In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonpe-riodic function, defined on an interval x ∈ [−χ,χ], to a function f ̃ which is periodic on the larger interval x ∈ [−Θ,Θ]. We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval x ∈ [−χ,χ], identically zero for |x|>Θ, and varies smoothly in between. Such smoothed “top-hat ” functions are “bells ” in wavelet theory. Our bell is (for x≥ 0) T (x;L,χ,Θ) = (1 + erf(z))/2 where z = Lξ/ 1 − ξ2 where ξ ≡ −1 + 2(Θ − x)/(Θ − χ). By applying steepest descents to approximate the coefficient inte-grals in the limit of large degree j, we show that when the width L is fixed, the Fourier cosine coefficients aj of T on x ∈ [−Θ,Θ] are proportional to aj ∼ (1/j) exp(−Lπ1/22−1/2(1 − χ/Θ)1/2j1/2)Λ(j) where Λ(j) is an oscillatory fac-tor of degree given in the text. We also show that to minimize error in a Fou-rier series truncated after the N th term, the width should be chosen to increase with N as L = 0.91√1−χ/ΘN1/2. We derive similar asymptotics for the func-tion f (x) = x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergence.

Wilson Bases are constituted by trigonometric functions multiplied by translates of a window function with good time frequency localization. In this paper we investigate the approximation of functions from Sobolev spaces by partial sums of the Wilson basis expansion. In particular, we show that the approximation can be improved if polynomials are reproduced. We give examples of Wilson bases, which reproduce linear functions with the lowest frequency term only. Key Words. Local trigonometric bases, Wilson bases, biorthogonality, Riesz basis, unconditional bases. AMS subject classifications. 41A25,41A30, 42A10. 1 Introduction Bases of L 2 (R) consisting of functions with good time frequency localization have gained great importance in signal and image processing. One possibility to construct such bases is to consider trigonometric functions multiplied by smooth, well localized window functions. One example of such bases are the orthonormal Wilson bases of Daubechies, Ja#ard and Jour...