The Segal algebra S0(Rd) and norm summability of Fourier series and Fourier transforms Hans G. Feichtinger Numerical Harmonic Analysis Group Faculty of Mathematics University of Vienna Nordbergstraße 15, A-1090 Vienna, Austria email:
[email protected] Ferenc Weisz∗ Department of Numerical Analysis E¨otv¨os L. University P´azm´any P. s´et´any 1/C., H-1117 Budapest, Hungary e-mail:
[email protected]
Abstract A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier series. Equivalent conditions are derived for the uniform and L1 -norm convergence of the θ-means σnθ f to the function f . If f is in a homogeneous Banach space, then the preceding convergence holds in the norm of the space. In case θ is an element of the Feichtinger’s Segal algebra S0 (Rd ), then these convergence results hold. Some new sufficient conditions are given for θ to be in S0 (Rd ). A long list of concrete special cases of the θ-summation are listed. The same results are also provided in the context of Fourier transforms, indicating how proof have to be changed in this case.
2000 AMS subject classifications: Primary 42B08, 46E30, Secondary 42B30, 42A38. Key words and phrases: Wiener algebra, Feichtinger’s algebra, homogeneous Banach space, θ-summability of Fourier series, Besov-, Sobolev-, fractional Sobolev spaces, amalgam spaces. ∗
This research was supported by Lise Meitner fellowship No M733-N04 and the Hungarian Scientific Research Funds (OTKA) No T043769, T047128, T047132.
1
1
Introduction
It is known for concrete summability methods, such as Fej´er, Riesz, Weierstrass, Abel, etc., that the corresponding means σn f of the Fourier series of f converge to f uniformly as n → ∞ if f is continuous, and in Lp norm if f ∈ Lp (Td ) for some 1 ≤ p < ∞ (see e.g. Fej´er [10], Zygmund [37], Butzer and Nessel [5], Stein and Weiss [25] or Trigub and Belinsky [30]). A general method of summation, the so called θ-summation, which is generated by a single function θ is intensively studied in the literature (see e.g. Butzer and Nessel [5], Trigub and Belinsky [2, 30, 31], Bokor, Schipp, Szili and V´ertesi [23, 27, 28] and Weisz [33, 34, 35]). The preceding convergence result holds for the θ-means of non-periodic functions f ∈ Lp (Rd ) (resp. f ∈ C(Rd )), too, if the Fourier transform of θ is integrable (see Butzer and Nessel [5] and Stein and Weiss [25]). In this paper we consider mainly Fourier series and the question: for what functions θ do we have the preceding convergence result. A natural choice of θ is a function from the Wiener algebra W (C, `1 )(Rd ). All concrete summability methods investigated in the literature satisfy this condition. In this case the θ-means σnθ f of the Fourier series of f ∈ L2 (Td ) converge to f in L2 norm as n → ∞. If θ is in the Wiener algebra then σnθ f → f uniformly (resp. at each point) for all f ∈ C(Td ) if and only if σnθ f → f in L1 norm for all f ∈ L1 (Td ) if and only if θˆ ∈ L1 (Rd ). If B is a homogeneous Banach space on Td and θˆ ∈ L1 (Rd ) then σnθ f → f in B norm for all f ∈ B. The pointwise convergence of the θ-means is studied in Feichtinger and Weisz [7]. We shall investigate some function spaces known from other topics of analysis which have not yet been considered in summability theory. The most important one is the Feichtinger’s Segal algebra S0 (Rd ) (nowadays often just called Feichtinger’s algebra in the time-frequency literature) which was introduced in [9] in this journal. Feichtinger‘s algebra is very intensively investigated in Gabor analysis (see e.g. Feichtinger and Zimmermann [8] and Gr¨ochenig [14]). It is the minimal (non-trivial) Banach space which is isometrically invariant under translation, modulation and Fourier transform. Moreover, S(Rd ) ,→ S0 (Rd ) ,→ W (C, `1 )(Rd ). Hence all the convergence results above hold if θ ∈ S0 (Rd ). We will show that if θ is continuous and has compact support then the uniform convergence of the θ-means is equivalent to the L1 norm convergence of the θ-means and this is equivalent to the condition θ ∈ S0 (Rd ). Next we give some sufficient conditions for θ such that θˆ ∈ L1 (Rd ), resp. θ ∈ S0 (Rd ). d/2 If θ is in the Besov space B2,1 (Rd ) then θˆ ∈ L1 (Rd ) and if s > d then the fractional Sobolev space Ls1 (Rd ) is embedded into S0 (Rd ). Moreover, if d0 denotes the smallest 0 even integer which is larger than d then the Sobolev space W1d (Rd ) is also embedded into S0 (Rd ). Using the usual derivative we introduce a Sobolev-like space V12 (Rd ) and prove that V12 (Rd ) ⊂ S0 (Rd ). This space is very useful for finding special cases of θ-summation. In this paper 17 such examples are investigated. Finally the same convergence results are proved for Fourier transforms. We thank the referees for their useful suggestions and for calling our attention to the recent book of Trigub and Belinsky [30].
