Lobachevskii Journal of Mathematics

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Vol. 14, 2004, 25–32 \begin{matrix}c \\ ◯ \end{matrix} A. I. Fedotov

A. I. Fedotov

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.

Introduction

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 m-dimensional Sobolev space H^{s} (s > m/2) the norm of n-order (n = (n_{1}, n_{2}, ..., n_{m})) Lagrange interpolation operator depends of the function M(n, s) which, w.r.t. the choice of the sequence of multiindices (n), n → ∞, is either bounded, or grows infinitely.

2000 Mathematical Subject Classification. 65D05.

Key words and phrases. Lagrange interpolation operator, Sobolev space.

1. Formulation of the problem

Let’s fix the natural m ∈ N and denote by N = N^{m}, N_{0} = N_{0}^{m}, Z = Z^{m}, R = R^{m}, Δ = Δ^{m} Cartesian degrees of the sets of natural N, natural with zero added N_{0}, integer Z, real R numbers and the interval Δ = (−π; π] ⊂ R correspondingly. For the elements of these sets (mcomponents vectors) besides the usual operations of addition, subtraction and multiplication to the number we’ll define the following operations

l · k = ∑_{j=1}^{m} l_{j}k_{j}, l^{2} = ∑_{j=1}^{m} l_{j}^{2}, l ∗ k = (l_{1}k_{1}, l_{2}k_{2}, ..., l_{m}k_{m}), [l] = ∏_{j=1}^{m} l_{j},

and the partial order

l < k ≡ &_{j=1}^{m}(l_{j} < k_{j}), l = (l_{1}, l_{2}, ..., l_{m}), k = (k_{1}, k_{2}, ..., k_{m}).

By n → ∞ we’ll mean, that n takes the values of sone sequence

(n_{j}), n_{j} ∈ N, n_{j} < n_{j+1}, j = 1, 2, ... .

Furthermore, in a sake of simplicity we’ll write

min(n) instead of min_{1≤j≤m}{n_{j} | n = (n_{1}, n_{2}, ..., n_{m}) ∈ N}

and

max(n) instead of max_{1≤j≤m}{n_{j} | n = (n_{1}, n_{2}, ..., n_{m}) ∈ N}.

For the fixed s ∈ R let H^{s} denote m-dimensional Sobolev space, i.e. the closure of all m-dimensional smooth 2π-periodic by every variable complex-valued functions w.r.t. the norm

∥u∥_{s} = ∥u∥_{H^{s}} = (∑_{l∈Z}(1 + l^{2})^{s} | û(l) |^{2})^{1/2},

where

û(l) = (2π)^{−m} ∫_{Δ} u(τ)ē_{l}(τ)dτ, l ∈ Z,

are the complex-valued Fourier coefficients of the function u ∈ H^{s} w.r.t the trigonometric monomials

e_{l}(τ) = exp(il · τ),  l ∈ Z,  τ ∈ Δ,  i = \sqrt{−1}.

It is known that, being equipped with the inner product

< u, v >_{s}= ∑_{l∈Z}(1 + l^{2})^{s}û(l)¯v̂(l), u, v ∈ H^{s},

H^{s} becomes Hilbert space. For the following we’ll assume that s > m/2, providing (see e.g. [2]) the embedding of H^{s} in the space of continuous functions.

Let’s fix n = (n_{1}, n_{2}, ..., n_{m}) ∈ N, denote by

I_{n} = I_{n_{1}}×I_{n_{2}}×···×I_{n_{m}}, I_{n_{j}} = {k_{j} | k_{j} ∈ Z, | k_{j} |≤ n_{j}}, j = 1, 2, ..., m,

the set of indices and define uniform partition

Δ_{n} = {t_{k} = (t_{k_{1}}, t_{k_{2}}, ..., t_{k_{m}}) | k = (k_{1}, k_{2}, ..., k_{m}) ∈ I_{n},
t_{k_{j}} = k_{j}h_{j}, h_{j} = 2π/(2n_{j} + 1), j = 1, 2, ..., m},

on Δ. By P_{n} we denote Lagrange interpolation operator that assigns to every function u ∈ H^{s} polynomial

