We estimate the error in approximating a function f by rational functions of degree n in the norm of Lq(Q), Q := [0,1]d. Among other things, we prove that if f is in the Sobolev space Wk(Q) and if k/d l/p + l/q > O, then f can be approximated by rational functions of degree n to an order O(n-k/d).

In this paper, the problem of approximating a function defined on a finite subset of the real line by a family of generalized rational functions whose numerator and denominator spaces satisfy the Haar conditions on some closed interval [a, b] containing the finite set is considered. The pointwise closure of the family restricted to the finite set is explicitly determined. The representation obt...

In this paper, a generalized rational Chebyshev approximation problem is considered. The problem is this: To minimize the maximum absolute value of the "criterion function" of the error. By imposing a rather natural restriction on the criterion function, the problem is solved completely; the existence, the uniqueness and the characterization of the best approximation are clarified and interesti...

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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 p...