Wiener–Beurling spaces and their properties

Journal Pre-proof Wiener–Beurling spaces and their properties

Erlan Nursultanov, Sergey Tikhonov

PII:

S0007-4497(19)30101-0

DOI:

https://doi.org/10.1016/j.bulsci.2019.102825

Reference:

BULSCI 102825

To appear in:

Bulletin des Sciences Mathématiques

Received date:

8 October 2019

Please cite this article as: E. Nursultanov, S. Tikhonov, Wiener–Beurling spaces and their properties, Bull. Sci. math. (2019), 102825, doi: https://doi.org/10.1016/j.bulsci.2019.102825.

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Wiener–Beurling spaces and their properties Erlan Nursultanov and Sergey Tikhonov Abstract. In this paper, we introduce the Wiener–Beurling space, which generalize the Wiener space of absolutely convergent Fourier series, the Berling space as well as the Ces` aro and Copson spaces. We investigate their main properties: embeddings, duality, and interpolation. We also discuss the product and convolution operators as well as various average operators in the Wiener–Beurling spaces.

1. Introduction The Wiener algebra A = A1 of absolutely convergent Fourier series on the circle T1 (see, e.g., [Ka]) and its β -versions, the β-Wiener spaces Aβ , are defined as follows ⎧ ⎫ ⎛ ⎞1/β ⎪ ⎪ ∞ ⎨ ⎬  β β 1 1 1 β⎠ ⎝ A = A (T ) = f ∈ L (T ) : f Aβ (T1 ) = |f (j)| <∞ . ⎪ ⎪ ⎩ ⎭ j=−∞ Beurling [Be] introduced the spaces A∗ , nowadays called the Beurling space, to study contraction properties of functions. By definition, ⎧ ⎫ ∞ ⎨ ⎬  A∗ (T1 ) = f ∈ L1 (T1 ) : f A∗ (T1 ) = sup |f (k)| < ∞ . ⎩ ⎭ j|k| j=0

Important properties of the Wiener and Beurling spaces and their comparison can be found in [BLT, Ka]. In the case of function on R the Wiener algebra was studied in [LST]. Recently [NRT], the β-Beurling spaces have been introduced: ⎧ ⎫ ∞ 1/β ⎨ ⎬  A∗,β = A∗,β (T1 ) = f ∈ L1 (T1 ) : f A∗,β (T1 ) = 2s sup |f (k)|β <∞ . ⎩ ⎭ 2s |k| s=0

It is easy to see that A∗,1 = A∗ . Both Wiener and Beurling function spaces play an important role in harmonic analysis, in particular, in the summability theory and in the Fourier synthesis (see, e.g., the texts [Ben], [SW, Theorems 1.25 and 1.16], and [TB, Theorem 8.1.3, Ch. 6]). The main goal of this paper is to introduce a more general scale of spaces, which we call the Wiener–Beurling spaces, and investigate their basic properties. 2000 Mathematics Subject Classification. Primary 46E30; Secondary 46A45, 44A35, 46B70. Key words and phrases. Wiener–Beurling spaces, Embeddings, Convolution, Interpolation, Duality. This research was partially supported by MTM 2017-87409-P, 2017 SGR 358, the CERCA Programme of the Generalitat de Catalunya, and Ministry of Education and Science of the Republic of Kazakhstan (AP051 32071, AP051 32590). 1

2

ERLAN NURSULTANOV AND SERGEY TIKHONOV

As usual, we will write F1  F2 if c−1 F1  F2  cF1 for some positive constant c independent of essential parameters. Moreover, F1  F2 stands for F1  cF2 . We will assume that p1 + p1 = 1 for any p ∈ (0, ∞]. By [a] we denote the integer part of a. α to be the collection Definition 1.1. Let α ∈ R and 0 < p, q  ∞. We define the space W Bp,q of formal trigonometric series (not necessarily the Fourier series)  cm eimx , x ∈ T, f (x) = m∈Z

such that (1.1)

α = f W Bp,q

∞  

2αk





k=0

|cm |p

1/p q 1/q

< ∞,

[2k−1 ]|m|<2k

α with the usual modification for p = ∞ or/and q = ∞. Moreover, f W Bp,q is a (quasi-) norm (a norm for 1  p, q  ∞).

