# Strong Summability of Fourier Transforms at Lebesgue Points and Wiener Amalgam Spaces

Authors: Ferenc Weisz
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We characterize the set of functions for which strong summability holds at each Lebesgue point. More exactly, iff is in the Wiener amalgam space W(L1,lq)(R) and f is almost everywhere locally bounded, or f∈W(Lp,lq)(R)(1<p<∞,1≤q<∞), then strong θ-summability holds at each Lebesgue point of f. The analogous results are given for Fourier series, too.


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## Body

## 1. Introduction

It was proved by Lebesgue [1] that the Fejér means [2] of the trigonometric Fourier series of an integrable function converge almost everywhere to the function; that is,(1)1n+1∑k=0nskfx-fx⟶0asn⟶∞for almost every x∈T, where T denotes the torus and skf the kth partial sum of the Fourier series of the one-dimensional function f. The set of convergence is characterized as the Lebesgue points of f.Hardy and Littlewood [3] considered the so-called strong summability and verified that the strong means(2)1n+1∑k=0nskfx-fxqtend to 0 at each Lebesgue point of f, as n→∞, whenever f∈Lp(T) (1<p<∞) (for Fourier transforms see Giang and Móricz [4]). This result does not hold for p=1 (see Hardy and Littlewood [5]). However, the strong means tend to 0 almost everywhere for all f∈L1(T). This is because of Marcinkiewicz [6] for q=2 and Zygmund [7] for all q>0 (see also Bary [8]). Later Gabisoniya [9] characterized the set of convergence as the so-called Gabisoniya points. Strong summability with lacunary partial sums and Lebesgue points are investigated by Belinsky et al. [10–13].In a general method of summation, the so-calledθ-summation method, which is generated by a single function θ and which includes the well-known Fejér, Riesz, Weierstrass, and Abel summability methods, is studied intensively in the literature (see, e.g., Butzer and Nessel [14], Trigub and Belinsky [15–17], Liflyand [18], and Weisz [19, 20]). In this paper we generalize some of the above-mentioned results for strong θ-summability of Fourier transforms and for Wiener amalgam spaces. We characterize the set of functions for which strong summability holds at each Lebesgue point. More exactly, we will show that(3)limT→∞⁡-1T∫0∞θ′tTstfx-fxrdt=0at each Lebesgue point x of f∈W(L1,lq)(R)⊃Lq(R) (1≤q<∞) when f is locally bounded at x, where r>0. Moreover, the convergence holds at each Lebesgue point of f if f∈W(Lp,lq)(R)⊃Lp(R) (1<p<∞,1≤q<∞). Here W(Lp,lq)(R) denotes the Wiener amalgam spaces. Gabisoniya’s result was generalized in [21]. The analogous results are given for Fourier series, too.


## 2. Wiener Amalgam Spaces and Lebesgue Points

Let us fixd≥1, d∈N. For a set Y≠∅ let Yd be its Cartesian product Y×⋯×Y taken with itself d-times. We briefly write Lp(Rd) instead of the Lp(Rd,λ) space equipped with the norm(4)fp≔∫Rdfxpdλx1/p1≤p<∞,with the usual modification for p=∞, where λ is the Lebesgue measure. Lploc(Rd) (1≤p<∞) denotes the space of measurable functions f for which |f|p is locally integrable. We say that f is locally bounded at x if there exists a neighborhood of x such that f is bounded on this neighborhood.Now we generalize theLp spaces. A measurable function f belongs to theWiener amalgam spaceW(Lp,lq)(Rd) (1≤p,q≤∞) if(5)fWLp,lq≔∑k∈Zdf·+kLp0,1dq1/q<∞,with the obvious modification for q=∞. It is easy to see that W(Lp,lp)(Rd)=Lp(Rd) and the following continuous embeddings hold true:(6)WLp1,lqRd⊃WLp2,lqRdp1≤p2,WLp,lq1Rd⊂WLp,lq2Rdq1≤q2,(1≤p1,p2,q1,q2≤∞). Thus(7)WL∞,l1Rd⊂LpRd⊂WL1,l∞Rd1≤p≤∞.A pointx∈Rd is called a p-Lebesgue point (or a Lebesgue point of order p) of f∈Lploc(Rd) if(8)limh→0⁡12hd∫-hh⋯∫-hhfx-s-fxpds1/p=0.It was proved by Feichtinger and Weisz [22, 23] that almost every point x∈Rd is a p-Lebesgue point of f∈W(Lp,l∞)(Rd) (1≤p<∞). In context of Lebesgue points of Lp-functions we call also for the earlier papers of Belinsky et al. [12, 13].In this paper the constantsC and Cp may vary from line to line and the constants Cp are depending only on p.


