Open Access

On degree of approximation of the Gauss-Weierstrass means for smooth L p ( R n ) functions

Journal of Inequalities and Applications20132013:428

https://doi.org/10.1186/1029-242X-2013-428

Received: 13 May 2013

Accepted: 23 August 2013

Published: 11 September 2013

Abstract

The notion of μ-smooth point of an L p ( R n ) -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.

Keywords

Gauss-Weierstrass integralGauss-Weierstrass meansapproximationinverse Fourier transformmaximal function

1 Introduction and formulations of main results

Let Φ C 0 ( R n ) L 1 ( R n ) and Φ ( 0 ) = 1 . The Φ-means of the integral R n f ( x ) d x are defined as [[1], p.6]
M ε , Φ ( f ) = R n Φ ( ε x ) f ( x ) d x ( ε > 0 ) .
If lim ε 0 + M ε , Φ ( f ) = l , then it is said that the (divergent) integral R n f ( x ) d x is summable to l. It is possible to obtain various summability methods by choosing a suitable function Φ. For example, by letting Φ ( x ) = e | x | , Φ ( x ) = e | x | 2 or for δ > 0 , Φ ( x ) = { ( 1 | x | 2 ) δ ; | x | 1 0 ; | x | > 1 } , 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 F ( f ) defined as
F ( f ) ( x ) = R n f ( t ) e 2 π i x t d t .
However, F ( f ) needs not be integrable for some f L p ( R n ) , and hence the formula
f ( x ) = R n F ( f ) ( t ) e 2 π i x t d t

becomes incorrect. To overcome this difficulty, one may apply suitable summability methods (see, e.g., [13]).

Whenever a function Φ is radial, it is well known that [[1], p.8] for the Φ-means of the convergent or divergent integral R n F ( f ) ( t ) e 2 π i x t d t , the following equality holds:
R n F ( f ) ( x ) e 2 π i x t Φ ( ε x ) d x = R n f ( x ) φ ε ( t x ) d x ,
(1.1)

where φ ε ( x ) = ( 1 / ε ) n φ ε ( x / ε ) and φ ( x ) = F ( Φ ) .

In particular, putting the function e | x | 2 instead of Φ ( x ) in (1.1), the following formula for the Gauss-Weierstrass means of the integral R n F ( f ) ( t ) e 2 π i x t d t
S ( x , ε ) = R n f ( t ) φ ε ( x t ) d t ( ε > 0 )
(1.2)
is obtained. Here, the function φ ε is defined as
φ ε ( x ) W ( x , ε ) = ( 4 π ε ) ( n / 2 ) e | x | 2 / 4 ε ,
(1.3)

and called the Gauss-Weierstrass kernel.

One of the well-known and basic results for the Gauss-Weierstrass means is the following ([[4], p.5], [[5], p.223]).

Proposition 1.1 Let f L p ( R n ) ( 1 p < ), and let the Gauss-Weierstrass means of f be defined as in (1.2). Then
  1. (a)

    lim ε 0 S ( x , ε ) f L p = 0 ;

     
  2. (b)

    lim ε 0 S ( x , ε ) = f ( x ) at each x belonging to the Lebesgue set of f;

     
  3. (c)

    sup ε > 0 | S ( x , ε ) | c ( M f ) ( x ) , where ( M f ) ( x ) 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 [1], Golubov [6, 7] and Gorodetskii [8]; see also Weisz [2] and [9] and references therein.

The aim of the paper is to investigate the error of approximation of f ( x ) by its Gauss-Weierstrass means S ( x , ε ) as ε 0 at the so-called μ-smoothness point of f. Note that some problems of the Bochner-Riesz summability of Fourier transform of f L p ( R n ) at the Dini-like points was studied in [10]. Also, the rate of convergence of the Gauss-Weierstrass means of relevant Fourier series at some kind of smoothness points was studied in [9].

Definition 1.2 Let μ ( r ) be a positive function on ( 0 , ) , and assume that lim r 0 + μ ( r ) = 0 . If ψ ( t , x ) , defined on R n × R n , is measurable, we define its μ-maximal function by
( M μ ψ ) ( x ) = sup r > 0 1 μ ( r ) r n | t | < r | ψ ( t , x ) | d t .
(1.4)
Definition 1.3 Let, for a constant ρ < 1 , a function μ ( r ) be a continuous and positive function on the interval ( 0 , ρ ) , and assume that lim r 0 + μ ( r ) = μ ( 0 ) = 0 . We say that a function f L l o c 1 ( R n ) is μ-smooth of order μ ( r ) at x R n if
D μ ( x ) = sup 0 < r < 1 1 r n μ ( r ) | t | < r | f ( x t ) f ( x ) | d t < .
(1.5)

The points x R n , 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
w f ( r ) = sup | x | r f ( x ) f ( )
satisfies the inequality w f ( r ) c μ ( r ) for r 0 , then every point x R n 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
| f ( x t ) f ( x ) | c | t | α , 0 < α 1 ,

then x is a μ-smoothness point of f, provided μ ( r ) = r α .