Several remarks are in order. (i) A similar definition (with specific values of the parameters α, p, and q) has first appeared in the paper by D. Borwein [Bo] devoted to summability problems. Various applications related to some special cases of Wiener–Beurling norm can be found in, e.g., [AM2, Ben, He, Ke, LL, Li, Sl, Su].    1/q ∞  α  p 1/p q α can be equivalently defined by (ii) Clearly, f W Bp,q |c | , γ k=0 [γk ]|m|<γk+1 m k where 0 < γ0 < 1 such that 1 < γ < γk+1 /γk < γ ∗ < ∞. (iii) The parameter α in definition (1.1) can be seen as the smoothness parameter since f ∈  α+β α , where f (β) is the Weyl fractional with f (x) = m∈Z cm eimx if and only if f (β) ∈ W Bp,q W Bp,q derivative of f [Zy, Ch. XII, §8]. Examples. It is clear that both Wiener and Beurling spaces are particular cases of the 0 = Aβ and W B 1/β = A∗,β . The latter follows from the Wiener–Beurling spaces, namely, W Bβ,β ∞,β fact that  f A∗,β 

∞ 

1/β 2s

s=0

sup 2s |k|<2s+1

|ck |β

.

It is also easy to see that, for 1 < q  ∞, we have f 

1 −1

q W B1,q

 {cn }ces(q) ,

where the discrete Ces` aro spaces ces(q) is given by ∞    , c ∈ R : {c } = ces(q) = {cm }∞ n ces(q) m=0 m s=0

s q 1/q   1 |cm | <∞ . 2s + 1 m=−s

Indeed, using for α < 0 Hardy’s inequality (1.2)

∞  k=0

2αk

k  m=0

γ Am



∞  k=0

2αk Aγk ,

Am  0,

γ > 0,

WIENER–BEURLING SPACES

3

gives (1.3)

α  f W Bp,q

∞ 

2αkq

k=0

For the case p = 1 and α = α f W Bp,q

1 q





|cm |p

q/p 1/q

.

0|m|<2k

− 1 < 0 we then have

∞   1  k+1 k=0



|cm |

q 1/q

 {cn }ces(q) .

0|m|
Similarly, one can show that for any 0 < q < ∞ we have f 

1 −1

q W B1,q

= {cn }cop(q) ,

where the discrete Copson spaces cop(q) is given by ∞  ∞    |cm | + |c−m | q 1/q ∞ <∞ . cop(q) = {cm }m=0 , cm ∈ R : m m=s s=0