## 3. The Kernel Functions

TheFourier transform of f∈L1(R) is given by(9)f^x=12π1/2∫Rfue-ıxudux∈R,where ı=-1. Suppose first that f∈Lp(R) for some 1≤p≤2. The Fourier inversion formula(10)fx=12π1/2∫Rf^ueıxudux∈R,f^∈L1Rmotivates the definition of the Dirichlet integral stf (t>0) introduced by(11)stfx≔12π1/2∫-ttf^ueıxudu=12π∫Rfx-uDtudu,where theDirichlet kernel is defined by(12)Dtx≔∫-tteıxudu=2sin⁡txx.Obviously, |Dt|≤Ct.It is easy to see that, with the help of the integral in (11), the definition of stf can be extended to all f∈W(L1,lq)(R) with 1≤q<∞. Note that W(L1,lp)(R)⊃Lp(R), where 1≤p<∞. It is known (see, e.g., Grafakos [24] or [20]) that, for f∈Lp(R), 1<p<∞,(13)limT→∞⁡sTf=fin  theLpR-norm  anda.e.This convergence does not hold forp=1. However, using a summability method, we can generalize these results. We may take a general summability method, the so-called θ-summation defined by a function θ:R+→R. This summation contains all well-known summability methods, such as the Marcinkiewicz-Fejér, Riesz, Weierstrass, Abel, Picard, and Bessel summations.Suppose thatθ is continuous on R+; the support of θ is [0,c] for some 0<c≤∞ and θ is differentiable on (0,c). Suppose further that(14)θ0=1,∫0∞t∨1dθ′tdt<∞,limt→∞⁡tdθt=0,where d∈N, ∨ denotes the maximum, and ∧ denotes the minimum.ForT>0 the θ-means of a function f∈Lp(R) (1≤p≤2) are defined by(15)σTθfx≔12π1/2∫RθuTf^ueıxudu.It is easy to see that(16)σTθfx=12π1/2∫Rfx-uKTθudu.Note that this formula is well defined for all f∈W(L1,l∞)(R). Here(17)KTθx≔12π1/2∫RθuTeıxudu=-12π1/2T∫R∫u∞θ′tTdteıxudu=-12π1/2T∫0∞θ′tT∫-tteıxududt=-12π1/2T∫0∞θ′tTDtxdtdenotes the θ-kernel. Thus(18)σTθfx=-1T∫0∞θ′tTstfxdtfor all f∈W(L1,lq)(R) with 1≤q<∞. Note that for the Fejér means (i.e., for θ(t)=max⁡((1-|t|),0)) we get the usual definition(19)σTθfx=1T∫0Tstfxdt.In Feichtinger and Weisz [22, 23] we have proved that, under conditions (14) and (25) with d=1,(20)limT→∞⁡σTθfx=fxfor all Lebesgue points of f∈W(L1,l∞)(R). In this paper, we investigate the problem of the strong summability, that is, whether the convergence(21)limT→∞⁡-1T∫0∞θ′tTstfx-fxrdt=0holds for Lebesgue points and some r>0. Usually θ is increasing; then we can take the absolute value of θ′ in the integral.To this end we have to introduce somed-dimensional definitions. In the d-dimensional case we define theDirichlet kernel by(22)Dtx≔∏i=1dDtxi=2d∏i=1dsin⁡txixit>0and the so calledMarcinkiewicz-θ-kernel by(23)KTθx=-12πd/2T∫0∞θ′tTDtxdtT>0,where x=(x1,…,xd)∈Rd. In [21], we have seen that we may suppose that x1>x2>⋯>xd>0 and x1-∑j=2dxj>0 and we proved the next lemma. Denote by(24)soc⁡t≔cos⁡t,ifdis  even;sin⁡t,ifdis  odd.Lemma 1.
Let(25)∫0∞θ′ttisocitudt≤Cu-αi=0,…,d-1for some 0<α<∞. Then(26)KTθx≤CTd-jx1-1⋯xj-1j=0,…,d.If in addition x1-∑j=2d‍xj>1/T and xj+1<1/T, where xd+1=0, then(27)KTθx≤CT-α+d-jx1-1⋯xj-1x1-∑j=2dxj-αj=1,…,d.The next lemma is due to the author [25].Lemma 2.
If (14) and (25) are satisfied for some d∈N and 0<α<∞, then ∫RdKTθdλ≤C (T∈R+).


