Approximation by Rational Functions

Making use of the Hardy-Littlewood maximal function, we give a new proof of the following theorem of Pekarski: If f' is in L log L on a finite interval, then f can be approximated in the uniform norm by rational functions of degree n to an error 0(1/n) on that interval. It is well known that approximation by rational functions of degree n can produce a dramatically smaller error than that for polynomials of degree n. The best example of this is Newman's theorem [3] which shows that the function f(x) = IxI can be approximated on [-1,1] by rational functions of degree n to an error O(exp(-cV/_)), whereas for polynomials of degree n the error is known to be larger than c/n. Other authors have shown that such improvement also occurs for certain classes of func1,ions. For example, V. Popov [5] showed that if f' E Lp[0, 1], with p > 1, then rn(f) = O(n-1) where rn(f) is the error in approximating f by rational functions R of degree at most n in the uniform norm: rn(f): = di(nRf Ilf -Rllco [0? 1]. deg(R)=n To obtain this order of approximation for polynomials requires roughly speaking that f' E Lo. A striking limiting version of Popov's result was given by A. A. Pekarski [4], who showwed that the same conclusion holds when f' E L log L, i.e. if If'I log(1 + If'I) is integrable. The Popov and Pekarski proofs of these theorems are quite technical, and it was the purpose of [2] to introduce an elementary technique using maximal functions and partitions of unity for rational functions in order to give a simpler proof of Popov's results. The point of this note is to show that a modification of the technique in [2], albeit a little tricky, will also prove Pekarski's theorem. The idea in [2] is to partition [0, 1] into a set I of disjoint intervals I and construct associated rational functions ,I which form a partition of unity: EIEI =_-1. Our rational approximation R is then given by ( 1) R(x) : = E f (xi),i (x) IEI with xI the center of I. Of course, the intervals I depend on f. The rational functions ,i are constructed using a standard method for partitions of unity. Namely, 4i := Oi/(D with (D := E 4I. In the case of Popov's theorem, the XI depend only on the interval I and all can be taken of degree 4. The intervals Received by the editors October 3, 1985. 1980 Matematics Subject CZamifwation (1985 Revision). Primary 41A20, 41A25, 41A63, 42B25. Supported by the National Science Foundation Grant DMS 8320562. (? 1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page