In particular, this immediately implies the following important fact due to Bennett [Ben, (6.8)]: ces(q) = cop(q) for 1 < q < ∞. For the survey of Ces`aro and Copson spaces see [AM2]. Structure of the paper. Section 2 consists of the key embeddings between Wiener– Beurling spaces with various parameters and Lebesgue and Besov spaces. In Section 3 we prove α is W B −α for α ∈ R and 1  p, q  ∞. This, in particular, that the K¨othe dual space of W Bp,q p ,q  extends the results on duality of Cesaro space obtained by Bennett. As a corollary, we derive −1/β that (A∗,β ) = W B1,β  , cf. [BLT, Prop. 2] for the case β = 1. Section 4 studies the convolution and product operator for functions from Wiener–Beurling α is an algebra for 1  p  ∞ and α  1 , significantly extending spaces. We claim that W Bp,1 p the results by Belinskii, Liflyand and Trigub [BLT]. In Section 5 two interpolation theorems of Wiener–Beurling spaces are obtained. We prove α 0 , W B α1 ) α that (W Bp,q p,q1 θ,q = W Bp,q for 0 < p, q, q0 , q1  ∞, −∞ < α1 < α0 < ∞ and α = 0 θ 1−θ θ (1 − θ)α0 + θα1 , 0 < θ < 1. In the diagonal case, i.e., 1q = 1−θ q0 + q1 = p0 + p1 , we derive that α α α (W Bp00,q0 , W Bp11,q1 )θ,q = W Bq,q for 0 < p0 , p1  ∞, 0 < q, q0 , q1 < ∞, and α = (1 − θ)α0 + θα1 , 0 < θ < 1. In particular, we obtain the interpolation properties of Cesaro and Copson spaces, investigated recently by Astashkin and Maligranda [AM1, AM2], as well as properties of the Ap,q spaces studied by Lerner and Liflyand [LL]. Recall that the {cm }Ap,q norm is given by   1/q ∞  1  p q/p |c | . m k=0 k+1 k|m|<2k Finally, in Section 6 we obtain Bernstein and Nikol’skii type polynomial inequalities in Wiener–Beurling spaces. Moreover, we study the Fourier series with averages of Fourier coefficients substantially improving the remarkable result by Bennett [Ben, Theorem 20.31] for Ces´ aro and Copson spaces. 2. Embedding properties of Wiener–Beurling spaces 2.1. Embeddings between Wiener–Beurling and Lebesgue spaces. We start with the simple embeddings, which follow from the properties of lp spaces and H¨older’s inequality. Let α ∈ R, ε > 0, 0 < p  ∞, and 0 < q, q1  ∞. Then

4

ERLAN NURSULTANOV AND SERGEY TIKHONOV

(1) (2) (3) (4)

α → W B α W Bp,q p,q+ε , α α+ε , 0 < p  ∞, 0 < q, q1  ∞, W Bp,q → W Bp,q 1 α α , W Bp,q → W Bp+ε,q if −∞ < α1 < α2 < ∞ and α1 + p11 = α2 + p12 , then

W Bpα22,q → W Bpα11,q .

(2.1)

α → Aβ with The latter implies that W Bp,β

1 β

= α + p1 .

Let us now recall Pitt’s inequality for Fourier coefficients [Pi, St]:     q 1/q p 1/p  −β C f W B −β  (|n| + 1) |fn | |x|α f (x) dx q,q

T

n∈Z

provided 1 1 1 , β = α + + − 1  0.  p p q The reverse inequality is given as follows: suppose 1 1 1 β = α − − + 1  0, 1 < p  q < ∞, 0α< , q p q then     q 1/q p 1/p  C  f W B β . |x|−α |f (x)| dx (|n| + 1)β |f n | 1 < p  q < ∞,

0α<

T

p,p

n∈Z

1 1  −p

p In particular, we derive that Lp → W Bp,p

1 1  −p

p for 1 < p  2 and W Bp,p

→ Lp for 2  p < ∞.

However, the latter two embeddings can be strengthened using the results of Kellogg [Ke]. Namely, we have (5) if 1 < p  2, then Lp → W Bp0 ,2 , (6)

if p  2, then

W Bp0 ,2 → Lp .

The next theorem discusses the well-known Wiener’s problem on functions with positive Fourier coefficients. Recall solid if it satisfies the following  thatinθa space of functions X is called  property: For every f = cn e in X, if another function g = dn einθ satisfies |dn |  |cn | for every n, then g is also in X. Theorem 2.1. [ATT] Let X be a space of functions. If Lp → X and X is solid, then Lploc+ → X, where Lploc+

 =

 f : all fˆ (n)  0 and

α W Bp,q

δ −δ

|f |p dx < ∞ for some δ = δ (f ) > 0 .

The space is clearly solid, which, together with embedding (5), implies that (5’) if 1 < p  2, then Lploc+ → W Bp0 ,2 .

