3. θ-summability of Fourier series

32(2005) pp. 167–186.

Wiener amalgams and summability of Fourier series ^∗

Ferenc Weisz

Department of Numerical Analysis Eötvös L. University e-mail: weisz@ludens.elte.hu

Abstract

Some recent results on a general summability method, on the so-called θ-summability is summarized. New spaces, such as Wiener amalgams, Fe- ichtinger’s algebra and modulation spaces are investigated in summability theory. Sufficient and necessary conditions are given for the norm and a.e.

convergence of theθ-means.

Key Words: Wiener amalgam spaces, Feichtinger’s algebra, homogeneous Banach spaces, Besov-, Sobolev-, fractional Sobolev spaces, modulation spa- ces, Herz spaces, Hardy-Littlewood maximal function,θ-summability of Fou- rier series, Lebesgue points.

AMS Classification Number: Primary 42B08, 46E30, Secondary 42B30, 42A38.

1. Introduction

In this paper we consider a general method of summation, the so called θ- summation, which is generated by a single functionθ. A natural choice ofθ is a function from the Wiener algebraW(C, `1)(R^d). All concrete summability methods investigated in the literature satisfy this condition.

We shall investigate some function spaces known from other topics of analysis, for example Wiener amalgam spaces, Feichtinger’s Segal algebra S0(R^d), mod- ulation and Herz spaces. Feichtinger‘s algebra and modulation spaces are very intensively investigated in Gabor analysis (see e.g. Feichtinger and Zimmermann [6] and Gröchenig [13]). S₀(R^d)is the minimal (non-trivial) Banach space which is isometrically invariant under translation, modulation and Fourier transform.

∗This research was supported by Lise Meitner fellowship No M733-N04 and the Hungarian Scientific Research Funds (OTKA) No T043769, T047128, T047132.

167

In Sections 4 and 5 we deal with norm convergence of the θ-means of multi- dimensional Fourier series and Fourier transforms. We will show that ifθ is in the Wiener algebra then theθ-meansσ^θ_nf of the Fourier series off ∈L2(T^d)converge to f in L2 norm asn→ ∞. Moreover, σ^θ_nf →f uniformly (resp. at each point) for allf ∈C(T^d)if and only ifσ^θ_nf →f inL1 norm for allf ∈L1(T^d)if and only ifθˆ∈ L1(R^d). If B is a homogeneous Banach space on T^d and θˆ∈L1(R^d) then σ_n^θf →f inB norm for allf ∈B. Ifθis continuous and has compact support then the uniform convergence of theθ-means is equivalent to the L1 norm convergence of theθ-means and this is equivalent to the conditionθ∈S0(R^d). In all cases we investigate convergence over the diagonal.

In Sections 7 and 8 the a.e. convergence of the θ-means is considered. We show thatθˆis in the homogeneous Herz spaceE˙q(R^d)for some1< q6∞ if and only if the maximal operator of theθ-means of the Fourier transform off can be estimated by the modified Hardy-Littlewood maximal functionMpf, wherepis the dual index toq. SinceMpis of weak type(p, p)we obtainσ^θ_Tf →f a.e. asT → ∞ for allf ∈L_r(R^d),p6r <∞. Under the conditionθˆ∈E˙_q(R^d)this convergence holds also for functions from the Wiener amalgam spaceW(Lp, `∞)(R<sup>d</sup>). The set of convergence is also characterized, the convergence holds at everyp-Lebesgue point of f. The converse holds also, more exactly,σ<sup>θ</sup><sub>T</sub>f(x)→ f(x) at eachp-Lebesgue point off ∈Lp(R<sup>d</sup>)(resp. of f ∈W(Lp, `∞)(R^d)) if and only ifθˆ∈E˙q(R^d).

In Sections 6 and 9 we give some sufficient conditions for θ such that θˆ ∈ L1(R^d), or θ ∈ S0(R^d)or θˆ∈ E˙q(R^d). More exactly, ifθ is in a suitable Besov, Sobolev, fractional Sobolev, weighted Wiener amalgam or modulation space then all convergence results above hold.

Most of the proofs of the results of this survey paper can be found in Feichtinger and Weisz [5, 4]. This paper was the base of my talk given at the Fejér-Riesz Conference, June 2005, in Eger (Hungary).

2. Wiener amalgams and Feichtinger’s algebra

Let us fix d > 1, d ∈ N. For a set Y 6= ∅ let Y^d be its Cartesian product Y×. . .×Ytaken with itself d-times. We shall prove results forR^dorT^d, therefore it is convenient to use sometimes the symbol Xfor either Ror T, where Tis the torus. Forx= (x1, . . . , xd)∈R^d andu= (u1, . . . , ud)∈R^d set

u·x:=

Xd

k=1

ukxk, kxkp:=

³X^d

k=1

|xk|^p

´_1/p

, |x|:=kxk2.

