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Let $$G$$ be a locally compact bounded Vilenkin group, $$\Gamma$$ be the dual group of $$G$$ . Suppose that a function $$f(x)$$ belongs to the the Lebesgue class $$L^p(G)$$ , $$10$$ , $$\beta\in{\mathbb R}$$ , then for which values of $$r$$ we can guarantee that $$\widehat{f}\in L^r(\Gamma)$$ ? The result is an analogue of one classical theorem of E. Titchmarsh about the Fourier transform of functions from the Lipschitz classes on $${\mathbb R}$$ .

Let ℚp be the field of p-adic numbers, a function f(x) belongs to the the Lebesgue class Lρ(ℚp), 1 ρ ≤ 2, and let $$\hat{f}(\xi)$$ be the Fourier transform of f. In this paper we give an answer to the next problem: if the function f belongs to the Dini-Lipschitz class DLip(α, β, ρ; ℚp), α > 0, β ∈ ℝ, then for which values of r we can guarantee that $$\hat{f} \in {L^r}(\mathbb{Q}_p)$$? The result is an analogue of one classical theorem of E. Titchmarsh about the Fourier transform of functions from the Lipschitz classes on ℝ.

Some problems in the theory of approximation of complex-valued functions on locally compact Vilenkin groups in the metric of Lp, 1 ≤ p ≤ ∞, by functions with bounded spectrum, are investigated. A description of certain function spaces in terms of the best approximations are obtained and some imbedding theorems are proved.

Some problems in the theory of approximation of complex-valued functions on the group Q p in the metric of L ρ , 1 ≤ ρ≤∞ by functions with bounded spectrum, are investigated. A description of certain function spaces in terms of the best approximations are obtained and some imbedding theorems are proved.

We consider some problems of spectral analysis and spectral synthesis in the topological vector space $${{\mathcal {M}}}(G)$$ of tempered functions on a discrete Abelian group G. It is proved that spectral analysis holds in the space $${{\mathcal {M}}}(G)$$ on every Abelian group G, that is, every nonzero closed linear translation invariant subspace of $${{\mathcal {M}}}(G)$$ contains an exponential. For any finitely generated Abelian group G it is proved, that spectral synthesis holds in $${{\mathcal {M}}}(G)$$ , that is, every closed linear translation invariant subspace $${{\mathscr {H}}}$$ of $${{\mathcal {M}}}(G)$$ coincides with the closed linear span of all exponential monomials belonging to $${{\mathscr {H}}}$$ . For any Abelian group G with infinite torsion free rank it is proved that spectral synthesis fails to hold in the space $${{\mathcal {M}}}(G)$$ .

In this paper for functions on locally compact Vilenkin groups, we prove an analogue of one classical Titchmarsh theorem on the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L2.

In this paper, for functions on the group Q p , we prove an analogue of the classical Titchmarsh theorem on description of the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L 2.