WIENER–BEURLING SPACES

5

2.2. Embeddings between Besov and Wiener–Beurling spaces. We define the Besov α , α ∈ R, 0 < p, q  ∞, to be the collection of formal trigonometric series (not space Bp,q  necessarily the Fourier series) f (x) = m∈Z cm eimx such that ⎛ ⎞1/q ∞ ! !q   ! ! α = ⎝ 2αqk ! cm eimx ! ⎠ < ∞. f Bp,q k=0

p

[2k−1 ]|m|<2k −1

It is well known (see, e.g., [Ni]) that for α > 0 and 1 < p < ∞ this definition is equivalent to the classical definitions of the Besov space given in terms of moduli of smoothness or Fourieranalytical decompositions. In light of Nikol’skii’s inequality for trigonometric polynomials it is easy to see the following (Jawerth–Franke-type) embedding: if −∞ < α1 < α2 < ∞, 0 < p2 < p1  ∞, and α1 − p11 = α2 − p12 , then Bpα22,q → Bpα11,q , cf. embedding (4) for Wiener-Beurling spaces. To compare Besov and Wiener–Beurling spaces, we start with the simple application of Parseval’s identity: α α B2,q = W B2,q ,

(2.2)

α ∈ R,

0 < q  ∞. 1/p−1/2

, Let us recall the classical Bernstein theorem and its generalization by Szasz [Sz]: if f ∈ B2,p 1  p  2, then {fn } ∈ lp . In fact, using (2.2) this result trivially follows from the embedding properties of Wiener-Beurling spaces (see (2.1)): 1/p−1/2

B2,p

1/p−1/2

= W B2,p

0 → W Bp,p = Ap .

Further, the Hausdorff–Young and Nikol’skii’s inequalities immediately give ⎧ α+1− 1 ⎪ ⎨ W B∞,q p , 0 < p  1; α Bp,q → W Bpα ,q , 1 < p  2; ⎪ ⎩ α , W B2,q 2 < p  ∞; and α W Bp,q

⎧ α , 0 < p  1; ⎪ ⎨ B∞,q α , B 1 < p  2;  p ,q → 1 1 ⎪ α+ − ⎩ B2,q p 2 , 2 < p  ∞.

Pitt’s inequalities imply the following embeddings α+β α Bp,q → W Br,q ,

1 < p  r  p ,

β=

1 1 + − 1  0, 0 < q  ∞, p r

∀α ∈ R

and α+β β → Br,q , W Bp,q

1 < r  p  r < ∞,

α=1−

1 1 −  0, 0 < q  ∞, p r

∀ β ∈ R.

Sharpness of these embeddings can be seen with the help of functions with lacunary and monotone Fourier coefficients. For trigonometric series with lacunary coefficients, i.e., f (x) =  imx such that c = 0 for |k| = n with n c e m s s+1 /ns  λ > 1, we have k m∈Z f W Bpα ,q  f Bpα ,q , 1

2

α ∈ R,

Now we consider series with monotone coefficients.

0 < p1 , p2 , q  ∞.

6

ERLAN NURSULTANOV AND SERGEY TIKHONOV

Theorem 2.2. Let 1 < p < ∞, α ∈ R, and 0 < q, q1  ∞. Let f (x) = such that ck  cm for |m|  |k|, then



m∈Z cm e

imx

be

α  f B α . f W Bp,q p ,q

Proof. Since the coefficients ck are decreasing, then by the Hardy–Littlewood theorem on monotone Fourier coefficients [Zy, V.2, XII, §6], we have that ⎞1/q ⎛ ∞  ! ! q   ! ! ⎠ 2αk ! f Bpα ,q = ⎝ cm eimx ! ⎛

k=0

[2k−1 ]|m|<2k

k=0

m=1

Lp

⎞q/p ⎞1/q ⎛ k−1 ∞ 2  p  1  ⎜ ⎠ ⎟ m1/p (|c2k−1 +m | + |c−2k−1 −m |)  ⎝ 2αqk ⎝ ⎠ m  

∞ 

(α+1/p)qk

2



|c2k | + |c−2k |

q

1/q ,

k=0

where in the last step we have used the monotonicity of ck . On the other hand, the monotonicity of coefficients implies that ⎛ ⎛ ⎞q/p ⎞1/q ∞  ⎜ αqk ⎝ ⎟ α = ⎝ f W Bp,q 2 |cm |p ⎠ ⎠ k=0