(P_{n}u)(τ) = ∑_{k∈I_{n}} u(t_{k})ξ_{n}(τ, t_{k}), τ = (τ_{1}, τ_{2}, ..., τ_{m}) ∈ Δ,

where t_{k} = (t_{k_{1}}, t_{k_{2}}, ..., t_{k_{m}}) ∈ Δ_{n}, coinciding with u in the nodes Δ_{n}. Here

ξ_{n}(τ, t_{k}) = ∏_{j=1}^{m} \frac{sin((2n_{j} + 1)(τ_{j} − t_{k_{j}})/2)}{(2n_{j} + 1) sin((τ_{j} − t_{k_{j}})/2)} = [2n + 1]^{−1} ∑_{l∈I_{n}} e_{l}(τ − t_{k}),

1 = (1, 1, ..., 1) ∈ N, τ ∈ Δ, t_{k} ∈ Δ_{n},

are fundamental polynomials satisfying

ξ_{n}(t_{l}, t_{k}) = \{ \begin{matrix}1, & l = k, \\ 0, & l ≠ k, & l, k ∈ I_{n}. \end{matrix}

We have to estimate the norm of the operator P_{n} : H^{s} → H^{s}.

2. Preliminaries

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 m ∈ N, s ∈ R, s > m/2 and n ∈ N

∑_{j∈Z}((n + j ∗ (2n + 1))^{2})^{−s} ≤ 2^{m} ∑_{l∈N}((n ∗ (2l − 1))^{2})^{−s}.

Proof. To change the set of sum indices from Z to N let’s represent Z as a merge of two sets: {−l | l ∈ N} and {l − 1 | l ∈ N}. For the k-th, 1 ≤ k ≤ m, component of the vector n + j ∗ (2n + 1) we’ll obtain

(n_{k} + j_{k}(2n_{k} + 1))^{2} = (l_{k}(2n_{k} + 1) − n_{k})^{2}

= (n_{k}(2l_{k} + \frac{l_{k}}{n_{k}} − 1))^{2} ≥ (n_{k}(2l_{k} − 1))^{2},

j_{k} ∈ Z, j_{k} < 0, l_{k} = −j_{k} ∈ N;

(n_{k}+j_{k}(2n_{k}+1))^{2} = (n_{k}+(l_{k}−1)(2n_{k}+1))^{2} = (n_{k}+2n_{k}l_{k}+l_{k}−2n_{k}−1)^{2})

= (n_{k}(2l_{k} + \frac{l_{k}}{n_{k}} − 1) − 1)^{2} ≥ (n_{k}(2l_{k} − 1))^{2},

j_{k} ∈ Z, j_{k} ≥ 0, l_{k} = j_{k} + 1 ∈ N.

As to each summand of index l ∈ N correspond 2^{m} summands when adding by Z then

∑_{j∈Z}((n + j ∗ (2n + 1))^{2})^{−s} ≤ 2^{m} ∑_{l∈N}((n ∗ (2l − 1))^{2})^{−s}.□

Let

A_{p}^{m} = {k | k = (k_{1}, k_{2}, ..., k_{m}) ∈ N_{0}, ∑_{j=1}^{m} k_{j} = p}

be the set of vectors from N_{0} which component’s sum is p ∈ N_{0}. By R(A_{p}^{m}) we denote the number of elements of A_{p}^{m}.

Lemma 2. For every p, m ∈ N

R(A_{p}^{m}) ≤ mp^{m−1}.

Proof. We’ll show first that

R(A_{p}^{m}) = C_{m+p−1}^{p} = \frac{(m + p − 1)!}{p!(m − 1)!}, p ∈ N_{0}, m ∈ N,  (1)

and then that

C_{m+p−1}^{p} ≤ mp^{m−1}, p, m ∈ N.  (2)

Let m = 1, then for every p ∈ N_{0} the set A_{p}^{1} contains only one vector, and hence R(A_{p}^{1}) = C_{p}^{p} = p^{0} = 1. Assume that (1) is valid for some m ∈ N, and prove that it is valid then for m + 1. We’ll construct the set A_{p}^{m+1} as a merge of the sets A_{j}^{m}, j = 0, 1, ..., p, adding to each element of the set A_{j}^{m} m + 1-th component equal to p − j, j = 0, 1, ..., p. Then

R(A_{p}^{m+1}) = ∑_{j=0}^{p} R(A_{j}^{m}) = ∑_{j=0}^{p} C_{m+j−1}^{j} = \frac{(m + p)!}{m!p!} = C_{m+1}^{p},

and hence (1) is valid for all p ∈ N_{0} and m ∈ N.

