48 Normat 59:1, 48 (2011)

Summary in English

Erwan Brugallé, En smula tropisk

geometri. (Swedish from French).

The aim of the article is to provide a

very elementary introduction to tropical

geometry and its basic objects, starting

by explaining the particular meaning of

the words polynomial, line, and curve in

the tropical algebra setup. The stage is

then set to explore some aspects of the

tropical world, in connection to classi-

cal algebraic geometry via Bezout Theo-

rem, Hilbert’s 16th problem (concerning

conﬁgurations of ovals of real plane cur-

ves), amoebas, and patchworking. Each

new deﬁnition is illustrated with concre-

te examples and pictures, and the artic-

le contains a fair amount of elementary

exercises for the beneﬁt of the serious

reader.

Ola Christensen, Mads Sielemann

Jakobsen Rækkefremstillinger i L

2

(R)

frembragt af trigonometriske funktioner

(Danish)

As is well known the exponential fun-

ctions e

2ﬁinx

with n œ Z make up an

orthonormal basis for L

2

(0, 1) but not

for L

2

(R). An obvious way of ﬁnding a

basis for the whole space is to look at

the functions e

2ﬁimx

‰(x ≠ n) where ‰ is

the characteristic equation for the inter-

val (0, 1).If‰ is replaced by a general

function g and m, n by ma, nb for ﬁxed

real numbers a, b we talk about a Gabor

system. It turns out to be impossible to

for a function g with reasonable decay in

time and frequency to generate an ort-

honormal basis, and instead the authors

consider a weaker condition, namely s o

called Gabor frames. In general, fram-

es lead to series expansion that are very

similar to the ones that are known for

orthonormal bases. The authors present

a case when g is given by a linear combi-

nation of products of trigonometric fun-

ctions with characteristic ones.

Frank Bengtsson,Tilføjelse til ’Om

Christoﬀer Dybvad..’(Danish)

This is a short addition, including a

facsimile, to a previous article of the

author in Normat dealing with 17th cen-

tury quadratures of the circle.