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\(M_{0}\) measures for the Walsh system. (English) Zbl 1182.42027

Let \(\mathbb{T}\) be the compact group \(\mathbb{R}/\mathbb{Z}\) and let \(M(\mathbb{T})\) be the space of all complex measures on \(\mathbb{T}\). The Fourier coefficients of \(\mu\in M(\mathbb{T})\) are \[ \hat{\mu}(n)=\int_\mathbb{T}e^{-2i\pi nt}d\mu(t)\quad(n\in\mathbb{Z}). \] Moreover, let \(M_0(\mathbb{T})\) be the subspace of measures from \(M(\mathbb{T})\) whose Fourier coefficients tend to zero at infinity.
On the other hand, \(\mathcal{D}=\prod(\mathbb{Z}/2\mathbb{Z})\) is the Cantor group equipped with the addition modulo \(1\) componentwise, without carry over. One can see that there is a one-to-one and onto mapping between \(\mathbb{T}\) and \(\mathcal{D}\) except on a countable set. Thus complex measures can be naturally transported from \(\mathbb{T}\) to \(\mathcal{D}\).
The corresponding space of \(M_0(\mathbb{T})\) on the Cantor group, denoted by \(M_0(\mathcal{D})\), contains the measures on \(\mathbb{T}\) for which \[ \lim_{k\rightarrow\infty}\int w_kd\mu=0, \] where \((w_k)_{k=0}^\infty\) is the Walsh system, an orthonormal and complete set on \(\mathcal{D}\).
The authors establish the surprising fact, that \(M_0(\mathbb{T})\not\subset M_0(\mathcal{D})\) and \(M_0(\mathcal{D})\not\subset M_0(\mathbb{T})\).
The paper ends with some remarks on normality theory (especially the theorem of H. Davenport, P. Erdős and W. J. LeVeque [Mich. Math. J. 10, 311–314 (1963; Zbl 0119.28201)]).

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42A55 Lacunary series of trigonometric and other functions; Riesz products
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

Citations:

Zbl 0119.28201
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References:

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