Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth

Abstract

This work deals with basic approximation problems such as direct, inverse and simultaneous theorems of trigonometric approximation of functions of weighted Lebesgue spaces with a variable exponent on weights satisfying a variable Muckenhoupt A p(·) type condition. Several applications of these results help us transfer the approximation results for weighted variable Smirnov spaces of functions defined on sufficiently smooth finite domains of complex plane ℂ.

Approximation theorems in weighted Lebesgue spaces with variable exponent

“…In case k = 1, Theorem 1.11 was proved in [15]. Moreover, in the case k ∈ R + , Theorem 1.11 was proved in [2] where…”

“…In case r = 1, Theorem 1.3 was proved in [14], Theorem 1.7 was proved in [15], Theorems 1.4 and 1.5 were proved in [29]. In case r ∈ R + , Theorems 1.3 and 1.7 were proved in [2] by different type modulus of smoothness where ω…”

“…Previously, the approximation properties of different summation methods were investigated in [11,19,20]. On the other hand, by different type modulus of smoothness appropriate approximation theorems in L p(•) ω (T) were proved in [2,3], where ω −p 0 ∈ A q * (•) (T), 1 (p(•)/p0) + 1 q * (•) = 1, for some p 0 ∈ 1, p − . Nevertheless, the A p(•) (T) class is more intelligible than the class of weights considered in these work.…”

Multiplier and approximation theorems in Smirnov classes with variable exponent

“…Theorems 4 and 5 in the case of r = 1 were proved in [16] (see also [17]). In the variable exponent Lebesgue spaces L p(•) ([0, 2π]) these theorems in the case of r=1 and p(•) ∈ P 0 ([0, 2π]) , using some other modulus of smoothness, were proved in [1][2][3]10,20,21]. For a wider class of the exponents p(•) , namely when…”

Convolution and Jackson inequalities in Musielak–Orlicz spaces

“…In general, Musielak-Orlicz spaces may not attain the translation invariance property, as can be seen in the case of variable exponent Lebesgue spaces L p (x) . Several inequalities of trigonometric polynomial approximation in L p (x) were obtained in [2,4,14,19,31,33]. Note that, under the translation invariance hypothesis on Musielak-Orlicz space, Musielak obtained some trigonometric approximation inequalities in [27].…”