 

∞ 

[2k−1 ]|m|<2k −1

(α+1/p)qk

2



|c2k | + |c−2k |

q

1/q ,

k=0

completing the proof.  Remark 2.1. Analyzing the proof of this theorem, we observe that the same result holds under less restrictive conditions on the coefficients {cn }. In particular, monotonicity can be assumed only on each dyadic block (that is, {cn }2k−1 |n|<2k is decreasing or increasing) or, what is more, monotonicity can be replaced by the general monotonicity, see, e.g., [Ti]. 3. Duality of Wiener-Beurling spaces The problem to describe the dual of the Cesaro space ces(q), which is a particular case of the Wiener-Beurling space, has a long history (see, e.g., [Ben, Sect. 2]). Bennett proved that, for 1 < q < ∞,       (3.1) sup |ck |q < ∞ , ces(q) = {cn } : n kn

  where the K¨othe dual of a space E is given by E  = {dn } : n |cn dn | < ∞ We extend this result to the Wiener-Beurling space. Theorem 3.1. Let α ∈ R and 1  p, q  ∞. We have   α  = W Bp−α W Bp,q  ,q  ,

for all

 {cn } ∈ E .

WIENER–BEURLING SPACES

where f X  := supgX =1

$

7

R f (x)g(x)dx.

In particular, if p = 1, α = 1q − 1, and 1 < q < ∞ we recover (3.1). For the Beurling space A∗,1 (the case p = ∞ and α = q = 1), this result was proved in [BLT, Proposition 2].   Proof. For f = n ξn einx and g = n∈Z cn einx , we have   sup f (x)g(x)dx = sup ξn c n . f (W B α ) := p,q

α =1 R gW Bp,q

α =1 gW Bp,q n∈Z

Then H¨ older’s inequality implies sup f (W B α )  p,q α g

W Bp,q =1



α = f  f W B −α gW Bp,q W B −α . p ,q 

p ,q 

 inx ∈ W B −α , we define g = To prove the reverse, for a given function f = n ξn e p ,q  inx with c e n∈Z n ⎛ ⎞−1/p    cn = dk ξn |ξn |p −1 ⎝ |ξm |p ⎠ for [2k−1 ]  |n| < 2k , k ∈ Z+ , [2k−1 ]|m|<2k

⎛ −αkq 

dk = 2

⎝

⎞ q −1  p



p

|ξm | ⎠

 f W B −α

α gW Bp,q

q

p ,q 

[2k−1 ]|m|<2k

Then we have

− q  .

⎛ ⎛  1/p ⎞q ⎞1/q   (p −1)p ∞ ∞  q 1/q |ξ | k−1 ]|n|<2k   n [2 ⎜ ⎜ αk ⎟ ⎟ 2αk dk = ⎝ ⎝2 dk  1/p ⎠ ⎠ =  p k=0 k=0 [2k−1 ]|n|<2k |ξn | ⎛ ⎛ ⎞q ⎞1/q ⎛ ⎞ q −1  p  − q ∞  q ⎜ ⎜ αk −αkq ⎟ ⎟  p ⎜2 2 ⎟ ⎟ ⎝ ⎠ f  = ⎜ |ξ | = 1. −α m W B ⎝ ⎝ ⎠ ⎠ p ,q  k=0

[2k−1 ]|m|<2k

Thus, f (W B α )  p,q



ξn cn = f W B −α , p ,q 

n∈Z

where the last equality follows by the choice of cn with the help of the Parseval identity.



