A new concept of smoothness in Orlicz spaces

Ferreyra, D. E. ^[1] ; Levis, F. E. ^[1] ; Roldán, M. V. ^[2]
1. [1] Universidad Nacional de Río Cuarto
  
  Universidad Nacional de Río Cuarto
  
  Argentina
2. [2] Universidad Nacional de La Pampa
  
  Universidad Nacional de La Pampa
  
  Argentina
Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 73, Fasc. 3, 2022, págs. 505-520
Idioma: inglés
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Resumen
- In a 2015 article Cuenya and Ferreyra defined a class of functions in Lp-spaces, denoted by cpn(x). The class cpn(x) contains the class of Lp-differentiability functions, denoted by tpn(x), introduced in a 1961 article by Calderón-Zygmund. A more recent paper by Acinas, Favier and Zó introduced a new class of functions in Orlicz spaces LΦ, called LΦ-differentiable functions in the present article. The class of LΦ-differentiable functions is closely related to the class tpn(x). In this work, we define a class of functions in LΦ, denoted by cΦn(x). The class cΦn(x) is more general than the class of LΦ-differentiable functions. We prove the existence of the best local Φ-approximation for functions in cΦn(x) and study the convexity of the set of cluster points of the set of best Φ-approximations to a function on an interval when their measures tend to zero.