2
2
Wiener’s and Feichtinger’s algebra
Let us fix d ≥ 1, d ∈ N. For a set Y 6= ∅ let Yd be its Cartesian product Y × . . . × Y taken with itself d-times. We shall prove results for Rd or Td , therefore it is convenient to use sometimes the symbol X for either R or T, where T is the torus. For x = (x1 , . . . , xd ) ∈ Rd and u = (u1 , . . . , ud ) ∈ Rd set u · x :=
d X
uk x k
and
kxkp :=
k=1
d ³X
|xk |p
´1/p
(1 ≤ p ≤ ∞).
k=1
We use also the shorter notation |x| := kxk2 . We briefly write Lp or RLp (Xd ) instead of Lp (Xd , λ) space equipped with the norm (or quasi-norm) kf kp := ( Xd |f |p dλ)1/p (0 < p ≤ ∞), where X = R or T and λ is the Lebesgue measure. We identify T with [−1/2, 1/2). The space of continuous functions with the supremum norm is denoted by C = C(Xd ) and we will use C0 = C0 (Rd ) for the space of continuous functions vanishing at infinity. Translation and modulation of a function f are defined, respectively, by Tx f (t) := f (t − x) and Mω f (t) := e2πıω·t f (t)
(x, ω ∈ Rd ).
A measurable function f belongs to the Wiener amalgam space W (L∞ , `1 )(Rd ) if X X kf kW (L∞ ,`1 ) := sup |f (x + k)| = kf · Tk 1[0,1)d k∞ < ∞. k∈Zd
x∈[0,1)d
k∈Zd
It is easy to see that W (L∞ , `1 )(Rd ) ⊂ Lp (Rd ) for all 1 ≤ p ≤ ∞. The closed subspace of W (L∞ , `1 )(Rd ) containing continuous functions is denoted by W (C, `1 )(Rd ) and is called Wiener algebra. It is used quite often in Gabor analysis, because it provides a convenient and general class of windows (see e.g. Walnut [32] and Gr¨ochenig [13]). A Banach space B consisting of Lebesgue measurable functions on Xd is called a homogeneous Banach space, if (i) for all f ∈ B and x ∈ Xd , Tx f ∈ B and kTx f kB = kf kB , (ii) the function x 7→ Tx f from Xd to B is continuous for all f ∈ B, (iii) the functions in B are uniformly locally integrable, i.e. for every compact set K ⊂ Xd there exists a constant CK such that Z |f | dλ ≤ CK kf kB (f ∈ B). K
If furthermore B is a dense subspace of L1 (Xd ) it is called a Segal algebra (cf. Reiter [21]). Note that the continuous embedding into L1 (Xd ) is a consequence of the closed graph theorem. Again, X = T or X = R. For an introduction to homogeneous Banach spaces see Katznelson [16] or Shapiro [24]. It is easy to see that the spaces Lp (Xd ) (1 ≤ p < ∞), C(Td ), C0 (Rd ), Lorentz spaces Lp,q (Xd ) (1 < p < ∞, 1 ≤ q < ∞) and Hardy spaces H1 (Xd ) (for the definitions see e.g. Weisz [35]) are homogeneous Banach spaces. 3
The Fourier transform of f ∈ L1 (Rd ) is Z ˆ f (t)e−2πıx·t dt Ff := f (x) :=
(x ∈ Rd , ı =
√
−1).
