On a question of Brézis and Nirenberg concerning the degree of circle maps (Q1291798)

This article deals with conditions under which the following formula \[ \deg F=\sum^\infty_{n= -\infty}n |c_n|^2 \tag{1} \] holds. Here \(F\) is a map of the unit circle \(S\) into itself, deg is its degree or winding number, \(c_n\) are the Fourier coefficients of \(f(t)=F(e^{it})\). H. Brézis and L. Nirenberg proved that (1) holds if \(F\in H^{1/2} (S, S)\) and formulate the conjecture that (1) holds always when \(\deg F\) is ``well-defined''. The basic result of the article is the negative answer to this conjecture. More precisely, for continuous maps \(F\) the righthand side of (1) may fail to exist, or may have any value different from \(\deg F\), including \(\pm\infty\).