Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 14, 2004, 25–32
© A. I. Fedotov
ABSTRACT. The norm estimation of the Lagrange interpolation operator is obtained. It is shown that the rate of convergence of the interpolative polynomials depends on the choice of the sequence of multiindices and, for some sequences, is equal to the rate of the best approximation of the interpolated function.
2000 Mathematical Subject Classification. 65D05.
Key words and phrases. Lagrange interpolation operator, Sobolev space.
In the paper  the collocation method for singular integral equations and periodic pseudodifferential equations in 1-dimensional Sobolev space was justified. The crucial role in the justification and error estimation plays the fact (Lemma 4) that the Lagrange interpolation operator in this space is bounded. To generalize this results for the multidimensional case the norm estimation (i.e. estimation of the Lebesgue constant) of the Lagrange interpolation operator in multidimensional Sobolev spaces is needed.
Here, we show that in -dimensional Sobolev space the norm of -order Lagrange interpolation operator depends of the function which, w.r.t. the choice of the sequence of multiindices , , is either bounded, or grows infinitely.
Let’s fix the natural and denote by , , , , Cartesian degrees of the sets of natural , natural with zero added , integer , real numbers and the interval correspondingly. For the elements of these sets (-components vectors) besides the usual operations of addition, subtraction and multiplication to the number we’ll define the following operations
and the partial order
By we’ll mean, that takes the values of sone sequence
Furthermore, in a sake of simplicity we’ll write
For the fixed let denote -dimensional Sobolev space, i.e. the closure of all -dimensional smooth -periodic by every variable complex-valued functions w.r.t. the norm
are the complex-valued Fourier coefficients of the function w.r.t the trigonometric monomials
It is known that, being equipped with the inner product
becomes Hilbert space. For the following we’ll assume that , providing (see e.g. ) the embedding of in the space of continuous functions.
Let’s fix , denote by
the set of indices and define uniform partition
on . By we denote Lagrange interpolation operator that assigns to every function polynomial
where , coinciding with in the nodes . Here
are fundamental polynomials satisfying
We have to estimate the norm of the operator .
The results of this section are technical ones. They are gathered in 2 lemmas to exclude less important details from the proof of the main result.
Proof. To change the set of sum indices from to let’s represent as a merge of two sets: and . For the -th, , component of the vector we’ll obtain
As to each summand of index correspond summands when adding by then
be the set of vectors from which component’s sum is . By we denote the number of elements of .
Proof. We’ll show first that
and then that
Let , then for every the set contains only one vector, and hence . Assume that (1) is valid for some , and prove that it is valid then for . We’ll construct the set as a merge of the sets , adding to each element of the set -th component equal to . Then
and hence (1) is valid for all and .
Now assume that estimation (2) is valid for some , and prove that it is valid then for . Indeed,
and - is Riemann’s -function bounded and decreasing for .
Proof. Let’s fix , choose an arbitrary function and write Lagrange interpolative polynomial w.r.t. the nodes for it
It’s Fourier coefficients are
Substituting the values of function in the nodes by its Fourier series expansion we’ll obtain
Further, according to the proof of Lemma 2 , we get
It is easy to check that sum
reaches maximum when , so using Lemma 1 we have
Summands could be estimated as
and using Lemma 2 we obtain
Theorem is proved.
Denote polynomial of the best approximation to of degree not higher than and the corresponding best approximation
where is the -th partial sum of Fourier series of .
The proof is obvious.
sequence of polynomials converges to function with the error estimate
Proof follows directly from Corollary 1.
Proof follows from properties of the best approximation and definitions of norms in and .
Corollary 3 allows to generalize Corollary 2.
sequence of polynomials converges to function with error estimate
Proof follows from Corollaries 1 - 3.
Remark 1. For any constant the set is a cone in . Choosing indices from this cone we’ll obtain sequence of interpolation polynomials converging with estimation (4) where is substituted by . The minimal possible value of will be on the set
of indices with equal components.
 Fedotov A.I. On the asymptotic convergence of the polynomial collocation method for singular integral equations and periodic pseudodifferential equations// Archivum mathematicum. 2002. V.1. P.1-13.
RESEARCH INSTITUTE OF MATHEMATICS AND MECHANICS, KAZAN STATE UNIVERSITY, UNIVERSITETSKAYA STR. 17, KAZAN:420008, RUSSIA
E-mail address: email@example.com
Received December 5, 2003