From now on, we will assume that the function μ ( r ) is continued as a constant from [ 0 , ρ ] to [ ρ , ) , that is, μ ( r ) = μ ( ρ ) , r ρ .

Now, we state the main results of the paper.

Theorem 1.5 Let f L p ( R n ) , 1 < p < , be μ-smooth at x R n . Then the following estimate holds:
| S ( x , ε ) f ( x ) | c 1 0 r n + 1 e r 2 / 4 μ ( ε r ) d r + c 2 ε n / 2 e 1 / 4 ε ( ε 0 + ) ,
(1.6)

where c 1 and c 2 are constants independent of ε.

Corollary 1.6 Let f L p ( R n ) , 1 p < , have the μ-smoothness property at x, and let μ ( r ) be a modulus of continuity (see [[11], p.40]) on [ 0 , ρ ] and continued as a constant to [ ρ , ) , i.e., μ ( r ) = μ ( ρ ) , r ρ ( 0 < ρ < 1 ). Then, under the conditions of Theorem  1.5, we have
| S ( x , ε ) f ( x ) | c μ ( ε ) ( ε 0 ) .
(1.7)
Corollary 1.7 Let α > 0 and μ ( r ) = ( 1 ln 1 r ) α , then
| S ( x , ε ) f ( x ) | c ( 1 ln 1 ε ) α ( ε 0 ) .
(1.8)
Corollary 1.8 Let α > 0 and < β < be fixed parameters. If we take μ ( r ) = r α | ln r | β for 0 < r ρ < 1 and μ ( r ) = μ ( ρ ) for ρ < r < , then under the conditions of Theorem  1.5,
| S ( x , ε ) f ( x ) | c ε α | ln ε | β ( ε 0 ) .
(1.9)

In particular, for β = 0 in (1.9), we obtain | S ( x , ε ) f ( x ) | c ε α as ε 0 .

The following lemma plays a crucial role in the proof of the main results.

Lemma A (cf. [[9], Lemma A])

Suppose that φ is differentiable on ( 0 , ) , and that the following limits exist:
lim r r n μ ( r ) φ ( r ) = 0 and lim r 0 + r n μ ( r ) φ ( r ) = 0 .
(1.10)
Let ψ ( t , x ) be measurable on R n × R n and ( M μ ψ ) ( x ) < , then
R n | ψ ( t , x ) φ ( | t | ) | d t ( M μ ψ ) ( x ) 0 r n μ ( r ) | φ ( r ) | d r ,
(1.11)

where ( M μ ψ ) ( x ) is a μ-maximal function defined by (1.4) and φ 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 [[1], p.9]

The Gauss-Weierstrass kernel, W ( t , ε ) = ( 4 π ε ) ( n / 2 ) e | t | 2 / 4 ε , has the following property:
R n W ( t , ε ) d t = 1 ( for all  ε > 0 ) .
(1.12)

Lemma 1.10 [[12], p.948]

The upper incomplete gamma function, defined as
Γ ( s , τ ) = τ u s 1 e u d u ( s > 0 , τ > 0 ) ,
has the following asymptotic property:
Γ ( s , τ ) = O ( 1 ) τ s 1 e τ as  τ .
(1.13)

2 Proof of the main results

Proof of Lemma A Changing variables to polar coordinates t ( r , θ ) , 0 < r < , θ S n 1 ( S n 1 is the unite sphere of R n ), the left side of (1.11) becomes
I ( x ) = R n | ψ ( t , x ) φ ( | t | ) | d t = 0 r n 1 [ S n 1 | ψ ( r θ , x ) φ ( r ) | d σ ( θ ) ] d r = 0 r n 1 | φ ( r ) | [ S n 1 | ψ ( r θ , x ) | d σ ( θ ) ] d r .
Now, denoting
λ ( t ) = S n 1 | ψ ( t θ , x ) | d σ ( θ ) , 0 t < ,
(2.1)
Λ ( r ) = 0 r λ ( t ) t n 1 d t , 0 r ,
(2.2)
we get
I ( x ) = 0 r n 1 | φ ( r ) | λ ( r ) d r = 0 | φ ( r ) | d Λ ( r ) = | φ ( r ) | Λ ( r ) | 0 0 Λ ( r ) sgn φ ( r ) φ ( r ) d r .
Using (1.10) and considering the inequality
Λ ( r ) = 0 r λ ( t ) t n 1 = | t | r | ψ ( t , x ) | d x r n μ ( r ) ( M μ ψ ) ( x ) ,
(2.3)
we have
| φ ( r ) | Λ ( r ) | 0 = 0 .
Thus,
I ( x ) = 0 Λ ( r ) sgn φ ( r ) φ ( r ) d r 0 Λ ( r ) | φ ( r ) | d r ( 2.3 ) ( M μ ψ ) ( x ) 0 r n μ ( r ) | φ ( r ) | d r .

