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Oscillations of anharmonic Fourier series and the wave equation.

Alain HarauxVilmos Komornik — 1985

Revista Matemática Iberoamericana

In this paper we have collected some partial results on the sign of u(t,x) where u is a (sufficiently regular) solution of ⎧     utt + (-1)m Δmu = 0     (t,x) ∈ R x Ω ⎨ ⎩     u = ... = Δm-1 u = 0     t ∈ R. These results rely on the study of a sign of almost periodic functions of a special form restricted...

Ingham type theorems and applications to control theory

Claudio BaiocchiVilmos KomornikPaola Loreti — 1999

Bollettino dell'Unione Matematica Italiana

Ingham [6] ha migliorato un risultato precedente di Wiener [23] sulle serie di Fourier non armoniche. Modificando la sua funzione di peso noi otteniamo risultati ottimali, migliorando precedenti teoremi di Kahane [9], Castro e Zuazua [3], Jaffard, Tucsnak e Zuazua [7] e di Ullrich [21]. Applichiamo poi questi risultati a problemi di osservabilità simultanea.

A two-dimensional univoque set

Martijn de VrieVilmos Komornik — 2011

Fundamenta Mathematicae

Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form x = i = 1 c i q - i with integer coefficients c i satisfying 0 c i < q , i ≥ 1. In this case we say that ( c i ) = c c . . . is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also prove new properties...

A simplified multidimensional integral

Ágnes M. BackhauszVilmos KomornikTivadar Szilágyi — 2009

Czechoslovak Mathematical Journal

We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained more quickly. We also give a characterization of the integrable functions and their primitives.

Generalized golden ratios of ternary alphabets

Vilmos KomornikAnna Chiara LaiMarco Pedicini — 2011

Journal of the European Mathematical Society

Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in dependence...

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