We briefly write Lp or Lp(X^d) instead of Lp(X^d, λ) space equipped with the norm (or quasi-norm) kfkp := (R

X^d|f|^pdλ)^1/p (0 < p 6 ∞), where X =R or T andλis the Lebesgue measure. We use the notation|I|for the Lebesgue measure of the setI.

TheweakLp space,Lp,∞(X^d) (0< p <∞)consists of all measurable functions f for which

kfkLp,∞ := sup

ρ>0ρλ(|f|> ρ)^1/p<∞,

while we setL∞,∞(X^d) =L∞(X^d). Note that Lp,∞(X^d)is a quasi-normed space (see Bergh and Löfström [1]). It is easy to see that for each0< p6∞,

Lp(X^d)⊂Lp,∞(X^d) and k · kLp,∞ 6k · kp.

The space of continuous functions with the supremum norm is denoted byC(X^d) and we will useC0(R^d)for the space of continuous functions vanishing at infinity.

Cc(R^d)denotes the space of continuous functions having compact support.

A measurable functionf belongs to theWiener amalgam spaceW(Lp, `^v_q^s)(R^d) (16p, q6∞)if

kfk_W(Lp,`^vs_{q )}:=³ X

k∈Z^d

kf(·+k)k^q_L

p[0,1)^dvs(k)^q

´_1/q

<∞

with the obvious modification for q=∞, where the weight function vs is defined by vs(ω) := (1 +|ω|)^s (ω ∈ R^d). If s = 0 then we write simply W(Lp, `q)(R^d).

W(Lp, c0)(R^d) is defined analogously, wherec0 denotes the space of sequences of complex numbers having 0 limit, equipped with the supremum norm. If we replace the spaceLp[0,1)^d byLp,∞[0,1)^d then we get the definition of W(Lp,∞, `q)(R^d).

The closed subspace ofW(L∞, `q)(R<sup>d</sup>)containing continuous functions is denoted byW(C, `q)(R^d) (16q6∞). The spaceW(C, `1)(R^d)is calledWiener algebra. It is used quite often in Gabor analysis, because it provides a convenient and general class of windows (see e.g. Walnut [33] and Gröchenig [12]). As we can see later, it plays an important rule in summability theory, too.

It is easy to see thatW(Lp, `p)(R^d) =Lp(R^d)and

W(L∞, `1)(R<sup>d</sup>)⊂Lp(R<sup>d</sup>)⊂W(L1, `∞)(R^d) (16p6∞).

For more about amalgam spaces see e.g. Heil [14].

Translation andmodulation of a functionf are defined, respectively, by Txf(t) :=f(t−x) and Mωf(t) :=e^2πıω·tf(t) (x, ω∈R^d).

Recall that theFourier transformand theshort-time Fourier transform(STFT) with respect to a window functiongare defined by

Ff(x) := ˆf(x) :=

Z

R^d

f(t)e^−2πıx·tdt (x∈R^d, ı=√

−1)

and

Sgf(x, ω) :=

Z

R^d

f(t)g(t−x)e^−2πıωtdt=hf, MωTxgi (x, ω ∈R^d),

respectively, whenever the integrals do exist.

Feichtinger’s algebraS₀(R^d)and themodulation spacesM₁^vs(R^d)(see e.g. Feich- tinger [7] and Gröchenig [13]) are intoduced by

M₁^vs(R^d) :=

n

f ∈L²(R^d) :kfk_M1^vs :=kSg0f·vsk_L1(R^2d₎<∞ o

(s>0),

whereg0(x) :=e^−π|x|2 is the Gauss function andvs(x, ω) := (1 +|ω|)^s(x, ω∈R^d).

In cases= 0 we writeS₀(R^d) :=M₁(R^d).

It is known that S0(R^d) is isometrically invariant under translation, modula- tion and Fourier transform. Actually,S0is the minimal space having this property (see Feichtinger [7]). Moreover, the embeddings S(R^d) ,→ M₁^vs(R^d) (s >0) and S0(R^d),→W(C, `1)(R^d)are dense and continuous (see e.g. Feichtinger and Zim- mermann [6] and Gröchenig [13]), whereS denotes the Schwartz functions.

A Banach spaceB consisting of Lebesgue measurable functions onX^dis called ahomogeneous Banach space, if

(i) for allf ∈B andx∈X^d,Txf ∈B andkTxfkB =kfkB, (ii) the functionx7→Txf from X^d to B is continuous for allf ∈B, (iii) the functions inBare uniformly locally integrable, i.e. for every compact

setK⊂X^d there exists a constantCK such that Z

K

|f|dλ6CKkfkB (f ∈B).

If furthermore B is a dense subspace of L1(X^d) it is called a Segal algebra (cf.

