Representation of Functions as Walsh Series to Different Bases and an Application to the Numerical Integration of High-dimensional Walsh Series

We will prove the following theorem on Walsh series, and we will derive from this theorem an effective and constructive method for the numerical integration of Walsh series by number-theoretic methods. Further, concrete computer calculations are given. Theorem. For base b > 2, dimension s > 1, and o > 1, c > 0 {b, s € N; c, a e R), let i,E~°(c) be the class of all functions f: [0, l)s -» C which are representable by absolutely convergent Walsh series to base b with Walsh coefficients W(h\,... ,hs) with the following property: \W(h\,... , hs)\ < c ' (Ai •• • hs)~a for all h\, ... ,hs, where h := max(l, |A|). We show that if f € iË"(c), then f € lhE~°~ßh{c • 2hsa) for all h > 2, provided that a > 1 + ß/,, where fc-*5i+ ££:02logsin I 4 + f j"ÏÏ2*+Tj J 2h Ä-log2 The "exponent" a ßh is best possible for all h , and /?/, is monotonically increasing with 1 log sin H ß:= lim ßh = + JL—JL =0.4499.... A-»oo l log 2