## 4. Strong Summability of Fourier Transforms

In this section we characterize a wide set of functions for which strong summability holds at each Lebesgue point. For the convergence off∈W(Lp,lq)(R) (1<p<∞,1≤q<∞) at p-Lebesgue points we proved the following result in [21]. Note that W(Lp,lp)(R)=Lp(R).Theorem 3.
Suppose that (14) and (25) hold for some d∈N and 0<α<∞. Let f∈W(Lp,lq)(R) for some 1<p<∞ and 1≤q<∞. If xj is a p-Lebesgue point of f for all j=1,…,d, then(28)limT→∞⁡-1T∫0∞θ′tT∏j=1dstfxj-fxjdt=0.If allxj (j=1,…,d) are equal, then we obtain the following.Corollary 4.
Suppose that (14) and (25) hold for some even d∈N and 0<α<∞. Let f∈W(Lp,lq)(R) for some 1<p<∞ and 1≤q<∞. If x∈R is a p-Lebesgue point of f, then(29)limT→∞⁡-1T∫0∞θ′tTstfx-fxddt=0.Obviously, the convergence holds almost everywhere. Corollary4 does not hold for p=1 (see Hardy and Littlewood [5]). However, we [21] extended it for p=1, but for much more specialized points than the Lebesgue points, for the so-called Gabisoniya points, which were introduced in [9]. In the next theorem we generalize Theorem 3 and Corollary 4 for p=1 and for a subspace of W(L1,lq)(R).Theorem 5.
Suppose that (14) and (25) hold for some d∈N and 0<α<∞. Let f∈W(L1,lq)(R) for some 1≤q<∞. If xj is a Lebesgue point of f and f is locally bounded at xj for all j=1,…,d, then(30)limT→∞⁡-1T∫0∞θ′tT∏j=1dstfxj-fxjdt=0.Proof.
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For simplicity we will prove the rest of the theorem ford=3, only. It can be proved for higher dimensions similarly. As we mentioned earlier, we may suppose that s1>s2>s3>0 and s1-s2-s3>0. Let us fix a small r>0 and denote the square [0,r/2]3 by Sr/2. We can prove in the same way as we did in Theorem 4 of [21] that(34)∫Sr/2∏j=13fxj-sj-fxjKTθsds<ϵif r is small enough and T is large enough. The estimation of this integral on the set Sr/2c of that proof does not work now, because there we used a modified maximal function which is not necessarily bounded in our case. So we have to show here that(35)limT→∞⁡∫Sr/2c∏j=13fxj-sj-fxjKTθsds=0.