Reiter [20]). Note that the continuous embedding into L1(X^d) is a consequence of the closed graph theorem. For an introduction to homogeneous Banach spaces see Katznelson [16] or Shapiro [24]. It is easy to see that the spacesLp(X^d) (16 p < ∞), C(T^d), C0(R^d), Lorentz spaces Lp,q(X^d) (1 < p < ∞,1 6 q < ∞), Hardy spacesH1(X^d) (for the definitions see e.g. Weisz [37]), Wiener amalgams W(Lp, `q)(R<sup>d</sup>) (16p, q <∞),W(Lp, c0)(R<sup>d</sup>) (16p <∞),W(C, `q)(R^d) (16q <

∞)andS0(R^d)are homogeneous Banach spaces. Note that ifBis a homogeneous Banach space onR^d thenB ,→W(L1, `∞)(R^d)(see Katznelson [16]).

3. θ-summability of Fourier series

Theθ-summation was considered in a great number of papers and books, such as Butzer and Nessel [3], Trigub and Belinsky [32], Bokor, Schipp, Szili and Vértesi [22, 2, 23, 28, 29], Natanson and Zuk [18], Weisz [35, 36, 37, 38] and Feichtinger and Weisz [5, 4]. We assume that the function θ is from the Wiener algebra W(C, `1)(R^d). We have seen in Feichtinger and Weisz [5, 4] that this is a natural choice ofθ and all summability methods considered in Butzer and Nessel [3] and Weisz [37] satisfy this condition.

Recall that for a distribution f ∈ S⁰(T^d)thenthFourier coefficient is defined byfˆ(n) :=f(e^−2πın·x) (n∈Z^d). In special case, iff ∈L1(T^d)then

fˆ(n) = Z

T^d

f(t)e^−2πın·tdt (n∈Z^d).

Given a functionθ∈W(C, `1)(R^d)theθ-means of a distributionf are defined by σ_n^θf(x) :=

Xd

j=1

X∞

kj=−∞

θ

³ −k₁

n1+ 1, . . . , −k_d nd+ 1

´fˆ(k)e^2πık·x= Z

T^d

f(x−t)K_n^θ(t)dt,

wherex∈T^d, n∈N^d and theθ-kernelsK_n^θ are given by K_n^θ(t) :=

Xd

j=1

X∞

kj=−∞

θ³ −k1

n1+ 1, . . . , −kd

nd+ 1

´

e^2πık·t (t∈T^d).

Under P_d

j=1

P_∞

kj=−∞ we mean the sum P_∞

k1=−∞. . .P_∞

kd=−∞. It is easy to see that

Xd

j=1

X∞

kj=−∞

¯¯

¯θ³ k1

n1+ 1, . . . , kd

nd+ 1

´¯¯

¯ 6 X

l∈Z^d

³Y^d

j=1

(n_j+ 1)´ sup

x∈[0,1)^d

|θ(x+l)|

= ³Y^d

j=1

(nj+ 1)´

kθk_W(C,`1)<∞,

and henceK_n^θ∈L1(T^d). We will always suppose thatθ(0) = 1.

Now we present some well known one-dimensional summability methods as special cases of the θ-summation. For more examples see Feichtinger and Weisz [5, 4].

Example 3.1(Fejér summation). Let θ(x) :=

(1− |x| if06|x|61, 0 if|x|>1, σ^θ_nf(x) :=

Xn

k=−n

(1− |k|

n+ 1) ˆf(k)e^2πık·x. Example 3.2(Riesz summation). Let

θ(x) :=

(

(1− |x|^γ)^α if|x|61, 0 if|x|>1, for some06α, γ <∞. The Riesz operators are given by

σ_n^θf(x) :=

Xn

k=−n

³ 1−

¯¯

¯ k n+ 1

¯¯

¯^γ

´_α

fˆ(k)e^2πık·x.

Example 3.3(Weierstrass summation). Let

θ(x) =e^−|x|γ (0< γ <∞),

σ^θ_nf(x) :=

X∞

k=−∞

e^{−(n+1|k| )γ}fˆ(k)e^2πık·x.

The most known form of the Weierstrass means are W_r^θf(x) :=

X∞

k=−∞

r^|k|γfˆ(k)e^2πık·x (0< r <1).

Example 3.4(Generalized Picar and Bessel summations). Let

θ(x) = 1

(1 +|x|^γ)^α

for some0< α, γ <∞such thatαγ >1. Theθ-means are given by σ_n^θf(x) :=

X∞

k=−∞

³ 1

1 + (_n+1^|k| )^γ´αfˆ(k)e^2πık·x. Example 3.5(de La Vallée-Poussin summation). Let

θ(x) =







1 if06x61/2,

−2x+ 2 if1/2< x61, 0 ifx >1.

