On Best Approximations from RS–sets in Complex Banach Spaces

Strong Uniqueness of a Class of Best Simultaneous Approximation From Rs-Sets in Normed Spaces

“…The notion of an RS-set in the following definition was introduced by Amir in [2] for the case when F = R and by Li in [7,8] for the case when F = C. Definition 2.2. Let X be a normed linear space on the field F and let y 1 , y 2 , .…”

“…The family of RS-sets in normed linear spaces, which was introduced in [2] for the real case and in [7] for the complex case, plays a key role in characterizing the uniqueness of the best approximation and/or the Chebyshev centers; see for example [7,8,12,13]. In [8], Li established the strong and/or the generalized strong uniqueness of the Chebyshev center relative to RS-sets.…”

Approximation by trigonometric polynomials in weighted rearrangement invariant spaces

“…In the non-weighted rearrangement invariant spaces the direct theorems can be found in [8]. Some other aspects of the approximation theory in the more general spaces were investigated by many authors (see, for example: [28]). …”

Nonlinear weighted best simultaneous approximation in Banach spaces

“…This section is mainly concerned with the uniqueness question of best simultaneous approximations from a special kind of sets, RS-sets, which was respectively introduced by Amir in [2] in the case when F = R and by Li in [16] in the case when F = C. Throughout this section, we assume that G is defined by (4.1). Recall that the notations I E , I S and I N are explained in the beginning of the previous section.…”

“…The problem of approximating simultaneously two continuous functions on a finite closed interval was first studied by Dunham in [8], where the results on characterization and uniqueness of best simultaneous approximation were obtained. Since then, such problems have been extended extensively, see, for example, [13][14][15][16][18][19][20][21][22][24][25][26]28,29]. In particular, characterization and uniqueness results were given in [26] for a class of problems involving L p norms, while a general treatment of a class of problems, which includes these problems in [8,26] as special cases, was given in [18].…”