Rd
Feichtinger’s algebra S0 (Rd ) has a number of different characterizations. We define S0 (Rd ) via the short-time Fourier transform (STFT). Recall that for f ∈ L2 (Rd ) the STFT with respect to a window function g ∈ L2 (Rd ) is defined by Z Sg f (x, ω) := (x, ω ∈ Rd ). f (t)g(t − x)e−2πıωt dt = hf, Mω Tx gi Rd 2
Using the STFT with respect to the Gauss function g0 (x) := e−π|x| we define S0 (Rd ) by © ª S0 (Rd ) := f ∈ L2 (Rd ) : kf kS0 := kSg0 f kL1 (R2d ) < ∞ . Any other non-zero Schwartz function defines the same space and an equivalent norm. It is known that S0 (Rd ) is a Segal algebra and it is isometrically invariant under translation, modulation and Fourier transform (see Feichtinger [9]). Furthermore, the embedding S0 (Rd ) ,→ W (C, `1 )(Rd ) is dense and continuous and S0 (Rd ) ( W (C, `1 )(Rd ) ∩ F (W (C, `1 )(Rd )) (see Feichtinger and Zimmermann [8], Losert [17] and Gr¨ochenig [14]).
3
θ-summability of Fourier series
The θ-summation (or sometimes called φ-summation) was considered in a great number of papers and books, such as Butzer and Nessel [5], Trigub and Belinsky [2, 30, 31], Bokor, Schipp, Szili and V´ertesi [23, 27, 28], Natanson and Zuk [19] and Weisz [33, 34, 35]. For a more detailed historical background see the book of Trigub and Belinsky [30] and the references therein. In these investigations P usually kit was supposed that θ ∈ L1 (R) is an even continuous function satisfying ∞ k=−∞ |θ( n+1 )| < ∞ (n ∈ N). In d dimension this condition reads as d ∞ ¯ ³ k X X kd ´¯¯ ¯ 1 θ , . . . , ¯ 1/2 then θˆ ∈ L1 (R) (see Natanson ˆ s ∈ L2 (Rd ) for some s > d or if and Zuk [19, p. 176]) and so θ ∈ S0 (R). If θvs , θv ˆ s ∈ L∞ (Rd ) for some s > 3d/2 then θ ∈ S0 (Rd ). The weight function vs is given θvs , θv by vs (ω) := (1 + |ω|)s (ω ∈ Rd , s ∈ R). 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 [29], Runst and Sickel [22], Stein [26] and Grafakos [12]. The Sobolev space Wpk (Rd ) (1 ≤ p ≤ ∞, k ∈ N) is defined by Wpk (Rd ) := {θ ∈ Lp (Rd ) : Dα θ ∈ Lp (Rd ), |α| ≤ k} and endowed with the norm kθkWpk :=
X |α|≤k
10
kDα θkp ,
where D denotes the distributional derivative. This definition is extended to every real s in the following way. The fractional Sobolev space Lsp (Rd ) (1 ≤ p ≤ ∞, s ∈ R) consists of all tempered distribution θ for which ˆ p < ∞. kθkLsp := kF −1 ((1 + | · |2 )s/2 θ)k It is known that Lsp (Rd ) = Wpk (Rd ) if s = k ∈ N and 1 < p < ∞ with equivalent norms. In order to define the Besov spaces take a non-negative Schwartz function ψ ∈ S(R) with support [1/2, 2] which satisfies ∞ X
ψ(2−k s) = 1 for all s ∈ R \ {0}.
k=−∞
For x ∈ Rd let −k
φk (x) := ψ(2 |x|) for k ≥ 1 and φ0 (x) = 1 −
∞ X
φk (x).