Now assume that estimation (2) is valid for some m ∈ N, and prove that it is valid then for m + 1. Indeed,

C_{m+p}^{p} = \frac{(m + p)!}{m!p!}

= C_{m+p−1}^{p} \frac{m + p}{m} ≤ mp^{m−1} \frac{m + p}{m} = p^{m}(\frac{m}{p} + 1) ≤ (m + 1)p^{m}.□ (3)

3. Main results

Theorem 1. For every s ∈ R, m ∈ N, s > m/2 and n ∈ N following estimation is valid

∥P_{n}∥_{H^{s}→H^{s}} ≤ 2^{\frac{m−s}{2}} m^{\frac{s+1}{2}} M(n, s)\sqrt{1 + ζ(2s − m + 1)},

where

M(n, s) = (\frac{\sqrt{n^{2}}}{min(n)})^{s} ,

and ζ(t) = ∑_{j=1}^{∞} j^{−t} - is Riemann’s ζ-function bounded and decreasing for t > 1.

Proof. Let’s fix m ∈ N, s ∈ R, s > m/2, n ∈ N, choose an arbitrary function u ∈ H^{s} and write Lagrange interpolative polynomial w.r.t. the nodes Δ_{n} for it

(P_{n}u)(τ) = ∑_{k∈I_{n}} u(t_{k})ξ_{n}(τ, t_{k}).

It’s Fourier coefficients are

(P̂_{n}u)(l) = \{ \begin{matrix}[2n + 1]^{−1} ∑_{k∈I_{n}} u(t_{k})ē_{l}(t_{k}), & l ∈ I_{n}, \\ 0, & l ∉ I_{n}. \end{matrix}

Substituting the values of function u in the nodes Δ_{n} by its Fourier series expansion we’ll obtain

(P̂_{n}u)(l) = [2n + 1]^{−1} ∑_{k∈I_{n}}(∑_{j∈Z} û(j)e_{j}(t_{k}))ē_{l}(t_{k}) =

= [2n + 1]^{−1} ∑_{j∈Z} û(j) ∑_{k∈I_{n}} e_{j}(t_{k})ē_{l}(t_{k}) = ∑_{j∈Z} û(l + j ∗ (2n + 1)).

30 A. I. FEDOTOV Further, according to the proof of Lemma 2 [1], we get

∥P_{n}u∥_{s}^{2} = ∑_{l∈I_{n}}(1+l^{2})^{s} | (P̂_{n}u)(l) |^{2}= ∑_{l∈I_{n}}(1+l^{2})^{s} | ∑_{j∈Z} û(l+j∗(2n+1)) |^{2}

= ∑_{l∈I_{n}} | ∑_{j∈Z}(1 + l^{2})^{\frac{s}{2}} û(l + j ∗ (2n + 1)) |^{2}

= ∑_{l∈I_{n}}

| ∑_{j∈Z}(1+l^{2})^{\frac{s}{2}} (1+(l+j∗(2n+1))^{2})^{− \frac{s}{2}} û(l+j∗(2n+1))(1+(l+j∗(2n+1))^{2})^{\frac{s}{2}} |^{2}

≤ ∑_{l∈I_{n}}(∑_{j∈Z}((1 + l^{2})/(1 + (l + j ∗ (2n + 1))^{2}))^{s}

∑_{j∈Z} | û(l + j ∗ (2n + 1)) |^{2} (1 + (l + j ∗ (2n + 1))^{2})^{s}) ≤

≤ max_{l∈I_{n}}(∑_{j∈Z}((1 + l^{2})/(1 + (l + j ∗ (2n + 1))^{2}))^{s})∥u∥_{s}^{2}.

It is easy to check that sum

∑_{j∈Z}((1 + l^{2})/(1 + (l + j ∗ (2n + 1))^{2}))^{s}

reaches maximum when l = n, so using Lemma 1 we have

max_{l∈I_{n}}(∑_{j∈Z}((1 + l^{2})/(1 + (l + j ∗ (2n + 1))^{2}))^{s})

= ∑_{j∈Z}((1 + n^{2})/(1 + (n + j ∗ (2n + 1))^{2}))^{s}

≤ 2^{s}(n^{2})^{s} ∑_{j∈Z}((n + j ∗ (2n + 1))^{2})^{−s} ≤ 2^{s+m}(n^{2})^{s} ∑_{j∈Z}((n ∗ (2j − 1))^{2})^{−s}

≤ 2^{s+m}M^{2}(n, s) ∑_{j∈N}((2j − 1)^{2})^{−s}.