4. Convolution and product of function from Wiener–Beurling spaces For functions f = f · g as follows:



k ck e

ikx

f ∗g =

and g =

 k



ck dk eikx ,

k

dk eikx , we define their convolution f ∗g and product f ·g =

 k

cν dk−ν eikx .

ν

Holder’s inequality immediately yields the following result for the convolution of functions in the Beurling–Wiener space.

8

ERLAN NURSULTANOV AND SERGEY TIKHONOV

Theorem 4.1. Let α, αi ∈ R and 0 < p, pi , q, qi  ∞ for i = 1, 2. Let 1 1 1 1 1 1 α = α1 + α2 , + , + . = = p p1 p 2 q q1 q2 α α α 1 2 If f ∈ W Bp1 ,q1 and g ∈ W Bp2 ,q2 , then f ∗ g ∈ W Bp,q and α α α  f  f ∗ gW Bp,q W Bp 1,q gW Bp 2,q . 1

1

2

2

This result is closely connected to the multiplier theory of H p spaces; see [Ke, Sl]. The product of functions from the Beurling–Wiener space is treated by the following result. Theorem 4.2. Let 1  p  ∞, α > 0, and α ¯ = max( p1 , α). Then α  f  α ¯ gW B α ¯ . f · gW Bp,1 W Bp,1 p,1   Proof. Let f = k ck eikx and g = k dk eikx . First, we note that ⎛ % %p ⎞1/p ∞ % %   % % α = 2kα ⎝ cν dn−ν % ⎠ f · gW Bp,1 % % % k−1 k

k=0



∞ 

⎛ 2kα ⎝

[2

k −1 2

ν∈Z

]|n|<2

⎛ % %p ⎞1/p ∞ % %  % % cν dn−ν % ⎠ + 2kα ⎝ % % %

n=[2k−1 ] ν∈Z

k=0

k=0

k +1 −2 

% %p ⎞1/p % % % % cν dn−ν % ⎠ =: I1 + I2 . % % %

n=−[2k−1 ] ν∈Z

We will estimate only the first term, the second one can be treated similarly. We have % n %p ⎞1/p ⎛ k −1 %  % ∞ 2 ∞ 2   % % kα ⎝ % I1 = 2 cν dn−ν + cν dn−ν %% ⎠ % % k=0 ν= n n=[2k−1 ] %ν=−∞ 2

% n %p ⎞1/p % n %p ⎞1/p ⎛ k −1 %  % %  % ∞ 2 2 2  % % % % kα ⎝ kα ⎝ % % % ⎠  2 cν dn−ν % + 2 cn−ν dν %% ⎠ =: I11 + I12 . % % % % k=0 k=0 n=[2k−1 ] %ν=−∞ n=[2k−1 ] %ν=−∞ ∞ 

⎛

k −1 2

Changing variable and using Minkowski’s inequality, we derive % %p ⎞1/p ⎛ k −1 % % 2 % % % ⎝ cν dn−ν %% ⎠ % % n=[2k−1 ] %ν n 2 ⎞1/p ⎞1/p ⎛ ⎛ k −1 k −1 2k−2 2 2k−1 −1 2 −1   |cν | ⎝ |dn−ν |p ⎠ + |cν | ⎝ |dn−ν |p ⎠ ν=−∞



2k−2 −1 ν=−∞



 ν∈Z

⎛ |cν | ⎝ ⎛

|cν | ⎝

n=[2k−1 ]

 ξ2k−2



ξ2k−2

ν=2k−2

⎞1/p

|dξ |p ⎠

+

ν=2k−2

⎞1/p

|dξ |p ⎠

2k−1 −1

.

n=2ν

⎛

|cν | ⎝

k

2  ξ=2k−2

⎞1/p |dξ |p ⎠

WIENER–BEURLING SPACES

Thus we have I11 



|cν |

ν∈Z

∞ 

⎛ 2kα ⎝

k=0

9

⎞1/p



|dn |p ⎠

.

n[2k−2 ]