k=1 s The Besov space Bp,r (Rd ) (0 < p, r ≤ ∞, s ∈ R) is the space of all tempered distributions f for which
s kf kBp,r :=
∞ ³X
ksr
2
k(F
−1
φk ) ∗
f krp
´1/r
< ∞.
k=0
The Sobolev, fractional Sobolev and Besov spaces are all quasi Banach spaces and if 1 ≤ p, r ≤ ∞ then they are Banach spaces. All these spaces contain the Schwartz functions. The following facts are known: in case 1 ≤ p, r ≤ ∞ one has s Wpm (Rd ), Bp,r (Rd ) ,→ Lp (Rd ) s Wpm+1 (Rd ) ,→ Bp,r (Rd ) ,→ Wpm (Rd )
if s > 0, m ∈ N, if m < s < m + 1,
s s s+² s Bp,r (Rd ) ,→ Bp,r+² (Rd ), Bp,∞ (Rd ) ,→ Bp,r (Rd ) d/p
d/p
Bp1 ,11 (Rd ) ,→ Bp2 ,12 (Rd ) ,→ C(Rd )
(6)
if ² > 0,
(7)
if 1 ≤ p1 ≤ p2 < ∞.
(8)
The connection between Besov spaces and Feichtinger’s algebra is summarized in the next theorem. Theorem 6 d/p (i) If 1 ≤ p ≤ 2 and θ ∈ Bp,1 (Rd ) then θˆ ∈ L1 (Rd ) and
ˆ 1 ≤ Ckθk d/p . kθk B p,1
11
(ii) If s > d then Ls1 (Rd ) ,→ S0 (Rd ). (iii) If d0 denotes the smallest even integer which is larger than d and s > d0 then 0
s B1,∞ (Rd ) ,→ W1d (Rd ) ,→ S0 (Rd ).
Proof. (i) was proved in Girardi and Weis [11] and (ii) in Okoudjou [20]. The first embedding of (iii) follow from (6) and (7). If k is even then W1k (Rd ) ,→ Lk1 (Rd ) (see Stein [26, p. 160]). Then (ii) proves (iii). It follows from (i) and (6) that θ ∈ W j (Rd ) (j > d/p, j ∈ N) implies θˆ ∈ L1 (Rd ). If p
j ≥ d0 then even W1j (Rd ) ,→ S0 (Rd ) (see (iii)). Moreover, if s > d0 as in (iii) then d/p
s d B1,∞ (Rd ) ,→ B1,1 (Rd ) ,→ Bp,1 (Rd )
(1 < p < ∞)
s by (7) and (8). Theorem 6 says that B1,∞ (Rd ) ⊂ S0 (Rd ) (s > d0 ) and if we choose θ d/p from the larger space Bp,1 (Rd ) (1 ≤ p ≤ 2) then θˆ is still integrable. 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 in S0 (Rd ).
Definition 1 A function θ is in V12 (R), if there are numbers −∞ = a0 < a1 < . . . < an < an+1 = ∞ such that n = n(θ) is depending on θ and θ ∈ C(R),
θ ∈ C 2 (ai , ai+1 ),
θ0 , θ00 ∈ L1 (R)
for all i = 0, . . . , n. Here C k denotes the set of k-times continuously differentiable functions. The norm of this space is introduced by kθkV12 :=
2 X j=0
(j)
kθ k1 +
n X
|θ0 (ai + 0) − θ(k−1) (ai − 0)|,
i=1
where θ0 (ai ± 0) denote the right and left limits of θ0 . Note that these limits do exist and are finite because θ00 ∈ C(ai , ai+1 )∩L1 (R) implies Z x 0 0 θ (x) = θ (a) + θ00 (t) dt a
for some a ∈ (ai , ai+1 ). Since θ0 ∈ L1 (R) we establish that lim−∞ θ0 = lim∞ θ0 = 0. Similarly, we have θ ∈ C0 (R). Of course, W12 (R) and V12 (R) are not identical. For θ ∈ V12 (R) we have θ0 = Dθ, however, θ00 = D2 θ only if limai +0 θ0 = limai −0 θ0 (i = 1, . . . , n). Theorem 7 We have V12 (R) ,→ S0 (R).
