Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 14, 2004, 25–32

© A. I. Fedotov

A. I. Fedotov

LEBESGUE CONSTANT ESTIMATION IN MULTIDIMENSIONAL SOBOLEV SPACE

(submitted by F. Avkhadiev)

LEBESGUE CONSTANT ESTIMATION IN MULTIDIMENSIONAL SOBOLEV SPACE

(submitted by F. Avkhadiev)

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.

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2000 Mathematical Subject Classification. 65D05. Key words and phrases. Lagrange interpolation operator, Sobolev space. |

In the paper [1] 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

Let’s fix the natural

and the partial order

By

Furthermore, in a sake of simplicity we’ll write

and

For the fixed

where

are the complex-valued Fourier coefficients of the function

It is known that, being equipped with the inner product

Let’s fix

the set of indices and define uniform partition

on

where

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.

Lemma 1. For every

Proof. To change the set of sum indices from

As to each summand of index

Let

be the set of vectors from

Proof. We’ll show first that

| (1) |

and then that

| (2) |

Let

and hence (1) is valid for all

Now assume that estimation (2) is valid for some

Theorem 1. For every

where

and

Proof. Let’s fix

It’s Fourier coefficients are

Substituting the values of function

Further, according to the proof of Lemma 2 [1], we get

It is easy to check that sum

reaches maximum when

Summands could be estimated as

and using Lemma 2 we obtain

Theorem is proved.

Denote polynomial of the best approximation to

where

Corollary 1. For every

| (4) |

The proof is obvious.

Corollary 2. For every

sequence of polynomials

Proof follows directly from Corollary 1.

Corollary 3. For any

Proof follows from properties of the best approximation and definitions of norms
in

Corollary 3 allows to generalize Corollary 2.

Corollary 4. For any

sequence of polynomials

Proof follows from Corollaries 1 - 3.

Remark 1. For any constant

of indices with equal components.

[1] 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.

[2] Taylor, M.E. Pseudodifferential operators, Princeton University Press, Princeton 1981.

RESEARCH INSTITUTE OF MATHEMATICS AND MECHANICS, KAZAN STATE UNIVERSITY, UNIVERSITETSKAYA STR. 17, KAZAN:420008, RUSSIA

E-mail address: fedotov@mi.ru

Received December 5, 2003