Since, for α > 0, taking into account Hardy’s inequality ∞ ∞ ∞  γ   kα (4.1) 2 Am  2kα Aγk , Am  0, k=0

m=k

we have ∞ 

⎛

ν∈Z

|dn |p ⎠

[2k−2 ]n

and |cν | 

⎞1/p



2kα ⎝

k=0



γ > 0,

k=0

∞  n=0

⎛

k −1 2

⎝

α  gW Bp,1

⎞1/p |cν |p ⎠

k

2 p  f 

n=[2k−1 ]

1 

p W Bp,1

,

we finally have I11  f 

1/p

W Bp,1

α  f  α ¯ gW B α ¯ . gW Bp,1 W Bp,1 p,1

To conclude the proof, we note that I12 and I2 can be estimated similarly. Corollary 4.1. Let 1  p  ∞, α  α is an algebra. i.e., W Bp,1

1 p .



α provided that f, g ∈ W B α , Then f · g ∈ W Bp,1 p,1

When p = ∞ and α = q = 1 this result was obtained in [BLT, Proposition 1]. We also mention the recent paper [Ri], where the author studied the boundedness of the convolution operator from the Ces´aro space ces(q) into ces(q). 5. Interpolation properties of Wiener–Beurling spaces Consider a compatible couple (X0 , X1 ) of Banach spaces X0 and X1 (see [BS, Ch. 3]). Let K(f, t) := K(f, t; X0 , X1 ) :=

inf

f =f0 +f1

(f0 X0 + tf1 X1 ), f ∈ X0 + X1

be the K-functional. The interpolation space (0 < θ < 1) is given by 

(X0 , X1 )θ,q = f ∈ X0 + X1 : f (X0 ,X1 )θ,q

 ∞  dt  1q = (t−θ K(f, t))q <∞ t 0

for 0 < q < ∞, with the usual modification for q = ∞. The main interpolation properties of Cesaro and Copson sequence spaces have been recently studied in [AM1, AM2]: (5.1)

(ces(q0 ), ces(q1 ))θ,q = ces(q),

(5.2)

(cop(q0 ), cop(q1 ))θ,q = cop(q),

1 θ 1−θ + , 1 < q0 < q1  ∞, 0 < θ < 1, = q q0 q1 1 θ 1−θ + , 1  q0 < q1  ∞, 0 < θ < 1. = q q0 q1

The similar result for the weighted Ces´aro space was proved in [LM, Cor.5] for q ∈ [1, ∞].

10

ERLAN NURSULTANOV AND SERGEY TIKHONOV

In [LL], the authors considered the Ap,q spaces with the norm ∞    1 q/p 1/q |cm |p <∞ {cm }Ap,q = k+1 k=0

k|m|<2k

and proved that (5.3)

1 θ 1−θ + , = q q0 q1

(Ap,q0 , Ap,q1 )θ,q = Ap,q ,

0 < θ < 1,

provided 1  q0 < q1  ∞, 1  p  ∞ and 1 θ 1−θ + , 0 < θ < 1, = p p0 p1 spaces are used in various problems of Fourier analysis;

(5.4)

(Ap0 ,p , Ap1 ,p )θ,p = Ap,p = p ,

provided 1  p0 < p1  ∞. The Ap,q see, e.g., the recent monograph [Li].  1/q−1/p It is easy to see that cn Ap,q < ∞ if and only if f ∈ W Bp,q , where f = ck eikx . In  α , f = fact we have that f ∈ W Bp,q ck eikx , if and only if ∞ 

αqk

2

k=0





|cm |

 1/q p q/p



∞  k=0

[2k−1 ]|m|<2k

k αq−1





|cm |p

q/p 1/q

< ∞.

k|m|<2k

We extend (5.1)–(5.4) in the following Theorem 5.1. Let −∞ < α1 < α0 < ∞. (i) If 0 < p, q, q0 , q1  ∞, then α0 α1 α (W Bp,q , W Bp,q ) = W Bp,q , 0 1 θ,q

