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The construction of normal bases for the space of continuous functions on \(V_ q\), with the aid of operators. (English) Zbl 0828.46069

Summary: Let \(a\) and \(q\) be two units of \(\mathbb{Z}_p\), \(q\) not a root of unity, and let \(V_q\) be the closure of the set \(\{aq^n\): \(n=0, 1, 2,\dots\}\). \(K\) is a non-archimedean valued field, \(K\) contains \(\mathbb{Q}_p\), and \(K\) is complete for the valuation \(|\cdot |\), which extends the \(p\)-adic valuation. \(C(V_q\to K)\) is the Banach space of continuous functions from \(V_q\) to \(K\), equipped with the supremum norm. Let \({\mathcal E}\) and \(D_q\) be the operators on \(C(V_q\to K)\) defined by \(({\mathcal E} f)(x)= f(qx)\) and \((D_q f)(x)= (f(qx)- f(x))/ (x(q- 1))\). We find all linear and continuous operators that commute with \({\mathcal E}\) (resp. with \(D_q\)), and we use these operators to find normal bases \((r_n (x))\) for \(C(V_q\to K)\). If \(f\) is an element of \(C(V_q\to K)\), then there exist elements \(\alpha_n\) of \(K\) such that \(f(x)= \sum_{n=0}^\infty \alpha_n r_n (x)\) where the series on the right-hand-side is uniformly convergent. In some cases it is possible to give an expression for the coefficients \(\alpha_n\).

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46E15 Banach spaces of continuous, differentiable or analytic functions
47B38 Linear operators on function spaces (general)
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References:

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