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## fa.functional analysis – Continuity of Radon transform w.r.t the angle

let $$f \in L^1(\mathbb R^n)$$ (or in case it helps, actually a probability density on $$\mathbb R^n$$). fix $$b \in \mathbb R$$ and define the Radon transform $$R[f]:S_{n-1} \times \mathbb R \to \mathbb R$$ of $$f$$ by

$$R[f](w,b) := \int_{\mathbb R^n}\delta(x^\top w – b)f(x)\,dx,$$

where $$S_{n-1}$$ is the unit-sphere in $$\mathbb R^n$$.

question. Under what minimal additional conditions on $$f$$ is mapping $$w \mapsto R[f](w,b)$$ continuous continuous on $$S_{n-1}$$for each $$b \in \mathbb R$$ ?

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