Example 3.6. Let 0 = α0 < α1 < . . . < αm and β0, . . . , βm (m ∈ N) be real numbers,β0= 1, βm= 0. Suppose thatθ(αj) =βj (j= 0,1, . . . , m),θ(x) = 0for x>αm,θ is a polynomial on the interval[αj−1, αj] (j= 1, . . . , m).

4. Norm Convergence of the θ-means of Fourier se- ries

In this section we collect some results about the norm convergence of σ_n^θf as n → ∞. The proofs of the theorems can be found in Feichtinger and Weisz [5].

Note thatxdenotes the vector(x, . . . , x)∈R^d (x∈R).

Theorem 4.1. If θ∈W(C, `1)(R^d)andθ(0) = 1 then for allf ∈L2(T^d)

n→∞lim σ^θ_nf =f in L2(T^d)norm.

If the Fourier transform ofθis integrable then theθ-means can be written as a singular integral off and the Fourier transform ofθin the following way.

Theorem 4.2. If θ∈W(C, `1)(R^d)andθˆ∈L1(R^d)then σ^θ_nf(x) = (n+ 1)^d

Z

R^d

f(x−t) ˆθ¡

(n+ 1)t¢ dt

for allx∈T^d,n∈Nandf ∈L1(T^d).

For the uniform andL1norm convergence ofσ_n^θf →fa sufficient and necessary condition can be given.

Theorem 4.3. If θ ∈W(C, `1)(R^d) and θ(0) = 1 then the following conditions are equivalent:

(i) θˆ∈L1(R^d),

(ii) σ^θ_nf →f uniformly for all f ∈C(T^d) asn→ ∞, (iii) σ^θ_nf(x)→f(x) for allx∈T^d andf ∈C(T^d)asn→ ∞,

(iv) σ^θ_nf →f in L1(T^d)norm for all f ∈L1(T^d) asn→ ∞.

One part of the preceding result is generalized for homogeneous Banach spaces.

Theorem 4.4. Assume that B is a homogeneous Banach space on T^d. If θ ∈ W(C, `1)(R^d),θ(0) = 1 andθˆ∈L1(R^d)then for all f ∈B

n→∞lim σ_n^θf =f inB norm.

Sinceθ∈S0(R^d)impliesθ∈W(C, `1)(R^d)andθˆ∈S0(R^d)⊂L1(R^d), the next corollary follows from Theorems 4.3 and 4.4.

Corollary 4.5. If θ∈S0(R^d)andθ(0) = 1 then

(i) σ^θ_nf →f uniformly for all f ∈C(T^d) asn→ ∞, (ii) σ^θ_nf →f in L1(T^d)norm for all f ∈L1(T^d) asn→ ∞,

(iii) σ^θ_nf → f in B norm for all f ∈B as n→ ∞ if B is a homogeneous Banach space.

Ifθhas compact support thenθ∈S0(R^d)is equivalent to the conditionsθ,θˆ∈ L1(R^d)(see Feichtinger and Zimmermann [6]). This implies

Corollary 4.6. If θ∈C(R^d)has compact support and θ(0) = 1then the following conditions are equivalent:

(i) θ∈S0(R^d),

(ii) σ^θ_nf →f uniformly for all f ∈C(T^d) asn→ ∞, (iii) σ^θ_nf(x)→f(x) for allx∈T^d andf ∈C(T^d)asn→ ∞,

(iv) σ^θ_nf →f in L1(T^d)norm for all f ∈L1(T^d) asn→ ∞.

5. Norm convergence of the θ-means of Fourier transforms

All the results above can be shown for non-periodic functions f ∈ Lp(R^d).

Suppose first thatf ∈L_p(R^d)for some 16p62. The Fourier inversion formula f(x) =

Z

R^d

fˆ(u)e^2πıx·udu (x∈R^d)

holds iffˆ∈L1(R^d).

In the investigation of Fourier transforms we can take a larger space than W(C, `1)(R^d), we will assume that θ ∈ L1(R^d)∩C0(R^d). The θ-means of f ∈ Lp(R^d) (16p62)are defined by

σ_T^θf(x) :=

Z

R^d

θ

³−t1

T1 , . . . ,−td

Td

´fˆ(t)e^2πıx·tdt= Z

R^d

f(x−t)K_T^θ(t)dt

wherex∈R^d, T ∈R^d₊ and

K_T^θ(x) = Z

R^d

θ

³−t1

T1 , . . . ,−td

Td

´

e^2πıx·tdt=

³Y^d

j=1

Tj

´θ(Tˆ 1x1, . . . , Tdxd),

(x∈R^d). Thus theθ-means can rewritten as

σ_T^θf(x) =³Y^d

j=1

Tj

´ Z

R^d

f(x−t)ˆθ(T1t1, . . . , Tdtd)dt (5.1)

which is the analogue to Theorem 4.2. Note that θ ∈ L1(R^d)∩C0(R^d) implies θ∈Lp(R^d) (16p6∞). Now we formulate Theorem 4.1 for Fourier transforms.