Summands could be estimated as

((2j − 1)^{2})^{−s} = (∑_{k=1}^{m}(2j_{k} − 1)^{2})^{−s} ≤ (\frac{m}{(∑_{k=1}^{m}(2j_{k} − 1))^{2}} )^{s} ,

and using Lemma 2 we obtain

2^{s+m}M^{2}(n, s) ∑_{j∈N}((2j − 1)^{2})^{−s}

≤ 2^{s+m}m^{s}M^{2}(n, s) ∑_{j∈N}(∑_{k=1}^{m}(2j_{k} − 1))^{−2s}

= 2^{s+m}m^{s}M^{2}(n, s) ∑_{j∈N}(2 ∑_{k=1}^{m} j_{k}−m)^{−2s} = 2^{s+m}m^{s}M^{2}(n, s) ∑_{j∈N_{0}} \frac{R(A_{j}^{m})}{(m + 2j)^{2s}}

≤ 2^{s+m}m^{s}M^{2}(n, s) (m^{−2s} + ∑_{j∈n} \frac{mj^{m−1}}{(m + 2j)^{2s}} )

≤ 2^{m−s}m^{s+1}M^{2}(n, s)(1 + ∑_{j∈N} j^{−(2s−m+1)}).

Theorem is proved. □

Denote polynomial of the best approximation to u ∈ H^{s} of degree not higher than n ∈ N_{0} and the corresponding best approximation

(S_{n}u)(τ) = ∑_{l∈I_{n}} û(l)e(iτ · l), E_{n}(u)_{s} = ∥u − S_{n}u∥_{s},

where (S_{n}u)(τ) is the n-th partial sum of Fourier series of u.

Corollary 1. For every s ∈ R, m ∈ N, s > m/2, n ∈ N and arbitrary function u ∈ H^{s}

∥u − P_{n}∥_{s} ≤ (1 + 2^{\frac{m−s}{2}} m^{\frac{s+1}{2}} M(n, s)\sqrt{1 + ζ(2s − m + 1)})E_{n}(u)_{s}. (4)

The proof is obvious. □

Corollary 2. For every s ∈ R, m ∈ N, s > m/2, n ∈ N, arbitrary function u ∈ H^{s} and sequence of indices (n_{j})_{j∈N} satisfying

lim_{n→∞} M(n, s) < ∞,

sequence of polynomials (P_{n}u) converges to function u with the error estimate

∥u − P_{n}u∥_{s} = O(E_{n}(u)_{s}).

Proof follows directly from Corollary 1. □

Corollary 3. For any p, s ∈ R, m ∈ N, p ≥ s > m/2, n ∈ N and arbitrary function u ∈ H^{p} the following estimation is valid

E_{n}(u)_{s} ≤ (1 + n^{2})^{\frac{s−p}{2}} E_{n}(u)_{p}.

Proof follows from properties of the best approximation and definitions of norms in H^{s} and H^{p}. □

Corollary 3 allows to generalize Corollary 2.

Corollary 4. For any p, s ∈ R, m ∈ N, p ≥ s > m/2, n ∈ N, arbitrary function u ∈ H^{p} and sequence of indices (n_{j})_{j∈N} satisfying

lim_{n→∞} M(n, s)(n^{2})^{\frac{s−p}{2}} < ∞,

sequence of polynomials (P_{n}u) converges to function u with error estimate

∥u − P_{n}u∥_{s} = O(E_{n}(u)_{p}).

Proof follows from Corollaries 1 - 3.

Remark 1. For any constant C, C ≥ \sqrt{m^{s}}, the set \{n | M(n, s) ≤ C\} is a cone in N. Choosing indices from this cone we’ll obtain sequence of interpolation polynomials converging with estimation (4) where M(n, s) is substituted by C. The minimal possible value of M(n, s) = \sqrt{m^{s}} will be on the set

{n | n ∈ N, n = (n_{1}, n_{2}, ..., n_{m}), n_{k} = n_{l}, 1 ≤ k, l ≤ m}

of indices with equal components.

References

[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