Unique minimality of Fourier projections

Abstract

The question of when the Fourier projection is the only one of least norm from a space of continuous functions on the circle onto spaces spanned by trigonometric polynomials is studied in two settings.In the first the domain space is the disc algebra and the range is finite-dimensional.In the second the domain space consists of all real continuous functions and the range has finite codimension.Introduction.General background.Let Sj, be the Fourier projection of the continuous (real or complex) periodic functions, C2", onto irn the trigonometric polynomials of degree less than or equal to n.One of the extremal properties of Sj, is that it has the minimal operator norm along all the projections (i.e.idempotent operators) of C2w onto irn (Lozinski [8]).The numbers ||Sj,|| are known to be unbounded.A consequence of this, for example, is that for any sequence of projections {Ln} onto wn-such as interpolation at n + 1 points-there is an / in C2v. and a point x such that (L,/)(x) is unbounded (Cheney [2]).The minimal norm property of §j, follows from an averaging process due to Berman [1] which shows that for any projection P, ^-¿/" T.x°P°TJd\, 2-ïï J-w where T,J(x) = f(x + X).This equation is also attributed to Marcinkiewicz [9].The Berman formula immediately shows more than we stated.Let T he any set of integers and trr the closed linear span of {e,kx: k G T}. Let £ be a subspace of C2" which contains trT and which is the closed linear span of those functions e,nx contained in it.The Fourier projection is then the (obviously unique if it exists) bounded projection Sp of E onto wr such that (%p)(t) = 2 PÜ)eUl JGT for all trigonometric polynomials p in E. In this setting Berman's formula obtains (since its right-hand side satisfies the defining properties for Sr-) and so if P is any projection of E onto trr then H^jr-H < ||P||.More recently it was found that, for the classical case, Sj, is in fact the only projection of minimal norm [3].It is the extension of this uniqueness property to