \Pi
\Bigl(
2f \circ R+(\bfx ) -
\sqrt{}
1 - \| \bfx \| 2\Lambda (f1)(\bfx )
\Bigr)
.
References
1. T. Bloom, J.-P. Calvi, A continiuty property of multivariate Lagrange interpolation, Math. Comput., 66, 1561 – 1577
(1997).
2. B. Bojanov, H. Hakopian, A. Sahakian, Spline functions and multivariate interpolations, Springer-Verlag, Amsterdam
(1993).
3. L. Bos, On certain configurations of points in \BbbR n which are unisolvent for polynomial interpolation, J. Approx.
Theory, 64, 271 – 280 (1991).
4. L. Bos, J.-P.Calvi, Multipoint Taylor interpolation, Calcolo, 45, 35 – 51 (2008).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
COMBINING INTERPOLATION SCHEMES AND LAGRANGE INTERPOLATION . . . 559
5. L. Bos, J.-P.Calvi, Taylorian points of an algebraic curve and bivariate Hermite interpolation, Ann. Scuola Norm.
Super. Pisa Cl. Sci., 7, 545 – 577 (2008).
6. J.-P. Calvi, V. M. Phung, On the continuity of multivariate Lagrange interpolation at natural lattices, L.M.S. J.
Comput. Math., 6, 45 – 60 (2013).
7. J.-P. Calvi, V. M. Phung, Can we define Taylor polynomials on algebraic curves?, Ann. Polon. Math., 118, 1 – 24
(2016).
8. K. C. Chung, T. H. Yao, On lattices admitting unique Lagrange interpolation, SIAM J. Numer. Anal., 14, 735 – 743
(1977).
9. V. M. Phung, On bivariate Hermite interpolation and the limit of certain bivariate Lagrange projectors, Ann. Polon.
Math., 115, 1 – 21 (2015).
10. V. M. Phung, Polynomial interpolation in \BbbR 2 and on the unit sphere in \BbbR 3, Acta Math. Hung., 153, 289 – 317 (2017).
11. V. M. Phung, Hermite interpolation on the unit sphere and the limits of Lagrange projectors, IMA J. Numer. Anal.,
41, 1441 – 1464 (2021).
12. Y. Xu, Polynomial interpolation on the unit sphere and on the unit ball, Adv. Comput. Math., 20, 247 – 260 (2004).
Received 08.01.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
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| id | umjimathkievua-article-6512 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:28:32Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7e/9a96a92e9ce61c39d4d16c5d429f2c7e.pdf |
| spelling | umjimathkievua-article-65122022-07-06T16:22:31Z Combining interpolation schemes and Lagrange interpolation on the unit sphere in $\mathbb R^{N+1}$ Combining interpolation schemes and Lagrange interpolation on the unit sphere in $\mathbb R^{N+1}$ Phung, V. M. Nguyen, V. T. Dinh, H. L. Phung, V. M. Nguyen, V. T. Dinh, H. L. iнтерполяцiя Лагранжа Lagrange interpolation, Limits of Lagrange interpolation, Interpolation on the unit sphere UDC 517.5 We study Lagrange interpolation in $\mathbb R^N$ and on the unit sphere in $\mathbb R^{N+1}$. We show that sequences of unisolvent sets can be combined to get other sequences of unisolvent sets such that the existence of the limits is preserved. Moreover, the limiting operators keep the interpolation conditions under the combining process. УДК 517.5 Комбiнування iнтерполяцiйних схемта iнтерполяцiї лагранжа на одиничнiй сферi в $\mathbb R^{N+1}$ Вивчається iнтерполяцiя Лагранжа в $\mathbb R^N$&nbsp;та на одиничнiй сферi в $\mathbb R^{N+1}$. Доведено, що послiдовностi унiрозв’язних множин можна скомбiнувати в iншi послiдовностi таким чином, що iснування меж збiжностi буде збережено. I навiть бiльше, у такому комбiнуваннi граничнi оператори також зберiгають умови iнтерполяцiї. Institute of Mathematics, NAS of Ukraine 2022-05-23 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6512 10.37863/umzh.v74i4.6512 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 4 (2022); 542 - 559 Український математичний журнал; Том 74 № 4 (2022); 542 - 559 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6512/9223 Copyright (c) 2022 Van Manh Phung, Van Trao Nguyen, Huu Lam Dinh |
| spellingShingle | Phung, V. M. Nguyen, V. T. Dinh, H. L. Phung, V. M. Nguyen, V. T. Dinh, H. L. Combining interpolation schemes and Lagrange interpolation on the unit sphere in $\mathbb R^{N+1}$ |
| title | Combining interpolation schemes and Lagrange interpolation on the unit sphere in $\mathbb R^{N+1}$ |
| title_alt | Combining interpolation schemes and Lagrange interpolation on the unit sphere in $\mathbb R^{N+1}$ |
| title_full | Combining interpolation schemes and Lagrange interpolation on the unit sphere in $\mathbb R^{N+1}$ |
| title_fullStr | Combining interpolation schemes and Lagrange interpolation on the unit sphere in $\mathbb R^{N+1}$ |
| title_full_unstemmed | Combining interpolation schemes and Lagrange interpolation on the unit sphere in $\mathbb R^{N+1}$ |
| title_short | Combining interpolation schemes and Lagrange interpolation on the unit sphere in $\mathbb R^{N+1}$ |
| title_sort | combining interpolation schemes and lagrange interpolation on the unit sphere in $\mathbb r^{n+1}$ |
| topic_facet | iнтерполяцiя Лагранжа Lagrange interpolation Limits of Lagrange interpolation Interpolation on the unit sphere |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6512 |
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