document: Approximation and complexity . II . Iterated integration For pt . I. see ibid. , no. 1 , p. 289-95 -LRB- 2001 -RRB- . We introduce two classes of real analytic functions W contained in/implied by U on an interval . Starting with rational functions to construct functions in W we allow the application of three types of operations : addition , integration , and multiplication by a polynomial with rational coefficients . In a similar way , to construct functions in U we allow integration , addition , and multiplication of functions already constructed in U and multiplication by rational numbers . Thus , U is a subring of the ring of Pfaffian functions . Two lower bounds on the L/sub infinity / - norm are proved on a function f from W -LRB- or from U , respectively -RRB- in terms of the complexity of constructing f
doc_bio_tags: O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O B I O O O O O O O O O O O O O O O B O B O O B O O B O O O O O O O O O O O O O O O O B O B O O B O O O O O O O B O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
extractive_keyphrases: real analytic functions rational functions addition integration multiplication polynomial pfaffian functions lower bounds
abstractive_keyphrases: l/sub infinity /-norm