Geodesic
On the properties of orthorecursive expansions with respect to subspaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 138-141

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In a Hilbert space, for orthorecursive expansions with respect to closed subspaces, we establish a criterion for expansions of elements of a certain finite-dimensional subspace with respect to a finite sequence of subspaces to coincide with the expanded elements. This implies a criterion for an element to be equal to its orthorecursive expansion with respect to a finite sequence of subspaces. We also obtain a number of results related to the best approximations of elements by partial sums of their orthorecursive expansions with respect to a sequence of finite-dimensional subspaces. 
@article{TM_2014_284_a6,
     author = {V. V. Galatenko and T. P. Lukashenko and V. A. Sadovnichii},
     title = {On the properties of orthorecursive expansions with respect to subspaces},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {138--141},
     publisher = {mathdoc},
     volume = {284},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2014_284_a6/}
}
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V. V. Galatenko; T. P. Lukashenko; V. A. Sadovnichii. On the properties of orthorecursive expansions with respect to subspaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 138-141. http://geodesic.mathdoc.fr/item/TM_2014_284_a6/