## 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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