Theorem 5.1. If θ∈L1(R^d)∩C0(R^d)andθ(0) = 1 then for allf ∈L2(R^d)

Tlim→∞σ^θ_Tf =f in L₂(R^d)norm.

Sinceσ_T^θ is defined only forf ∈Lp(R^d) (16p62), instead of Theorem 4.3 we have

Theorem 5.2. Ifθ∈L1(R^d)∩C0(R^d)andθ(0) = 1then the following conditions are equivalent:

(i) θˆ∈L1(R^d),

(ii) σ^θ_Tf →f in L1(R^d)norm for allf ∈L1(R^d)asT → ∞.

If θhas compact support then θ∈S0(R^d)is also an equivalent condition.

Ifθˆ∈L1(R^d), the definition of theθ-means extends tof ∈W(L1, `∞)(R^d)by σ_T^θf :=f∗K_T^θ (T ∈R^d₊),

where ∗ denotes the convolution. Note that θ ∈ L1(R^d) and θˆ ∈ L1(R^d) imply θ∈C0(R^d).

The analogue of Theorem 4.4 follows in the same way:

Theorem 5.3. Assume that B is a homogeneous Banach space on R^d. If θ ∈ L1(R^d),θ(0) = 1 andθˆ∈L1(R^d) (e.g.θ∈S0(R^d)) then for allf ∈B

Tlim→∞σ_T^θf =f in B norm.

Since the space Cu(R^d) of uniformly continuous bounded functions endowed with the supremum norm is also a homogeneous Banach space, we have

Corollary 5.4. If f is a uniformly continuous and bounded function,θ∈L1(R^d), θ(0) = 1 andθˆ∈L1(R^d)then

Tlim→∞σ^θ_Tf =f uniformly.

6. Sufficient conditions

In this section we give some sufficient conditions for a functionθ, which ensures that θˆ∈L1(R^d), resp. θ∈S0(R^d). As mentioned beforeθ∈S0(R^d)implies also thatθˆ∈L1(R^d). Recall thatS0(R^d)contains all Schwartz functions. Ifθ∈L1(R^d) andθˆhas compact support or ifθ∈L1(R^d)has compact support andθˆ∈L1(R^d) thenθ∈S0(R^d).

Sufficient conditions can be given with the help of Sobolev, fractional Sobolev and Besov spaces, too. For a detailed description of these spaces see Triebel [31], Runst and Sickel [21], Stein [26] and Grafakos [11].

A function θ ∈Lp(R^d)is in the Sobolev space W_p^k(R^d) (16p6∞, k ∈N) if D^αθ∈Lp(R^d)for all|α|6kand

kθk_Wk

p := X

|α|6k

kD^αθkp<∞,

whereD denotes the distributional derivative.

This definition is extended to every realsin the following way. Thefractional Sobolev space L^s_p(R^d) (16p6∞, s∈ R) consists of all tempered distribution θ for which

kθkL^s_p:=kF⁻¹((1 +| · |²)^s/2θ)kˆ p<∞.

It is known that L^s_p(R^d) =W_p^k(R^d)if s=k ∈Nand1 < p <∞with equivalent norms.

In order to define the Besov spaces take a non-negative Schwartz functionψ∈ S(R)with support [1/2,2]which satisfiesP_∞

k=−∞ψ(2^−ks) = 1 for alls∈R\ {0}.

Forx∈R^d let

φk(x) :=ψ(2^−k|x|) for k>1 and φ0(x) = 1− X∞

k=1

φk(x).

The Besov space B_p,r^s (R^d) (0 < p, r 6 ∞, s ∈ R) is the space of all tempered distributionsf for which

kfkB_p,r^s :=

³X^∞

k=0

2^ksrk(F⁻¹φk)∗fk^r_p

´_1/r

<∞.

The Sobolev, fractional Sobolev and Besov spaces are all quasi Banach spaces and if16p, r6∞then they are Banach spaces. All these spaces contain the Schwartz functions. The following facts are known: in case16p, r6∞one has

W_p^m(R^d), B_p,r^s (R^d),→Lp(R^d) if s >0, m∈N, W_p^m+1(R^d),→B_p,r^s (R^d),→W_p^m(R^d) if m < s < m+ 1, B_p,r^s (R^d),→B_p,r+²^s (R^d), B_p,∞^s+²(R^d),→B_p,r^s (R^d) if ² >0, B_p^d/p_1,1¹(R^d),→B_p^d/p_2,1²(R^d),→C(R^d) if 16p16p2<∞.

Theorem 6.1.