To this end let us introduce the sets(36)A1≔s:s1>r2,0<s3<s2<1T,A2≔s:s1>r2,0<s3<1T<s2<δ,A3≔s:s1>r2,1T<s3<s2<δ,A4≔s:s1>r2,0<s3<1T<δ<s2,A5≔s:s1>r2,1T<s3<δ<s2,A6≔s:s1>r2,1T<δ<s3<s2,B1≔s:0<s1-s2-s3<1T,B2≔s:1T<s1-s2-s3<δ,B3≔s:δ<s1-s2-s3<s1-s22,B4≔s:s1-s22∨δ<s1-s2-s3<s1-s2-δ,B5≔s:s1-s2-δ∨δ<s1-s2-s3<s1-s2,C1≔s:0<s1-s2<δ,C2≔s:δ<s1-s2<s12,C3≔s:s12<s1-s2<s1-δ,C4≔s:s1-δ<s1-s2<s1for a given small δ>0 and large T. Then we have to estimate the integral in (35) for Ai, i=1,…,6. First of all observe that Ai∩Bj=∅ for i=1,2,3 and j=1,2. On the set A1 we have s2,s3<1/T and so s1-s2-s3>s1/2. Hence(37)KTθs≤CT2-αs1-1s1-s2-s3-α≤CT2-αs1-1-α.Since f is locally bounded at xj, we get by (37) that(38)∫A1∩B3∪B4∪B5∏j=13fxj-sj-fxjKTθsds≤CT2-α∫r/2∞∫01/T∫01/Ts1-1-α∏j=13fxj-sjds≤CT-α∑i=0∞i∨1-1-α∫ii+1fx1-s1ds1≤CT-αfWL1,l∞⟶0,as T→∞. On A2 we have(39)KTθs≤CT1-αs1-1s2-1s1-s2-s3-α≤CT1-αs1-1-αs2-1and so(40)∫A2∩B3∪B4∪B5∏j=13fxj-sj-fxjKTθsds≤CT1-α∫r/2∞∫1/Tδ∫01/Ts1-1-αs2-1∏j=13fxj-sjds≤CT-αln⁡T∑i=0∞i∨1-1-α∫ii+1fx1-s1ds1≤CT-αln⁡TfWL1,l∞⟶0.Similarly,(41)∫A3∩B3∪B4∪B5∏j=13fxj-sj-fxjKTθsds≤CT-α∫r/2∞∫1/Tδ∫1/Tδs1-1-αs2-1s3-1∏j=13fxj-sjds≤CT-αln2⁡TfWL1,l∞⟶0.
On the setA4∩B1 we have s1-s2<2/T<δ if T is large enough and so s2>s1/2. Then(42)KTθs≤CTs1-1s2-1≤CTs1-2,(43)∫A4∩B1∏j=13fxj-sj-fxjKTθsds≤CT∫r/2∞∫s1-δs1∫01/Ts1-2∏j=13fxj-sjds≤C∑i=0N0-1i∨1-2·∫ii+1∫s1-δs1fx1-s1fx2-s2ds1ds2+C∑i=N0∞i-2·∫ii+1∫s1-δs1fx1-s1fx2-s2ds1ds2≤Cf×f1s1,s2:0<s1<N0,s1-δ<s2≤s1WL1,l∞+CN0-1f×fWL1,l∞.The second term is less than ϵ if N0 is large enough and the first term is less than ϵ if δ is small enough. The set A4∩B2 can be handled in the same way.
OnA4∩(B3∪B4∪B5) we have s1-s2>δ. Thus (B3∪B4∪B5)∩C1=∅ and s1-s2-s3>s1-s2-1/T>(s1-s2)/2. Then(44)KTθs≤CT1-αs1-1s2-1s1-s2-s3-α≤CT1-αs1-1s2-1s1-s2-α.On C2 we have(45)KTθs≤CT1-αs1-2s1-s2-α≤CT1-αs1-1-ηs1-s2-1-α+η,where η is chosen such that 0<η<1∧α. Hence(46)∫A4∩B3∪B4∪B5∩C2∏j=13fxj-sj-fxjKTθsds≤CT1-α∫r/2∞∫s1/2s1-δ∫01/Ts1-1-ηs1-s2-1-α+η·∏j=13fxj-sjds≤CT-α∑i=0∞i∨1-1-η·∑0≤j<i+1/2j∨1-1-α+η·∫ii+1∫s1-j-1s1-jfx1-s1·fx2-s2ds1ds2≤CT-αfWL1,l∞2⟶0.On C3, (44) implies that(47)KTθs≤CT1-αs1-1-αs2-1≤CT1-αs1-1-α+ηs2-1-η,∫A4∩B3∪B4∪B5∩C3∏j=13fxj-sj-fxjKTθsds≤CT1-α∫r/2∞∫δs1/2∫01/Ts1-1-α+ηs2-1-η∏j=13fxj-sjds≤CT-α∑i=0∞i∨1-1-α+η∑0≤j<i+1/2j∨1-1-η·∫ii+1∫jj+1fx1-s1fx2-s2ds1ds2≤CT-αfWL1,l∞2⟶0.Observe that s2>δ contradicts C4.