That completes the proof. □

Proof of Theorem 1.5 Let us fix x, a μ-smoothness point of f, and consider the difference
| S ( x , ε ) f ( x ) | = | R n [ f ( x t ) f ( x ) ] W ( t , ε ) d t | | t | 1 | f ( x t ) f ( x ) | W ( t , ε ) d t + | t | > 1 | f ( x t ) f ( x ) | W ( t , ε ) d t = A 1 ( ε ) + A 2 ( ε ) .
(2.4)
In order to estimate A 1 ( ε ) , we let
ψ ( t , x ) = { f ( x t ) f ( x ) , | t | 1 ; 0 , | t | > 1 ,
and then
A 1 ( ε ) = R n W ( t , ε ) | ψ ( t , x ) | d t , W ( t , ε ) = ( 4 π ε ) ( n / 2 ) e | t | 2 / 4 ε .
(2.5)
Now, by Lemma A, taking φ ( | t | ) = ( 4 π ε ) n / 2 e | t | 2 / 4 ε , we have
A 1 ( ε ) ( M μ ψ ) ( x ) 0 r n μ ( r ) | φ ( r ) | d r , where  φ ( r ) = ( 4 π ε ) ( n / 2 ) e r 2 / 4 ε .
(2.6)
Since f is μ-smooth at the point x R n , we have ( M μ ψ ) ( x ) D μ ( x ) < (see (1.5)). So we get
A 1 ( ε ) c 1 0 r n μ ( r ) | φ ( r ) | d r c 0 r n + 1 e r 2 / 4 μ ( ε r ) d r .
(2.7)
To estimate A 2 ( ε ) , we first apply Hölder’s inequality for p > 1 and observe that
A 2 ( ε ) | f ( x ) | | t | > 1 W ( t , ε ) d t + ( | t | > 1 | f ( x t ) | p d t ) 1 / p ( | t | > 1 | W ( t , ε ) | q d t ) 1 / q ( 1 p + 1 q = 1 ) .
(2.8)
Let us estimate the first term on the right of (2.8). Changing variables to polar coordinates yields
| t | > 1 W ( t , ε ) d t = k 1 1 r n 1 [ S n 1 ε n / 2 e r 2 / 4 ε d σ ( θ ) ] d r = k 2 1 r n 1 ε n / 2 e r 2 / 4 ε = k 3 ( 1 / 2 ε ) r n 1 e r 2 d r (we set  u = r 2 d u = 2 r d r ) = k 3 ( 1 / 4 ε ) u n / 2 1 e u d u = k 3 Γ ( n 2 , 1 4 ε ) ,
where Γ ( s , τ ) is the upper incomplete gamma function. Now, using asymptotic formula (1.13), we get
| t | > 1 W ( t , ε ) d t = O ( ε 1 n 2 e 1 / 4 ε ) as  ε 0 + .
(2.9)
The same is true for the second term of (2.8):
( | t | > 1 | W ( t , ε ) | q d t ) 1 / q = k 4 ( 1 r n 1 [ S n 1 ( ε n / 2 e r 2 / 4 ε ) q d σ ( θ ) ] d r ) 1 / q = k 5 ε n 2 q n 2 ( ( q / 2 ε ) r n 1 e r 2 d r ) 1 / q = k 5 ε n 2 q n 2 ( ( q / 4 ε ) u n / 2 1 e u d u ) 1 / q .
Now, using formula (1.13), we get
( | t | > 1 | W ( t , ε ) | q d t ) 1 / q = O ( ε 1 q n 2 e 1 / 4 ε ) as  ε 0 + .
(2.10)
Collecting estimates (2.9) and (2.10), and taking into account that
( | t | > 1 | f ( x t ) | p ) 1 / p f p < ,
we have
A 2 ( ε ) = O ( ε n / 2 e 1 / 4 ε ) as  ε 0 + .
(2.11)

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 φ ( r ) = ( 4 π ε ) n / 2 e r 2 / 4 ε ,
lim r r n μ ( r ) φ ( r ) = 0 and lim r 0 + r n μ ( r ) φ ( r ) = 0 .

