4 G. AMBARTSOUMIAN AND V.P. KRISHNAN Since both f(r, θ) and g(ρ, φ) are 2π-periodic in the second variable, one can expand these functions into Fourier series (1) f(r, θ) = n=−∞ fn(r) einθ and (2) g(ρ, φ) = n=−∞ gn(ρ) einφ, where fn(r) = 1 0 f(r, θ) e−inθdθ and gn(ρ) = 1 0 g(ρ, φ) e−inφdφ. We will use Cormack-type [8] inversion strategy to recover Fourier coefficients of f from those of g for limited values of ρ both for CRT and ERT in various setups of the support of f. In the statements below, we denote by A(r1,r2) the open annulus with radii 0 r1 r2 centered at the origin A(r1,r2) = {(r, θ) : r (r1,r2),θ [0, 2π]}. The disc of radius R centered at the origin is denoted by D(0,R). The k-th order Chebyshev polynomial of the first kind is denoted by Tk, i.e. Tk(t) = cos(k arccos t). 3. Main Results The first statement in this section is a generalization of Theorem 1 from [4], which was proved for CRT, to the case of ERT. Theorem 3.1. Let f(r, θ) be a continuous function supported inside the annulus A(ε, b). Suppose REf(ρ, φ) is known for all φ [0, 2π] and ρ (0,b−ε), then f(r, θ) can be uniquely recovered. In this and other theorems of this section, we require f to be continuous, which guarantees the convergence of the Fourier series (1) and (2) almost everywhere. If one needs to ensure convergence everywhere, then some additional conditions on f (e.g. bounded variation) should be added. At the same time, if we assume that f is only piecewise continuous with respect to r for each fixed θ, then we will recover f correctly at points of continuity. As a result, if the function f is not identically zero in D(0,ε1), then one can consider a modified function ˜ f such that ˜(r, f θ) = ⎪0, r ε f(r, θ), ε1 r b smooth cutoff , ε r ε1. It is easy to notice that if ε ε1 then ˜ f satisfies the hypothesis of the theorem, and by sending ε1 ε we also have REf(ρ, φ) = RE ˜(ρ, f φ) for all φ [0, 2π] and ρ (0,b ε1). Hence we get the following statement.
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