Consider the setA5∩B1. If s1-s2>2/T, then s3>s1-s2-1/T>(s1-s2)/2 and if s1-s2<2/T, then s3>1/T>(s1-s2)/2. Observe that s1-s2<1/T+δ<2δ. Hence(48)KTθs≤Cs1-1s2-1s3-1≤Cs1-2s1-s2-1/2s3-1/2.Since s3<δ, we can integrate in s3 to obtain(49)∫A5∩B1∏j=13fxj-sj-fxjKTθsds≤C∫r/2∞∫s1-2δs1∫s1-s2-1/Ts1-s2s1-2s1-s2-1/2·s3-1/2∏j=13fxj-sjds≤C∫r/2∞∫s1-2δs1s1-2fx1-s1fx2-s2ds1ds2≤Cf×f1s1,s2:0<s1<N0,s1-2δ<s2≤s1WL1,l∞+CN0-1f×fWL1,l∞as in (43). Similarly, s1-s2<2δ holds as well on A5∩B2. If s1-s2-s3>s3, then(50)KTθs≤CT-αs1-1s2-1s3-1s1-s2-s3-α≤CT-αs1-2s3-1-αand so(51)∫A5∩B2∏j=13fxj-sj-fxjKTθsds≤CT-α∫r/2∞∫s1-2δs1∫1/Tδs1-2s3-1-α∏j=13fxj-sjds≤C∫r/2∞∫s1-2δs1s1-2fx1-s1fx2-s2ds1ds2<ϵas before. If s3>s1-s2-s3, then(52)KTθs≤CT-αs1-2s1-s2-s3-1-α,∫A5∩B2∏j=13fxj-sj-fxjKTθsds≤CT-α∫r/2∞∫s1-2δs1∫s1-s2-δs1-s2-1/Ts1-2s1-s2-s3-1-α·∏j=13fxj-sjds≤C∫r/2∞∫s1-2δs1s1-2fx1-s1fx2-s2ds1ds2<ϵ.If s1-s2<2δ, then the integral on the set A5∩(B3∪B4∪B5) can be estimated in the same way. If s1-s2>2δ, then A5∩B3=∅ because s3>(s1-s2)/2 and s3<δ on this set. Writing 1/T instead of δ in the sets Bi (i=4,5), we obtain the definition of the sets Bi′. On B4′ we have by (50) that(53)KTθs≤CT-αs1-1s2-1s3-1s1-s2-α,∫A5∩B4′∩C2∏j=13fxj-sj-fxjKTθsds≤CT-α∫r/2∞∫s1/2s1-δ∫1/Tδs1-2s3-1s1-s2-α·∏j=13fxj-sjds≤CT-αln⁡T∑i=0∞i∨1-1-η·∑0≤j<i+1/2j∨1-1-α+η∫ii+1∫s1-j-1s1-jfx1-s1·fx2-s2ds1ds2≤CT-αln⁡TfWL1,l∞2⟶0,∫A5∩B4′∩C3∏j=13fxj-sj-fxjKTθsds≤CT-α∫r/2∞∫δs1/2∫1/Tδs1-1-αs2-1s3-1∏j=13fxj-sjds≤CT-αln⁡T∑i=0∞i∨1-1-α+η∑0≤j<i+1/2j∨1-1-η·∫ii+1∫jj+1fx1-s1fx2-s2ds1ds2≤CT-α·ln⁡TfWL1,l∞2⟶0.Moreover, B5′ contradicts A5, more exactly, s3>1/T.
OnA6s3>δ and s1-s2-s3>0, thus s1-s2>δ. Similar to (48),(54)KTθs≤Cs1-1s2-1s3-1≤Cs1-2s1-s2-1on A6∩B1∩C2. Then(55)
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x
2
-
s
2
·
f
x
3
-
s
3
d
s
1
d
s
2
d
s
3
≤
C
f
×
f
×
f
1
s
:
0
<
s
1
<
N
0
,
s
1
/
2
<
s
2
≤
s
1
-
δ
,
s
1
-
s
2
-
δ
<
s
3
<
s
1
-
s
2
W
L
1
,
l
∞
+
C
N
0
-
1
/
2
f
×
f
×
f
W
L
1
,
l
∞
,which is small enough if N0 is large and δ is small enough. On A6∩B1∩C3 we use the estimation(56)KTθs≤Cs1-1s2-1s3-1≤Cs1-2s2-1to obtain(57)
∫
A
6
∩
B
1
∩
C
3
∏
j
=
1
3
f
x
j
-
s
j
-
f
x
j
K
T
θ
s
d
s
≤
C
∫
r
/
2
∞
∫
δ
s
1
/
2
∫
s
1
-
s
2
-
1
/
T
s
1
-
s
2
s
1
-
2
s
2
-
1
∏
j
=
1
3
f
x
j
-
s
j
d
s
≤