(i) If16p62andθ∈B_p,1^d/p(R^d)thenθˆ∈L1(R^d)and kθkˆ 16Ckθk_Bd/p

p,1. (ii) If s > dthenL^s₁(R^d),→S0(R^d).

(iii) Ifd⁰ denotes the smallest even integer which is larger thandands > d⁰ then

B^s_1,∞(R^d),→W₁^d0(R^d),→S0(R^d).

The embedding W₁²(R),→S0(R)follows from (iii). With the help of the usual derivative we give another useful sufficient condition for a function to be inS0(R^d).

A functionθis inV₁^k(R) (k>2, k∈N), if there are numbers−∞=a0< a1<

. . . < an< an+1=∞such thatn=n(θ)is depending onθand θ∈C^k−2(R), θ∈C^k(ai, ai+1), θ^(j)∈L1(R)

for alli= 0, . . . , nandj= 0, . . . , k. HereC^kdenotes the set ofk-times continuously differentiable functions. The norm of this space is introduced by

kθk_V^k

1 :=

Xk

j=0

kθ^(j)k1+ Xn

i=1

|θ^(k−1)(ai+ 0)−θ^(k−1)(ai−0)|

whereθ^(k−1)(ai±0)denote the right and left limits ofθ^(k−1). These limits do exist and are finite becauseθ^(k)∈C(ai, ai+1)∩L1(R)implies

θ^(k−1)(x) =θ^(k−1)(a) + Z _x

a

θ^(k)(t)dt

for some a ∈ (ai, ai+1). Since θ^(k−1) ∈ L1(R) we establish that lim−∞θ^(k−1) = lim∞θ^(k−1)= 0. Similarly,θ^(j)∈C0(R)forj= 0, . . . , k−2.

Of course,W₁²(R)andV₁²(R)are not identical. Forθ∈V₁²(R)we haveθ⁰=Dθ, however,θ⁰⁰=D²θonly if limai+0θ⁰ = limai−0θ⁰ (i= 1, . . . , n).

We generalize the previous definition for thed-dimensional case as follows. For d >1andk>2 letθ∈V₁^k(R^d)ifθ is even in each variable and

θ∈C^k−2(R^d), θ∈C^k([0,∞)^d\ {(0, . . . ,0)}), ∂₁ⁱ¹· · ·∂_d^idθ(t)∈L1([0,∞)^l) for eachij = 0, . . . , k (j = 1, . . . , d)and fixed0< tm1, . . . , tmd−l <∞ (16m1 <

m2<· · ·< md−l6d)and16l6d.

Theorem 6.2. If θ∈V₁²(R^d)thenθ∈S0(R^d).

The next Corollary follows from the definition ofS0(R^d).

Corollary 6.3. If each θj∈V₁²(R) (j= 1, . . . , d)thenθ:=Q_d

j=1θj∈S0(R^d).

7. A.e. convergence of the θ-means of Fourier trans- forms

For the a.e. convergence we will investigate first Fourier transforms rather than Fourier series, because the theorems for Fourier transforms are more complicated.

The proofs of the results can be found in Feichtinger and Weisz [4].

L^loc_p (X^d) (16p6∞) denotes the space of measurable functions f for which

|f|^p is locally integrable, resp. f is locally bounded if p = ∞. For 1 6 p 6 ∞ and f ∈ L^loc_p (X^d) let us define a generalization of the Hardy-Littlewood maximal function by

Mpf(x) := sup

x∈I

³ 1

|I|

Z

I

|f|^pdλ

´_1/p

(x∈X^d)

with the usual modification forp=∞, where the supremum is taken over all cubes with sides parallel to the axes. Ifp= 1, this is the usual Hardy-Littlewood maximal function. The following inequalities follow easily from the case p= 1, which can be found in Stein [27] or Weisz [37]:

kMpfkLp,∞ 6Cpkfkp (f ∈Lp(X^d)) (7.1) and

kMpfkr6Crkfkr (f ∈Lr(X^d), p < r6∞). (7.2)

The first inequality holds also ifp=∞.

The spaceE˙q(R^d)contains all functionsf ∈L^loc_q (R^d)for which kfkE˙q:=

X∞

k=−∞

2^kd(1−1/q)kf1Pkkq <∞,

where Pk := {2^k−1 6 |x| <2^k}, (k ∈ Z). These spaces are special cases of the Herz spaces [15] (see also Garcia-Cuerva and Herrero [9]). The non-homogeneous version of the spaceE˙q(R^d)was used by Feichtinger [8] to prove some Tauberian theorems. It is easy to see that

L1(R^d) = ˙E1(R^d)←-E˙q(R^d)←-E˙q⁰(R^d)←-E˙∞(R^d), 1< q < q⁰<∞.

