Direct and inverse theorems of approximation theory in Banach function spaces

The paper deals with approximation of functions defined on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in spaces that are not translation invariant. The spaces under consideration are Banach function spaces in which Steklov averaging operators are uniformly bounded. It is proved that operators of convolution with a kernel whose bell shaped majorant is integrable are bounded in these spaces. With the help of convolution operators, direct and inverse theorems of the theory of approximation by trigonometric polynomials and entire functions of exponential type are established. As structural characteristics, the powers of deviations of Steklov averages are used, including nonintegral powers. Theorems for periodic and nonperiodic functions are obtained in a unified way. The results of the paper generalize and refine a lot of known theorems on approximation in specific spaces such as weighted spaces, Lebesgue variable exponent spaces and others.