C
f
×
f
×
f
1
s
:
0
<
s
1
<
N
0
,
δ
<
s
2
≤
s
1
/
2
,
s
1
-
s
2
-
δ
<
s
3
<
s
1
-
s
2
W
L
1
,
l
∞
+
C
N
0
-
1
/
2
f
×
f
×
f
W
L
1
,
l
∞
<
ϵ
,as just before.
OnA6∩B2s1-s2>δ again. If s1-s2>2δ, then s3>s1-s2-δ>(s1-s2)/2 and if s1-s2<2δ, then s3>δ>(s1-s2)/2. This case can be handled in the same way as the set A6∩B1.
OnA6∩B3 we get(58)KTθs≤CT-αs1-1s2-1s3-1s1-s2-s3-α≤CT-αs1-1s2-1s1-s2-1s1-s2-s3-α,which implies that(59)∫A6∩B3∩C2∏j=13fxj-sj-fxjKTθsds≤CT-α∫r/2∞∫s1/2s1-δ∫s1-s2/2s1-s2-δs1-2s1-s2-1·s1-s2-s3-α∏j=13fxj-sjds≤CT-α∑i=0∞i∨1-1-μ∑0≤j<i+1/2j∨1-1-μ·∑0≤k<j+1/2k∨1-1-α+2μ·∫ii+1∫s1-j-1s1-j∫s1-s2-k-1s1-s2-kfx1-s1fx2-s2·fx3-s3ds1ds2ds3≤CT-αfWL1,l∞3⟶0,where 0<μ<1/2∧α/2. Similarly,(60)∫A6∩B3∩C3∏j=13fxj-sj-fxjKTθsds≤CT-α∫r/2∞∫δs1/2∫s1-s2/2s1-s2-δs1-2s2-1s1-s2-s3-α·∏j=13fxj-sjds≤CT-α∑i=0∞i∨1-1-μ·∑0≤j<i+1/2j∨1-1-μ∑0≤k<j+1/2k∨1-1-α+2μ·∫ii+1∫jj+1∫s1-s2-k-1s1-s2-kfx1-s1fx2-s2·fx3-s3ds1ds2ds3≤CT-αfWL1,l∞3⟶0.
Moreover, onA6∩B4(61)KTθs≤CT-αs1-1s2-1s3-1s1-s2-α,∫A6∩B4∩C2∏j=13fxj-sj-fxjKTθsds≤CT-α∫r/2∞∫s1/2s1-δ∫δs1-s2/2s1-2s1-s2-α·s3-1∏j=13fxj-sjds≤CT-α∑i=0∞i∨1-1-μ·∑0≤j<i+1/2j∨1-1-α+2μ∑0≤k<j+1/2k∨1-1-μ·∫ii+1∫s1-j-1s1-j∫kk+1fx1-s1fx2-s2·fx3-s3ds1ds2ds3≤CT-αfWL1,l∞3⟶0,∫A6∩B4∩C3∏j=13fxj-sj-fxjKTθsds≤CT-α∫r/2∞∫δs1/2∫δs1-s2/2s1-1-αs2-1s3-1∏j=13fxj-sjds≤CT-α∑i=0∞i∨1-1-α/2∑0≤j<i+1/2j∨1-1-α/2·∑0≤k<j+1/2k∨1-1-α/2∫ii+1∫jj+1∫kk+1fx1-s1·fx2-s2fx3-s3ds1ds2ds3≤CT-αfWL1,l∞3⟶0.Since A6∩B5=∅, the proof of the theorem is complete.Corollary 6.
Suppose that (14) and (25) hold for some even d∈N and 0<α<∞. Let f∈W(L1,lq)(R) for some 1≤q<∞. If x∈R is a Lebesgue point of f and f is locally bounded at x, then(62)limT→∞-1T∫0∞θ′tTstfx-fxddt=0.Iff is almost everywhere locally bounded, then the corollary holds almost everywhere. Unfortunately, it is not true that an integrable function is almost everywhere locally bounded (see [21]). The strong summability can be extended to all exponents in the usual way (see [21]).Corollary 7.
Suppose that (14) holds for all d∈N, θ is nonincreasing, and r>0. Under the same conditions as in Corollaries 4 or 6, respectively, one gets that(63)limT→∞⁡-1T∫0∞θ′tTstfx-fxrdt=0.Note that under the same conditions we get for the Fejér summation that(64)limT→∞⁡1T∫0Tstfx-fxrdt=0.