12
Proof. Integrating by parts we have Z Sg0 θ(x, ω) = θ(t)g0 (t − x)e−2πıωt dt R
= =
n Z X
ai+1
2
θ(t)e−π(t−x) e−2πıωt dt
i=0 ai n h X
2
θ(t)e−π(t−x)
i=0
−
n Z X i=0
ai+1
³
ai
e−2πıωt iai+1 −2πıω ai
´ e−2πıωt 2 2 θ0 (t)e−π(t−x) − 2πθ(t)e−π(t−x) (t − x) dt. −2πıω
Observe that the first sum is 0. In the second sum we integrate by parts again to obtain n h³ ´ e−2πıωt iai+1 X 2 2 Sg0 θ(x, ω) = − θ0 (t)e−π(t−x) − 2πθ(t)e−π(t−x) (t − x) (2πıω)2 ai i=0 n Z ai+1 ³ X 2 2 + θ00 (t)e−π(t−x) − 4πθ0 (t)e−π(t−x) (t − x) i=0
ai
³ ´´ e−2πıωt 2 2 dt. − 2πθ(t) − 2πe−π(t−x) (t − x)2 + e−π(t−x) (2πıω)2
The first sum is equal to n ³ X i=1
Hence
´ −2πıωai 2e θ0 (ai + 0) − θ0 (ai − 0) e−π(ai −x) . (2πıω)2
Z Z R
{|ω|≥1}
|Sg0 θ(x, ω)| dx dω ≤ Cs kθkV12 .
On the other hand Z Z Z Z |Sg0 θ(x, ω)| dx dω ≤ Cs R
{|ω| 1) then θ ∈ S0 (Rd ). 13
6
Some summability methods
We consider some summability methods as special cases of the θ-summation. In the first 14 examples θ ∈ V12 (Rd ). γ
Example 1 (Weierstrass summation) θ(x) = e−|x| (x ∈ R, 1 ≤ γ < ∞). If γ = 1 then it is called Abel summation. q )γ
Example 2 θ(x) = e−(1+|x|
(x ∈ R, 1 ≤ q < ∞, 0 < γ < ∞).
q
Example 3 θ(x) = e−kxkq (x ∈ Rd , 1 ≤ q < ∞). q γ
Example 4 θ(x) = e−(1+kxkq ) (x ∈ Rd , 1 ≤ q < ∞, 0 < γ < ∞). ˆ Example 5 If θ(x) = e−2πkxk2 (x ∈ Rd ) then θ(x) = cd /(1 + kxk22 )(d+1)/2 (see Stein and Weiss [25, p. 6.]). ˆ Example 6 For θ(x) = 1/(1 + kxk22 )(d+1)/2 (x ∈ Rd ) we have θ(x) = cd e−2πkxk2 . Example 7 (Picard and Bessel summations) θ(x) = (1 + |x|γ )−α (x ∈ R, 0 < α < ∞, 1 ≤ γ < ∞, αγ > 1). Example 8 θ(x) = (1 + kxkγγ )−α (x ∈ Rd , 0 < α < ∞, 1 ≤ γ < ∞, αγ > d). Example 9 ½ θ(x) :=
1 kxk−α q
if kxkq ≤ 1 if kxkq > 1
(x ∈ Rd , 1 ≤ q < ∞, d < α < ∞).
Example 10 ½ θ(x) :=
1
α
1−e−|x| |x|α
if x = 0 if |x| > 0
(x ∈ R, 1 < α < ∞).
Example 11 (de La Vall´ ee-Poussin summation) if |x| ≤ 1/2 1 θ(x) = −2|x| + 2 if 1/2 < |x| ≤ 1 0 if |x| > 1
(x ∈ R).
