Geodesic
Methods of solving Fredholm equations optimal on classes of functions
Matematičeskie zametki, Tome 15 (1974) no. 4, pp. 595-602

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This paper is devoted to the solution of linear Fredholm equations in the unit $s$-dimensional cube for classes of functions with a dominant mixed derivative of order $r$ in each variable. We present an algorithm for obtaining the solution over the whole domain with an error $O(N^{-r}\ln^{2s-1}N)$ in the uniform metric using the values of the given functions at $O(N\ln^{2s-1}N)$ points and consisting of $O(N\ln^{2s-1}N)$ elementary operations. We show that these estimates can only be improved at the expense of the exponent of $\ln N$. 
@article{MZM_1974_15_4_a10,
     author = {A. F. Shapkin},
     title = {Methods of solving {Fredholm} equations optimal on classes of functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {595--602},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_15_4_a10/}
}
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A. F. Shapkin. Methods of solving Fredholm equations optimal on classes of functions. Matematičeskie zametki, Tome 15 (1974) no. 4, pp. 595-602. http://geodesic.mathdoc.fr/item/MZM_1974_15_4_a10/