On degree of approximation of the Gauss-Weierstrass means for smooth functions
© Eryigit; licensee Springer. 2013
Received: 13 May 2013
Accepted: 23 August 2013
Published: 11 September 2013
The notion of μ-smooth point of an -function f is introduced in terms of some ‘maximal function.’ Then the connection between the order of μ-smoothness of the function f and the rate of convergence of the Gauss-Weierstrass means to f, when ε tends to 0, is obtained.
MSC:41A25, 42B08, 26A33.
1 Introduction and formulations of main results
where and .
and called the Gauss-Weierstrass kernel.
at each x belonging to the Lebesgue set of f;
, where is the Hardy-Littlewood maximal function.
Various aspects of the Gauss-Weierstrass and Abel-Poisson type summability of the multiple Fourier series and integrals have been studied in Stein and Weiss , Golubov [6, 7] and Gorodetskii ; see also Weisz  and  and references therein.
The aim of the paper is to investigate the error of approximation of by its Gauss-Weierstrass means as at the so-called μ-smoothness point of f. Note that some problems of the Bochner-Riesz summability of Fourier transform of at the Dini-like points was studied in . Also, the rate of convergence of the Gauss-Weierstrass means of relevant Fourier series at some kind of smoothness points was studied in .
The points , for which (1.5) holds, are called μ-smoothness points of f.
then x is a μ-smoothness point of f, provided .
From now on, we will assume that the function is continued as a constant from to , that is, , .
Now, we state the main results of the paper.
where and are constants independent of ε.
In particular, for in (1.9), we obtain as .
The following lemma plays a crucial role in the proof of the main results.
Lemma A (cf. [, Lemma A])
where is a μ-maximal function defined by (1.4) and is a derivative of φ.
We need also the following lemmas on the well-known properties of the Gauss-Weierstrass kernel and the upper incomplete gamma function.
Lemma 1.9 [, p.9]
Lemma 1.10 [, p.948]
2 Proof of the main results
That completes the proof. □
By (2.7) and (2.11) we have shown that inequality (1.6) holds, as desired.
But this is obvious. □
Proof of Corollary 1.6 Let , be a modulus of continuity, i.e., (a) as ; (b) is non-negative and non-decreasing on ; (c) is continuous and subadditive .
where the constant c does not depend on . □
Simple calculations show that the above inequalities are fulfilled if one takes . □
in (1.6), where and are given numbers and .
which is the desired result. □
This paper was supported by the Scientific Research Project Administration Unit of Akdeniz University and TUBITAK (Turkey).
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