1. THE RESULT 5

where the infimum is taken for all polynomials in d-variables of total degree at most

n. This is the error in best polynomial approximation and this is what we would

like to characterize.

The main result of this paper is

Theorem 1.1. Let K ⊂ Rd be a d-dimensional convex polytope and r =

1, 2,.... Then, for n ≥ rd, we have

(1.14) En(f)K ≤ MωK

r

f,

1

n

,

where M depends only on K and r.

The matching weak converse

(1.15) ωK

r

f,

1

n

≤

M

nr

n

k=0

(k +

1)r−1Ek(f)K,

n = 1, 2,...,

is an immediate consequence of (1.4) if we apply it on every chord (considered as

[−1, 1]) of K. See [12, Theorem 12.2.3,(12.2.4)], which proof goes over to our case

without any change. Note also, that, exactly as in [12, Corollary 12.2.6], we get

the following consequence of (1.14)–(1.15).

Corollary 1.2. Let α 0 and let f be a continuous function on a d-

dimensional convex polytope K ⊂

Rd.

If f can be approximated with error

n−α

on any chord I of K by polynomials (of a single variable on I) of degree at most

n = 1, 2,..., then En(f)K ≤

Mn−α,

where M depends only on K and α.

This Corollary tells us that

n−α

rate of d-dimensional polynomial approxima-

tion is equivalent to

n−α-rate

of one-dimensional polynomial approximation along

every segment of K (note that if we restrict any function/polynomial of d-variables

to a chord I of K we get a function/polynomial of a single variable on I). This

corollary is true only on polytopes, see [12, Proposition 12.2.7].

In Chapter 13 the same problem in Lp spaces will be considered, and in the

second part of the paper we verify a complete analogue of Theorem 1.1 for Lp-

approximation. In Lp spaces one can even do somewhat better (see Chapter 20),

and we shall prove a stronger form of Theorem 1.1 and its converse (1.15).

There have been many works on polynomial approximation in several variables,

for some of the recent ones see e.g. [4]–[1], [6]–[8], [11], [17] and [21]–[23] and the

references there. In these works various moduli of smoothness are constructed for

special sets like balls and spheres which solve the approximation problem there. Of-

ten the moduli are shown to be equivalent to a K-functional, and the approximation

goes trough the use of that K-functional. These do not work on polytopes, and

precisely the absence of the relevant K-functional what makes the problem of the

present work diﬃcult. We also mention the paper [13] where global approximation

is characterized in terms of local ones.

For special polytopes Theorem 1.1 had a predecessor: call K ⊂

Rd

a simple

polytope if there are precisely d edges at every vertex of K. For example, simplices

and cubes/parallelepipeds are simple polytopes. Now it was proven in Theorem

[12, Theorem 12.2.3] that if K is a simple polytope, then

(1.16) En(f)K ≤ M ωK

r

f,

1

n

+

n−r

f

K

.