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Récurrences 2 - et 3 -mahlériennes

Bernard Randé (1993)

Journal de théorie des nombres de Bordeaux

On sait (Cobham) qu’une suite 2 - et 3 -automatique est une suite rationnelle. Une question de Loxton et van der Poorten étend ce résultat au cas 2 - et 3 -régulier. On montre dans cet article que, si une suite vérifie une récurrence 2 - et 3 -mahlérienne d’ordre un, elle est rationnelle.

Refinement type equations: sources and results

Rafał Kapica, Janusz Morawiec (2013)

Banach Center Publications

It has been proved recently that the two-direction refinement equation of the form f ( x ) = ∑ n ∈ c n , 1 f ( k x - n ) + ∑ n ∈ ℤ c n , - 1 f ( - k x - n ) can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation f ( x ) = ∑ n ∈ ℤ c ₙ f ( k x - n ) , which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation f ( x ) = ∫ ℝ c ( y ) f ( k x - y ) d y has also various interesting applications....

Regular fractional iteration of convex functions

Marek Kuczma (1980)

Annales Polonici Mathematici

The existence of a unique C 1 solution φ of equation (1) is proved under the condition that f: I → I is convex or concave and of class C 1 in I, 0 < f(x) < x in I*, and f’(x) > 0 in I. Here I = [0, a] or [0, a), 0 < a ≤ ∞, and I* = I 0.

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