Example 12 (Jackson-de La Vall´ ee-Poussin summation) 1 − 3x2 /2 + 3|x|3 /4 if |x| ≤ 1 θ(x) = (2 − |x|)3 /4 (x ∈ R). if 1 < |x| ≤ 2 0 if |x| > 2 The next example generalizes Examples 11 and 12.
14
Example 13 Let 0 = α0 < α1 < . . . < αm and β0 , . . . , βm (m ∈ N) be real numbers, β0 = 1, βm = 0. Suppose that θ is even, θ(αj ) = βj (j = 0, 1, . . . , m), θ(x) = 0 for x ≥ αm , θ9 is a polynomial on the interval [αj−1 , αj ] (j = 1, . . . , m). Example 14 (Rogosinski summation) ½ cos πx/2 if |x| ≤ 1 + 2j θ(x) = 0 if |x| > 1 + 2j
(j ∈ N).
This summation was originally defined for j = 0. In the next three examples θ ∈ S0 (Rd ). Example 15 (Riesz summation I.) ½ (1 − |x|γ )α if |x| ≤ 1 θ(x) := 0 if |x| > 1
(x ∈ R, 0 < α < ∞, 1 ≤ γ < ∞).
It is called Fej´er summation if α = γ = 1. We have proved in [35] that ˆ |θ(x)| ≤
C |x|(α∧1)+1
.
Example 16 (Riesz summation II.) ½ (1 − kxkk2 )α if kxk2 ≤ 1 θ(x) := (x ∈ Rd , (d − 1)/2 < α < ∞, k ∈ N, k > 0). 0 if kxk2 > 1 One can find in Stein and Weiss [25, p. 171] and Lu [18, p. 132] that ˆ |θ(x)| ≤ C|x|−d/2−α−1/2
(x 6= 0)
and this is integrable if α > (d − 1)/2. ³ Example 17 (Riemann summation) If θ(x) =
7
sin x/2 x/2
´2
then θˆ = max(0, 1 − |x|).
Convergence of the θ-means of Fourier transforms
All the results above can be shown for non-periodic functions f ∈ Lp (Rd ). Suppose first that f ∈ Lp (Rd ) for some 1 ≤ p ≤ 2. The Fourier inversion formula Z f (x) = fˆ(u)e2πıx·u du (x ∈ Rd ) Rd
holds if fˆ ∈ L1 (Rd ). In the investigation of Fourier transforms we can take a larger space than W (C, `1 )(Rd ), we will assume that θ ∈ L1 (Rd ) ∩ C0 (Rd ). The θ-means of f ∈ Lp (Rd ) (1 ≤ p ≤ 2) are defined by Z ³ −t −td ´ ˆ 2πıx·t 1 θ θ ,..., f (t)e dt. (9) σT f (x) := T1 Td Rd 15
Then
Z σTθ f (x)
= Rd
f (x − t)KTθ (t) dt
(x ∈ Rd , T ∈ Rd+ )
where Z KTθ (x)
=
θ
³ −t
Rd
T1
³Y ´ −td ´ 2πıx·t ˆ 1 x1 , . . . , Td xd ), e dt = Tj θ(T Td j=1 d
1
,...,
(10)
(x ∈ Rd ). Thus the θ-means can rewritten as σTθ f (x)
=
d ³Y j=1
´Z
ˆ 1 t1 , . . . , Td td ) dt f (x − t)θ(T
Tj Rd
which is the analogue to Theorem 3. Here we have used that fˆ ∈ Lp0 (Rd ) if f ∈ Lp (Rd ) (1 ≤ p ≤ 2, 1/p + 1/p0 = 1) and the Hausdorff-Young inequality. Note that θ ∈ L1 (Rd ) ∩ C0 (Rd ) implies θ ∈ Lp (Rd ) (1 ≤ p ≤ ∞). Now we formulate Theorem 2 for Fourier transforms. Theorem 9 If θ ∈ L1 (Rd ) ∩ C0 (Rd ) and θ(0) = 1 then σTθ f → f in L2 (Rd ) norm for all f ∈ L2 (Rd ) as T → ∞. Proof. We can prove as in Theorem 2 that the norm of the operator σTθ : L2 (Rd ) → L2 (Rd ) is kθk∞ . The set of the functions f ∈ L2 (Rd ) the Fourier transforms of which have compact supports is dense in L2 (Rd ). If f is such a function then (9) and the Fourier inversion formula imply Z ³ ³ ´ −td ´ −t1 θ ,..., − 1 fˆ(t)e2πıx·t dt σT f (x) − f (x) = θ T1 Td Rd ³³ ³ −t ´ ´ −td ´ 1 = F −1 θ ,..., − 1 fˆ(t) (x). T1 Td Thus
