Approximation theorems for multivariate Taylor-Abel-Poisson means

Juergen Prestin, Viktor Savchuk, Andrii Shidlich


It is well-known that any function $f \in L_p(\mathbb T^1)$  that is different from a constant can be approximated by its
Abel-Poisson means $f(\varrho,\cdot)$ with a precision not better than $1-\varrho$. It relates to the so-called saturation property of this approximation method.
From this property, it follows that for any $f\in L_p(\mathbb T^1)$, the relation $\|f-f(\varrho,\cdot)\|_{_{\scriptstyle p}} =\mbox{\tiny  $\mathcal O$}(1-\varrho)$, $\varrho\to 1-$,
only holds  in the trivial case when  $f$ is a constant  function. Therefore, any additional restrictions on the smoothness of functions do not give us any
order of approximation better than  $1-\varrho$. In this connection, a natural question is to find a linear operator,  constructed similarly to the Poisson operator,
which takes into account the smoothness properties of functions and at the same time, for a given functional class,  is the best in a certain sense.
In [\ref{Savchuk_Zastavnyi}], for classes of convolutions whose kernels were generated by some moment sequences, the authors proposed a general method
of construction of similar operators that take into account properties of such kernels and hence, the smoothness of functions from corresponding  classes.
One example of such operators are the operators $ A_{\varrho, r} $, which are the main subject of study in this paper.


direct approximation theorem; inverse approximation theorem; Taylor-Abel-Poisson means, $K$-functional, multiplier

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