To prove pointwise convergence of theθ-means we will investigate themaximal operator

σ_¤^θf := sup

T >0|σ_T^θf|.

Ifθˆ∈L1(R^d)then (5.1) implies

kσ^θ_¤fk∞6kθkˆ 1kfk∞ (f ∈L∞(R^d)).

In the one-dimensional case Torchinsky [30] proved that if there exists an even functionη such thatη is non-increasing onR+,|θ|ˆ 6η,η∈L1thenσ^θ_¤is of weak type(1,1) and a.e. convergence holds. Under similar conditions we will generalize this result for the multi-dimensional setting. First we introduce an equivalent condition.

Theorem 7.1. Forθ∈L1(R^d)letη(x) := sup_ktkr>kxkr|θ(t)|ˆ for some16r6∞.

Thenθˆ∈E˙∞(R^d)if and only ifη∈L1(R^d)and C⁻¹kηk₁6kθkˆ E˙∞ 6Ckηk₁.

Theorem 7.2. Letθ∈L1(R^d),16p6∞and1/p+ 1/q= 1. Ifθˆ∈E˙q(R^d)then kσ_¤^θfkLp,∞6Cpkθkˆ E˙qkfkp

for allf ∈Lp(R^d). Moreover, for everyp < r6∞,

kσ^θ_¤fkr6Crkθkˆ E˙qkfkr (f ∈Lr(R^d)).

The proof of this theorem follows from the pointwise inequality

σ_¤^θf(x)6Ckθkˆ E˙qMpf(x) (7.3) and from (7.1) and (7.2). Inequality (7.3) is proved in Feichtinger and Weisz [4].

Theorem 7.2 and the usual density argument due to Marcinkiewicz and Zyg- mund [17] imply

Corollary 7.3. Ifθ∈L1(R^d),θ(0) = 1,16p6∞,1/p+1/q= 1andθˆ∈E˙q(R^d) then

Tlim→∞σ_T^θf =f a.e.

iff ∈Lr(R^d)forp6r <∞ orf ∈C0(R^d).

Note that E˙q(R^d)⊃E˙q⁰(R^d) wheneverq < q⁰. If θˆis in a smaller space (say in E˙∞(R^d)) then we get convergence for a wider class of functions (namely for f ∈Lr(R^d),16r6∞).

In order to generalize the last theorem and corollary for the larger space W(L1, `∞)(R^d), we have to define the local Hardy-Littlewood maximal function by

mpf(x) := sup

0<r61

³ 1

|B(x, r)|

Z

B(x,r)

|f|^pdλ

´_1/p

(x∈R^d),

where f ∈ L^loc_p (R^d), 1 6p6 ∞ and B(x, r) denotes the ball with center x and radiusr. It is easy to see that inequalities (7.1) and (7.2) imply

kmpfk_{W(Lp,∞,`s)</sub>6Cpkfk<sub>W(Lp,`s)} (f ∈W(Lp, `s)(R^d)) (7.4) and

kmpfk_{W(Lr,`s)</sub>6Crkfk<sub>W(Lr,`s)} (f ∈W(Lr, `s)(R^d)) (7.5) for allp < r6∞and16s6∞. Recall that

kfk_{W(Lp,∞,`∞)}= sup

k∈Z^d

sup

ρ>0ρ λ(|f|> ρ,[k, k+ 1))^1/p.

Theorem 7.4. Letθ∈L1(R^d),16p6∞and1/p+ 1/q= 1. Ifθˆ∈E˙q(R^d)then kσ_¤^θfk_{W(Lp,∞,`∞)</sub>6Cpkθkˆ E˙qkfk<sub>W(Lp,`∞)}

for allf ∈W(Lp, `∞)(R^d). Moreover, for everyp < r6∞,

kσ^θ_¤fkW(Lr,`∞)6Crkθkˆ E˙qkfkW(Lr,`∞) (f ∈W(Lr, `∞)(R^d)).

It is easy to see that

Mpf 6Cmpf+CpkfkW(Lp,`∞) (16p6∞).

The proof of Theorem 7.4 follows from (7.3)–(7.5).

Corollary 7.5. Ifθ∈L1(R^d),θ(0) = 1,16p <∞,1/p+1/q= 1andθˆ∈E˙q(R^d) then

T→∞lim σ_T^θf =f a.e.

iff ∈W(Lp, c0)(R^d).

Note that W(Lp, c0)(R^d)contains allW(Lr, c0)(R^d)spaces forp6r6∞.

