On linear spline algorithms of computerized tomography in the space of n-orbits

Abstract

Abstract We fix a continuous linear operator A : H → M {A:H\rightarrow M} acting between the Hilbert spaces H and M that admits a singular value decomposition (SVD). We consider the following ill-posed problem: for an element f ∈ M {f\in M} , find u ∈ H {u\in H} such that A ⁢ u = f {Au=f} and a generalized solution in the sense of Moore–Penrose u is sought that satisfies the equation A * ⁢ A ⁢ u = A * ⁢ f {A^{*}Au=A^{*}f} . Moreover, we fix an integer n ∈ ℕ 0 = { 0 , 1 , 2 , … } {n\in\mathbb{N}_{0}=\{0,1,2,\dots\}} and transfer this equation to a special Hilbert space D ⁢ ( ( A * ⁢ A ) - n ) {D((A^{*}A)^{-n})} of n-orbits. For an approximate solution of this equation in the case of a nonadaptive information on the right-hand side f, a linear spline algorithm is constructed. The specificity of the considered norm is that the approximate solution is the truncated singular value decomposition (TSVD) and does not depend on n. In the case n = 0 {n=0} , the space D ⁢ ( ( A * ⁢ A ) - n ) {D((A^{*}A)^{-n})} coincides with H and we obtain the results for the latter space. In the limiting case of the Fréchet–Hilbert space of all orbits D ⁢ ( ( A * ⁢ A ) - ∞ ) {D((A^{*}A)^{-\infty})} , the equation A * ⁢ A ⁢ u = A * ⁢ f {A^{*}Au=A^{*}f} becomes well-posed and was considered in [D. Ugulava and D. Zarnadze, On a linear generalized central spline algorithm of computerized tomography, Proc. A. Razmadze Math. Inst. 168 2015, 129–148]. It is also noted that the space D ⁢ ( ( A * ⁢ A ) - ∞ ) {D((A^{*}A)^{-\infty})} is the projective limit of the sequence of Hilbert spaces { D ⁢ ( ( A * ⁢ A ) - n ) } {\{D((A^{*}A)^{-n})\}} . The application of the obtained results for the computerized tomography problem, i.e., for the inversion of the Radon transform ℜ {\mathfrak{R}} with the SVD of Louis [A. K. Louis, Orthogonal function series expansions and the null space of the Radon transform, SIAM J. Math. Anal. 15 1984, 3, 621–633] in the space D ( ( ℜ * ℜ ) - n ) ) {D((\mathfrak{R}^{*}\mathfrak{R})^{-n}))} is given.