A counterexample to a result concerning controlled approximation

A result of Strang and Fix states that if the order of controlled approximation from a collection of locally supported elements is k, then there is a linear combination Q of those elements and their translates such that any polynomial of degree less than k can be reproduced by Ql and its translates. This paper gives a counterexample to their result. This paper concerns the so-called controlled approximation. This concept was introduced by Strang [S] in 1970. In that paper, Strang described a systematic approach to the choice of trial functions used in finite element analysis. (Also see [FS] for some related results.) To describe his approach, we first introduce some notation. As usual, we mean by Wk?P(Rn) the usual Sobolev space with norm JJJk,p = E |D u||LP I&I?k (Here we use standard multi-index notation.) The seminorm I. Ik,p on Wk,P(Rn) iS defined to be Ulk,p= E ||DUIILP lIel=k When k = 0, Ik,p is a norm, and we write lluflp := jufo,p = IIUIILP. By WckP(Rn) we denote the space of all functions in Wk P(Rn) that have compact support. By lP(Zn) we mean the space of the mappings w from Zn to R which satisfy -\ /p llwllp:I MAW)P) 0 (i = 1, . . ., N), and translate the functions just constructed, replacing x by x jh. Thus we have a family of trial functions: hj(x) := h -n/20 (xlh j), i = 1,... , N, j E Zn. When h = 1, we omit the superscript h. Note that the coefficient h-n/2 normalizes (4 in the sense that /|/ 4jJI2 = |I'i /2. A principal question in finite element analysis is the degree of approximation which can be achieved by the span of q$i',. Concerning this problem, Strang and Fix state the following result (see [SF, Theorem II]): Received by the editors September 24, 1984 and, in revised form July 29, 1985. 1980 Mathematics Subject Classification. Primary 41A29. 'Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based upon work supported by the National Science Foundation under Grant No. MCS-8210950. (?)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page