English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We fix a continuous linear operatorA: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 elementf∈M{f\in M}, findu∈H{u\in H}such thatA⁢u=f{Au=f}and a generalized solution in the sense of Moore–Penrose  u  is sought that satisfies the equationA*⁢A⁢u=A*⁢f{A^{*}Au=A^{*}f}. Moreover, we fix an integern∈ℕ0={0,1,2,…}{n\in\mathbb{N}_{0}=\{0,1,2,\dots\}}and transfer this equation to a special Hilbert spaceD⁢((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 casen=0{n=0}, the spaceD⁢((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 orbitsD⁢((A*⁢A)-∞){D((A^{*}A)^{-\infty})}, the equationA*⁢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 spaceD⁢((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 transformℜ{\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 spaceD((ℜ*ℜ)-n)){D((\mathfrak{R}^{*}\mathfrak{R})^{-n}))}is given.

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