Above are papers related to the following paper: [Paper_ID] Paper #73712. 	[Title] Approximation from shift-invariant subspaces of L [Abstract] A complete characterization is given of closed shift-invariant subspaces of L 2 (IR d ) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #52000. 	[Title] Radial Basis Functions: L p -approximation orders with scattered centres [Abstract] In this paper we generalize several results on uniform approximation orders with radial basis functions in (Buhmann, Dyn and Levin, 1993) and (Dyn and Ron, 1993) to L p -approximation orders. These results apply, in particular, to approximants from spaces spanned by translates of radial basis functions by scattered centres. Examples to which our results apply include quasi-interpolation and least-squares approximation from radial function spaces. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1112194. 	[Title] Negative observations concerning approximations from spaces generated by scattered shifts of functions vanishing at 1 [Abstract] Approximation by scattered shifts f( ff)g ff2A of a basis function are considered, and different methods for localizing these translates are compared. It is argued in the note that the superior localization processes are those that employ the original translates only. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #52003. 	[Title] Radial basis function approximation: from gridded centers to scattered centers [Abstract] The paper studies L 1 (IR d )-norm approximations from a space spanned by a discrete set of translates of a basis function . Attention here is restricted to functions whose Fourier transform is smooth on IR d n0, and has a singularity at the origin. Examples of such basis functions are the thin-plate splines and the multiquadrics, as well as other types of radial basis functions that are employed in Approximation Theory. The above approximation problem is well-understood in case the set of points ffi used for translating forms a lattice in IR d , and many optimal and quasi-optimal approximation schemes can already be found in the literature. In contrast, only few, mostly specific, results are known for a set ffi of scattered points. The main objective of this paper is to provide a general tool for extending approximation schemes that use integer translates of a basis function to the non-uniform case. We introduce a single, relatively simple, conversion method that preserves the approximation orders provided by a large number of schemes presently in the literature (more precisely, to almost all "stationary schemes"). In anticipation of future introduction of new schemes for uniform grids, an effort is made to impose only a few mild conditions on the function , which still allow for a unified error analysis to hold. In the course of the discussion here, the recent results of [BuDL] on scattered center approximation are reproduced and improved upon. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1104809. 	[Title] APPROXIMATION IN L p (R d FROM SPACES SPANNED BY THE PERTURBED INTEGER TRANSLATES OF [Abstract] May 14, 1995 Abstract. The problem of approximating smooth L p -functions from spaces spanned by the integer translates of a radially symmetric function is very well understood. In case the points of translation, ffi, are scattered throughout R d , the approximation problem is only well understood in the "stationary" setting. In this work, we treat the "non-stationary" setting under the assumption that ffi is a small perturbation of Z d . Our results, which are similar in many respects to the known results for the case ffi = Z d , apply specifically to the examples of the Gauss kernel and the Generalized Multiquadric. Please summarise all the given information.

Paper #73712 has the title `Approximation from shift-invariant subspaces of L  
` and the abstract `A complete characterization is given of closed shift-invariant subspaces of L 2 (IR d ) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.
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Question: Which category should Paper #73712 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #73712 has the title `Approximation from shift-invariant subspaces of L  
` and the abstract `A complete characterization is given of closed shift-invariant subspaces of L 2 (IR d ) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.
`.
Question: Which category should Paper #73712 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Neural_Networks