6

AHMED I. ZAYED

to be the m-th rectangular partial sum of S. Let m be a nonnegative integer and

define

to be the m-th square partial sum of S. We say that Sis square convergent to

s

if limm-+oo Bm

=

S.

If

we let llmll

=

max:{mj}, then Sm.= L:llnll:::=;m

an

DEFINITION

0.6 (Unrestrictedly Rectangular Convergence). We say that Sis

unrestrictedly rectangularly convergent to s if

lim Sm

=

s.

min{m;}-+oo

The goal of this chapter is to extend Theorem 0.2 and Theorem 0.3 to higher

dimensions under the above five summation methods and discuss some open ques-

tions.

Charles Fefferman's article is a transcript of his talk which was for a general

audience. It is informative, non-technical, and based on a joint work with B.

Klartag. It relates two fields that are not commonly associated together, Fourier

analysis and computational geometry. The main problem posed in the article may

be described as follows. Fix two positive integers m and n and let E

C

IRn be a

finite set of cardinality

N,

and let

f :

E--+

IR be a given function. We are interested

in finding a smooth surface that passes through the points { ( x,

f (

x))}

xEE ,

where

the smoothness of the surface is measured by m. More precisely, we look for a

function F

E

em ( IRn) such that

(0.1)

with

(0.2)

IF(x)- f(x)l

~

Ma, for all x

E

E,

where

a :

E

--+

[0,

oo).

The significance of

a

is that it measures the closeness of

F

to

f.

For example, when a

=

0, we require that

F

and

f

coincide on

E.

Let

llfllcm(E,u) be the infimum of all M 0 for which there exists F

E

cm(IRn) such

that (0.1) and (0.2) hold. This is a quantitative measure of how randomly scattered

the data points are. The article provides answers to the following two questions:

• How to compute the order of magnitude of llfllcm{E,u) ?

• How to find a function F

E

cm(JRn) that satisfies Equations (0.1) and

(0.2) with

M

having the same order of magnitude as llfllcm{E,u)?

Chapter Eight by P. Hagelstein discusses other convergence questions of Fourier

series. First, let us denote the N-th partial sum of the Fourier series off by SN

f.

The following theorem is well known.

THEOREM

0.7.

Iff

E

LP (T),

1

p

oo,

then the Fourier series off converges

almost everywhere to f. Moreover,

lim IISN

f-

fllv

=

0,

N-+oo

and

if