## 5. Strong Summability of Fourier Series

In this section we formulate the above results for Fourier series. For an integrable functionf the kthFourier coefficient is defined by(65)f^k=12π∫Tfxe-ıkxdxk∈Z,where T denotes the torus. For f∈L1(T) and n∈N the nth partial sum snf is introduced by(66)snfx≔∑k=-nnf^keıkx=12π∫Tfx-uDnudu,where(67)Dnu≔∑k=-nneık·u=2sin⁡n+1/2xsin⁡x/2n∈Nis theDirichlet kernel. Let(68)θ=θk,n,k∈N,n∈N+be a two-parameter sequence of real numbers satisfying(69)θ0,n=1,limn→∞θk,n=1,limk→∞kdθk,n=0,∑k=0∞k+1βΔ1θk,n≤Cnβforβ=0,d,where(70)Δ1θk,n≔θk,n-θk+1,n.The θ-means of f∈L1(T) are defined by(71)σnθfx≔∑k=-∞∞θk,nf^keıkx=12π∫Tfx-uKnθudu,where the θ-kernel is given by(72)Knθx=∑j=0∞Δ1θj,nDjx.Hence(73)σnθfx=∑j=0∞Δ1θj,nsjfx.Basically, we will consider two types of examples. The first type of examples is the Cesàro summation, which is generated by a sequenceθ.Example 8 (Cesàro summation).
Fork∈N, α≠-1,-2,…, let(74)Akα≔k+αk=α+1α+2⋯α+kk!=Okαk∈N(see Zygmund [26, p. 77]). Let(75)θk,n=An-k-1αAn-1αifk≤n-10ifk≥nfor some 0<α<∞. Since Δ1θ(k,n)=An-k-1α-1/An-1α, the Cesàro operators can be given by(76)σnθfx=1An-1α∑k=0n-1An-1-kα-1skfx.If α=1, we get back the Fejér means.The other type is generated by a functionθ:R+→R (see all other examples in this paper). Let θ be continuous on R+ and(77)θk,n≔θknk∈N,n∈N+.Suppose that(78)θ0=1,limt→∞⁡tdθt=0,∑k=0∞k+1βΔ1θkn≤Cnβforβ=0,d.Note that for the Fejér means we get the usual definition(79)σnθfx=1n∑j=0n-1sjfx.All examples of this paper satisfy (26) and (27).Theorem 9.
Suppose that (26), (27), and (69) are satisfied for some d∈N and 0<α<∞. Let f∈Lp(T) for some 1<p<∞. If xj is a p-Lebesgue point of f for all j=1,…,d, then(80)limn→∞⁡∑k=0∞Δ1θk,n∏j=1dskfxj-fxj=0.Moreover, if d is even, then(81)limn→∞⁡∑k=0∞Δ1θk,nskfx1-fx1d=0.Theorem 10.
Suppose that (26), (27), and (69) are satisfied for some d∈N and 0<α<∞. If f∈L1(T), xj is a Lebesgue point of f, and f is locally bounded at xj for all j=1,…,d, then (80) and (81) hold.Corollary 11.
Suppose thatr>0, Δ1θ(k,n)≤0, and (69) is satisfied for all d. Under the same conditions as in Theorems 9 or 10, respectively, one gets that(82)limn→∞⁡∑k=0∞Δ1θk,nskfx-fxr=0.Note that under the same conditions we get for the Fejér summation that(83)limn→∞⁡1n∑k=0n-1skfx-fxr=0.Finally we note that the Weierstrass, Abel, Picard, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations can be considered as special cases ofθ-summation (see Weisz [21]).



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