We can characterize the set of convergence in the following way. Lebesgue differentiation theorem says that

h→0lim 1

|B(0, h)|

Z

B(0,h)

f(x+u)du=f(x)

for a.e.x∈X^d, wheref ∈L^loc₁ (X^d), X=Tor X=R. A point x∈X^d is called a p-Lebesgue point (or a Lebesgue point of orderp) off ∈L^loc_p (X^d)if

h→0lim

³ 1

|B(0, h)|

Z

B(0,h)

|f(x+u)−f(x)|^pdu

´_1/p

= 0 (16p <∞) resp.

h→0lim sup

u∈B(0,h)

|f(x+u)−f(x)|= 0 (p=∞).

Usually the 1-Lebesgue points, called simply Lebesgue points are considered (cf.

Stein and Weiss [25] or Butzer and Nessel [3]). One can show that almost every pointx∈X^dis ap-Lebesgue point off ∈L^loc_p (X^d)if16p <∞, which means that almost every pointx∈R^d is ap-Lebesgue point off ∈W(Lp, `∞)(R^d). x∈X^d is an∞-Lebesgue point off ∈L^loc_∞(X^d)if and only iff is continuous atx. Moreover, allr-Lebesgue points arep-Lebesgue points, wheneverp < r.

Stein and Weiss [25, p. 13] (see also Butzer and Nessel [3, pp. 132-134]) proved that if η(x) := sup_|t|>|x||θ(t)|ˆ and η ∈L1(R^d)then one has convergence at each Lebesgue point off ∈Lp(R^d) (16p6∞). Using theE˙q spaces we generalize this result.

Theorem 7.6. Let θ ∈ L1(R^d), θ(0) = 1, 1 6 p 6 ∞ and 1/p+ 1/q = 1. If θˆ∈E˙q(R^d)then

Tlim→∞σ_T^θf(x) =f(x) for allp-Lebesgue points off ∈W(Lp, `∞)(R^d).

Note that W(L1, `∞)(R<sup>d</sup>) contains all Lp(R<sup>d</sup>) spaces and amalgam spaces W(Lp, `q)(R^d)for the full range16p, q6∞.

If f is continuous at a point x then x is a p-Lebesgue point of f for every 16p6∞.

Corollary 7.7. Let θ ∈ L1(R^d), θ(0) = 1, 1 6 p 6 ∞ and 1/p+ 1/q = 1. If θˆ∈E˙q(R^d)andf ∈W(Lp, `∞)(R^d)is continuous at a pointxthen

T→∞lim σ_T^θf(x) =f(x).

Recall thatE˙1(R^d) =L1(R^d)andW(L∞, `∞)(R^d) =L∞(R^d). Iff is uniformly continuous then we have uniform convergence (see Corollary 5.4).

Let us consider converse-type problems. The partial converse of Theorem 7.2 is given in the next result.

Theorem 7.8. Let θ∈L1(R^d),θˆ∈L1(R^d),16p <∞ and1/p+ 1/q= 1. If

σ_¤^θf(x)6CMpf(x) (7.6)

for allf ∈Lp(R^d)andx∈R^d thenθˆ∈E˙q(R^d).

The converse of Theorem 7.6 reads as follows.

Theorem 7.9. Suppose that θ∈L1(R^d),θ(0) = 1, θˆ∈L1(R^d), 1 6p < ∞ and 1/p+ 1/q= 1. If

Tlim→∞σ_T^θf(x) =f(x) (7.7) for allp-Lebesgue points off ∈Lp(R^d)thenθˆ∈E˙q(R^d).

Corollary 7.10. Suppose that θ∈L1(R^d),θ(0) = 1,θˆ∈L1(R^d),16p <∞and 1/p+ 1/q= 1. Then

Tlim→∞σ_T^θf(x) =f(x)

for all p-Lebesgue points off ∈Lp(R^d) (resp. off ∈W(Lp, `∞)(R^d)) if and only ifθˆ∈E˙q(R^d).

If we take the supremum in the maximalθ-operator over a cone, say over{T ∈ R^d₊: 2^−τ6Ti/Tj 62^τ;i, j= 1, . . . , d} for some fixedτ>0:

σ_c^θf := sup

2−τ6Ti/Tj62τ i,j=1,...,d

|σ_T^θf|,

then all the results above can be shown for σ_c^θ. In this case, under the conditions above we obtain the convergenceσ_T^θf →f a.e. as T → ∞and 2^−τ 6Ti/Tj 62^τ (i, j = 1, . . . , d). This convergence has been investigated in a great number of papers (e.g. in Marcinkiewicz and Zygmund [17], Zygmund [39], Weisz [34, 36, 37]).

For more details see Feichtinger and Weisz [4]. The unrestricted convergence of σ_T^θf, i.e. asTj→ ∞for eachj= 1, . . . , d, is also investigated in that paper.

8. A.e. convergence of the θ-means of Fourier series

All the results of Section 7 holds also for Fourier series. In this case we define themaximal operatorof theθ-means by

σ^θ_¤f := sup

n∈N|σ^θ_nf|.

Similarly to Theorem 7.4 we have