°³ ³ −t ° ´ −td ´ ° ° 1 kσTθ f − f k2 = ° θ ,..., − 1 fˆ(t)° → 0 T1 Td 2 as T → ∞, because θ is continuous. The density theorem proves Theorem 9. Since σTθ is defined only for f ∈ Lp (Rd ) (1 ≤ p ≤ 2), instead of Theorem 4 we have Theorem 10 If θ ∈ L1 (Rd ) ∩ C0 (Rd ) and θ(0) = 1 then the following conditions are equivalent: (i) θˆ ∈ L1 (Rd ), (ii) σTθ f → f in L1 (Rd ) norm for all f ∈ L1 (Rd ) as T → ∞, θ (iii) σT f → f in L1 (Rd ) norm for all f ∈ L1 (Rd ) as T → ∞.
If θ has compact support then θ ∈ S0 (Rd ) is also an equivalent condition. 16
Because of (10) the proof of this theorem is simpler than that of Theorem 4. We will extend the definition of the θ-means. To this end we introduce the amalgam spaces. A measurable function f belongs to the Wiener amalgam space W (Lp , `q )(Rd ) (1 ≤ p, q ≤ ∞) if kf kW (Lp ,`q ) :=
³X
kf (· +
k)kqLp [0,1)d
´1/q
k∈Zd
=
³X
kf ·
Tk 1[0,1)d kqp
´1/q
< ∞,
k∈Zd
with the obvious modification for q = ∞. W (Lp , c0 )(Rd ) is defined analogously (1 ≤ p ≤ ∞), where c0 denotes the space of sequences of complex numbers having 0 limit equipped with the supremum norm. It is easy to see that W (Lp , `p )(Rd ) = Lp (Rd ) and W (L1 , `∞ )(Rd ) ⊃ W (Lp , `q )(Rd ) (1 ≤ p, q ≤ ∞). For more about amalgam spaces see e.g. Heil [15]. If θˆ ∈ L1 (Rd ), the definition of the θ-means extends to f ∈ W (L1 , `∞ )(Rd ) by σTθ f := f ∗ KTθ
(T ∈ Rd+ ).
Note that θ ∈ L1 (Rd ) and θˆ ∈ L1 (Rd ) imply θ ∈ C0 (Rd ). The analogue of Theorem 5 follows in the same way: Theorem 11 Assume that B is a homogeneous Banach space on Rd . If θ ∈ L1 (Rd ), θ(0) = 1 and θˆ ∈ L1 (Rd ) (e.g. θ ∈ S0 (Rd )) then σTθ f → f in B, for all f ∈ B as T → ∞. Note that if B is a homogeneous Banach space on Rd then B ,→ W (L1 , `∞ )(Rd ) (see Katznelson [16]). It is easy to see that the spaces W (Lp , `q )(Rd ) (1 ≤ p, q < ∞), W (Lp , c0 )(Rd ) (1 ≤ p < ∞) and W (C, `q )(Rd ) (1 ≤ q < ∞) are homogeneous Banach spaces (see e.g. Heil [15]). Moreover, the space Cu (Rd ) of uniformly continuous bounded functions endowed with the supremum norm is also a homogeneous Banach space. Corollary 4 If f is a uniformly continuous and bounded function, θ ∈ L1 (Rd ), θ(0) = 1 and θˆ ∈ L1 (Rd ) then σTθ f → f uniformly as T → ∞.
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