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

Abstract

One considers maps F of the unit circle into itself. For maps of class C 1 or in the Sobolev class H½, the degree or winding number is equal to the sum of the series $ \sum n|c_n|^2 $ in terms of the Fourier coefficients c n of F (e it ) = f (t). Here one studies the possible relation between this series (in symmetric arrangement) and the degree for arbitrary continuous maps F. It is shown that for such maps, the series may fail to be convergent or Abel summable. If the series does converge, the sum may have any value different from the degree of F.