Numerical experiments in Fourier asymptotics of Cantor measures and wavelets. (English) Zbl 0788.65129

Janardhan, Prem; Rosenblum, David; Strichartz, Robert S.

Exp. Math. 1, No. 4, 249-273 (1992).

A so-called multiperiodic function \(F\) fulfils a simple recursion relation involving additive and multiplicative structures. Such functions arise as Fourier transforms of self-similar objects, such as Cantor measures and wavelets. The authors are interested in the following problem: Find some conditions on the pointwise asymptotic behavior of \(F\) that would imply the known asymptotic behavior of \(\int^ x_ 0 | F(t)|^ q dt\) for \(x\to\infty\).
The results for the recursion \(F(x)= f(x/\rho) F(x/\rho)\), \(F(0)= 1\), where \(\rho>1\) and \(f\geq 0\) is 1-periodic with \(f(0)= 1\), are completely proven. For the general case, the authors present conjectures and interesting numerical experiments.

Reviewer: M.Tasche (Rostock)

MSC:

65Q05	Numerical methods for functional equations (MSC2000)
39B22	Functional equations for real functions
65T40	Numerical methods for trigonometric approximation and interpolation
28A80	Fractals
41A60	Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

fractals; multiperiodic function; Fourier transforms; Cantor measures; wavelets; asymptotic behavior; numerical experiments

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