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...

Abstract. We present results and views about a project in assisted living. The scenario is a room in which an elderly and/or disabled person lives who is not able to perform certain actions due to restricted mobility. We enable the person to express commands verbally that will then be executed automatically. There are several severe problems involved that complicate the situation. The person may utter the command in a rather unexpected way, the person makes an error or the action cannot be performed due to several reasons. In our approach we present an architecture with three components: The recognition component that contains novel features in the signal processing, the analysis component that logically analyzes the command, and the execution component that performs the action automatically. All three components communicate with each other.