But this is obvious. □

Proof of Corollary 1.6 Let μ ( r ) , r [ 0 , ) be a modulus of continuity, i.e., (a) μ ( r ) 0 as r 0 + ; (b) μ ( r ) is non-negative and non-decreasing on ( 0 , ) ; (c) μ ( r ) is continuous and subadditive ( 0 , ) .

It follows from the subadditivity of μ ( r ) that
μ ( ε r ) ( 1 + r ) μ ( ε ) for all  ε , r > 0 .
By employing this in (1.6), we get
| S ( x , ε ) f ( x ) | c 1 μ ( ε ) 0 ( 1 + r ) r n + 1 e r 2 d r + c 2 ε n / 2 e 1 / 4 ε c 3 μ ( ε ) + c 2 ε n / 2 e 1 / 4 ε .
(2.12)
Now, since the function μ ( r ) is a modulus of continuity, it cannot tend to zero too rapidly as ε 0 , that is, for instance, if μ ( ε ) ε 0 as ε 0 , then μ ( ε ) 0 . Therefore
ε n / 2 e 1 / 4 ε c 4 μ ( ε ) , ε 0
for some constant c 4 . Taking into account this in (2.12), we obtain
| S ( x , ε ) f ( x ) | c μ ( ε ) , ε 0 ,

where the constant c does not depend on ε > 0 . □

Proof of Corollary 1.7 Let us show that for some 0 < ρ < 1 the function
μ ( r ) = { 0 , r = 0 ( 1 / ln 1 / r ) α , 0 < r < ρ < 1 ( 1 / ln 1 / ρ ) α , ρ r < } ( 0 < α < )
is a modulus of continuity, i.e., it is continuous, non-decreasing, subadditive on [ 0 , ) and tends to zero as r 0 + . The continuity and lim r 0 μ ( r ) = 0 are obvious. To prove the other properties, it suffices to show that (see [11], p.41)
μ ( r ) 0 and ( μ ( r ) / r ) 0 ( 0 < r < ρ ) .

Simple calculations show that the above inequalities are fulfilled if one takes ρ = e α . □

Proof of Corollary 1.8 Let us substitute the function
μ ( r ) = { 0 , r = 0 r α | ln r | β , 0 < r ρ ρ α | ln ρ | β , r > ρ }

in (1.6), where α > 0 and β ( , ) are given numbers and 0 < ρ < 1 .

If β 0 , we have, for sufficiently small ε > 0 ,
μ ( ε r ) ε α | ln ε | β r α ( 1 + | ln r | | ln ε | ) β ε α | ln ε | β r α ( 1 + | ln r | ) β .
By making use of this estimate in (1.6), we have for ε 1 that
| S ( x , ε ) f ( x ) | c ε α | ln ε | β 0 r n + 1 e r 2 / 4 r α ( 1 + ln r ) β d r + O ( ε n / 2 e 1 / 4 ε ) c 1 ε α | ln ε | β + c 2 ε n / 2 e 1 / 4 ε c 3 ε α | ln ε | α , ε 0 .
Let now β < 0 . By setting δ = β > 0 , we have for ε 1
μ ( ε r ) = ε α | ln ε | β r α | ln ε r ln ε | β = ε α | ln ε | β r α | ln ε ln ε r | δ = ε α | ln ε | β r α | 1 ln r ln ε r | δ ε α | ln ε | β r α ( 1 + | ln r | | ln ε r | ) δ .
Since ε r < ρ < 1 , it follows that
μ ( ε r ) ε α | ln ε | β r α ( 1 + | ln r | | ln ρ | ) δ .
Using this in (1.6), we get
| S ( x , ε ) f ( x ) | c ε α | ln ε | β 0 r n + 1 e r 2 / 4 r α ( 1 + | ln r | | ln ρ | ) δ e r d r + c 2 ε n / 2 e 1 / 4 ε c ε α | ln ε | β ,

which is the desired result. □

Declarations

Acknowledgements

This paper was supported by the Scientific Research Project Administration Unit of Akdeniz University and TUBITAK (Turkey).

Authors’ Affiliations

(1)
Department of Mathematics, the Faculty of Science, Akdeniz University

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© Eryigit; licensee Springer. 2013

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