Anisotropic best τ_C -approximation in normed spaces

CHARACTERIZATIONS AND UNIQUENESS OF BEST SIMULTANEOUS $\tau_C$-APPROXIMATIONS

“…[6]) has played important roles in nonlinear approximation theory in Banach spaces; see, e.g., [2,3,4,6,17] and references therein, and an extension to the case of the best simultaneous approximation was done in [16]. Recently, it was extended in [14] to the setting of the best τ C -approximation. Below we further extend this notion to the case of the best simultaneous τ C -approximation.…”

“…Some uniqueness results of the best simultaneous approximation from a subspace and from a simultaneous sun were given in [1] and [9], respectively. While in the case when F is a singleton, the best simultaneous τ C -approximation is reduced to the generalized best approximation following [5] or the best τ C -approximation following [14]. This has been extensively studied; see, e.g., [5,11,13,14] and references therein.…”

“…While in the case when F is a singleton, the best simultaneous τ C -approximation is reduced to the generalized best approximation following [5] or the best τ C -approximation following [14]. This has been extensively studied; see, e.g., [5,11,13,14] and references therein. In particular, the generic well-posedness of the generalized best approximation problem in terms of the Baire category was investigated in [5,11]; the relationships between the existence of a generalized best approximation and the directional derivative of the function τ C (•; G) were studied in [13], while the characterizations of the generalized best approximation from a general set were given in [14].…”

“…Our main results are on twofolds: one is on the characterization and the other is on the uniqueness. For the first one, we introduce the notions of the simultaneous τ C -sun and the simultaneous regular set, and establish the equivalence between the above notions and the Kolomogorov type condition for the best simultaneous τ C -approximation, which are extensions of the corresponding ones due to [14,16,17]; while for the second one, we use the strict convexity with respect to the approximating set, together with the uniform convexity in every direction in the approximating set, to characterize the uniqueness of the best simultaneous τ C -approximation. It should be remarked that the uniqueness problem for the general case are different from the special case when C is the closed unit ball.…”

The Representation and Continuity of a Generalized Metric Projection onto a Closed Hyperplane in Banach Spaces

“…Thus, natural problems are that whether we can extend results in the classical approximation theory to the setting of the generalized approximation. In this direction, some meaning results, such as existences, characterizations, and well-posedness of this kind of approximation, have been established recently; see [1,[5][6][7]. In this paper, we will consider the problem of representation of generalized metric projection few authors; see [8][9][10][11][12].…”