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On degree of approximation of the Gauss-Weierstrass means for smooth functions
Journal of Inequalities and Applications volume 2013, Article number: 428 (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
Let and . The Φ-means of the integral are defined as [, p.6]
If , then it is said that the (divergent) integral is summable to l. It is possible to obtain various summability methods by choosing a suitable function Φ. For example, by letting , or for , , the classical Abel, Gauss-Weierstrass and Bochner-Riesz means and corresponding summability methods are obtained. One of the important problems in classical harmonic analysis is to construct an (unknown) function f by means of its Fourier transform defined as
However, needs not be integrable for some , and hence the formula
Whenever a function Φ is radial, it is well known that [, p.8] for the Φ-means of the convergent or divergent integral , the following equality holds:
where and .
In particular, putting the function instead of in (1.1), the following formula for the Gauss-Weierstrass means of the integral
is obtained. Here, the function is defined as
and called the Gauss-Weierstrass kernel.
Proposition 1.1 Let (), and let the Gauss-Weierstrass means of f be defined as in (1.2). Then
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 .
Definition 1.2 Let be a positive function on , and assume that . If , defined on , is measurable, we define its μ-maximal function by
Definition 1.3 Let, for a constant , a function be a continuous and positive function on the interval , and assume that . We say that a function is μ-smooth of order at if
The points , for which (1.5) holds, are called μ-smoothness points of f.
Remark 1.4 Simple characterization of a μ-smoothness point is not known. However, most of the classes of ‘smooth’ functions in a classical sense have the μ-smoothness property. For example, if the modulus of continuity of f
satisfies the inequality for , then every point is a μ-smoothness point of f, as can easily be seen from (1.5). In particular, if f satisfies the local Lipschitz (Hölder) condition
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.
Theorem 1.5 Let , , be μ-smooth at . Then the following estimate holds:
where and are constants independent of ε.
Corollary 1.6 Let , , have the μ-smoothness property at x, and let be a modulus of continuity (see [, p.40]) on and continued as a constant to , i.e., , (). Then, under the conditions of Theorem 1.5, we have
Corollary 1.7 Let and , then
Corollary 1.8 Let and be fixed parameters. If we take for and for , then under the conditions of Theorem 1.5,
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])
Suppose that φ is differentiable on , and that the following limits exist:
Let be measurable on and , then
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]
The Gauss-Weierstrass kernel, , has the following property:
Lemma 1.10 [, p.948]
The upper incomplete gamma function, defined as
has the following asymptotic property:
2 Proof of the main results
Proof of Lemma A Changing variables to polar coordinates , , ( is the unite sphere of ), the left side of (1.11) becomes
Using (1.10) and considering the inequality
That completes the proof. □
Proof of Theorem 1.5 Let us fix x, a μ-smoothness point of f, and consider the difference
In order to estimate , we let
Now, by Lemma A, taking , we have
Since f is μ-smooth at the point , we have (see (1.5)). So we get
To estimate , we first apply Hölder’s inequality for and observe that
Let us estimate the first term on the right of (2.8). Changing variables to polar coordinates yields
where is the upper incomplete gamma function. Now, using asymptotic formula (1.13), we get
The same is true for the second term of (2.8):
Now, using formula (1.13), we get
Collecting estimates (2.9) and (2.10), and taking into account that
By (2.7) and (2.11) we have shown that inequality (1.6) holds, as desired.
To complete the proof, we have to show that the conditions of Lemma A are satisfied; that is, for ,
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 .
It follows from the subadditivity of that
By employing this in (1.6), we get
Now, since the function is a modulus of continuity, it cannot tend to zero too rapidly as , that is, for instance, if as , then . Therefore
for some constant . Taking into account this in (2.12), we obtain
where the constant c does not depend on . □
Proof of Corollary 1.7 Let us show that for some the function
is a modulus of continuity, i.e., it is continuous, non-decreasing, subadditive on and tends to zero as . The continuity and are obvious. To prove the other properties, it suffices to show that (see , p.41)
Simple calculations show that the above inequalities are fulfilled if one takes . □
Proof of Corollary 1.8 Let us substitute the function
in (1.6), where and are given numbers and .
If , we have, for sufficiently small ,
By making use of this estimate in (1.6), we have for that
Let now . By setting , we have for
Since , it follows that
Using this in (1.6), we get
which is the desired result. □
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This paper was supported by the Scientific Research Project Administration Unit of Akdeniz University and TUBITAK (Turkey).
The author declares that they have no competing interests.
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Eryigit, M. On degree of approximation of the Gauss-Weierstrass means for smooth functions. J Inequal Appl 2013, 428 (2013). https://doi.org/10.1186/1029-242X-2013-428
- Gauss-Weierstrass integral
- Gauss-Weierstrass means
- inverse Fourier transform
- maximal function