(5.5)

where α = (1 − θ)α0 + θα1 and 0 < θ < 1. (ii) If 0 < p0 , p1  ∞ and 0 < q, q0 , q1 < ∞, then (5.6)

α (W Bpα00,q0 , W Bpα11,q1 )θ,q = W Bq,q ,

1 θ 1−θ θ 1−θ + = + , = q q0 q1 p0 p1

where α = (1 − θ)α0 + θα1 and 0 < θ < 1. In particular, if α0 = 1/q0 − 1/p and α1 = 1/q1 − 1/p we recover (5.3) and if q = q0 = q1 , α0 = 1/q − 1/p0 , and α1 = 1/q − 1/p1 we recover (5.4) with q in place of p. Note also that (5.6) with q = q0 = q1 = ∞ becomes a well-known result ([BL, Ch. 5]).  α0 , W B α 1 ) ikx ∈ Proof. Let f ∈ (W Bp,q p,q1 θq be such that f = f0 + f1 , where f0 = k dk e 0  α0 , f = ikx ∈ W B α1 , and c = d + ξ . Then for any k ∈ Z , we have W Bp,q 1 + k k k p,q1 k ξk e 0 ⎛ ⎞1/p   1/p   |cm |p ⎠  2(α−α0 )k 2α0 k |dm |p 2αk ⎝ [2k−1 ]|m|<2k

+ 2(α0 −α1 )k 2α1 k





|ξm |p

[2k−1 ]|m|<2k

Thus we derive that ⎛ 2αk ⎝



[2k−1 ]|m|<2k

1/p 

[2k−1 ]|m|<2k

  (α −α1 )k α0 + 2 0 α1 f1 W Bp,∞  2(α−α0 )k f0 W Bp,∞ .

⎞1/p |cm |p ⎠

α0 α1  2(α−α0 )k K(f, 2(α0 −α1 )k ; W Bp,∞ , W Bp,∞ ),

WIENER–BEURLING SPACES

11

which yields  α f W Bp,q



∞  

α0 α1 2(α−α0 )k K(f, 2(α0 −α1 )k ; W Bp,∞ , W Bp,∞ )

k=0

 

∞

(t

−θ

K(f, t))

q dt

0

q

1 q

1 q

t

α0 α1 = f (W Bp,∞ ,W Bp,∞ )θq .

Using now the embedding properties of Beurling–Wiener spaces, we obtain α0 α1 α0 α1 α (W Bp,q , W Bp,q ) → (W Bp,∞ , W Bp,∞ )θ,q → W Bp,q . 0 1 θ,q α and τ = min(q , q , q), r ∈ Z. Define functions To show the reverse embedding, let f ∈ W Bp,q 0 1 f0 and f1 as follows   imx for r ∈ Z+ = {z ∈ Z, z  0}, |m|<2r cm (f )e f0 = 0 for r ∈ / Z+ .

and f1 = f − f0 . Then, letting Δk := Δk (p) :=



[2k−1 ]|m|<2k

|cm |p

1/p

, we obtain

α0 α1 α0 α1 , W Bp,q )  K(f, 2(α0 −α1 )r ; W Bp,τ , W Bp,τ ) K(f, 2(α0 −α1 )r ; W Bp,q 0 1 (α −α1 )r α0 + 2 0 α1 f1 W Bp,τ  f0 W Bp,τ r ∞ 1/τ   1/τ  = 2α0 kτ Δτk + 2(α0 −α1 )r 2α1 kτ Δτk .

(5.7)

k=0

k=r+1

It follows that  f 

α0 α1 (W Bp,q ,W Bp,q ) 0 1 θ,q

 f 

α0 α1 (W Bp,τ ,W Bp,τ )θ,q

=

∞

(t

−θ

K(f, t))

q dt

1 q

t 0 q dt  1q t−θ(α0 −α1 ) K(f, t(α0 −α1 ) )  t 0  1  q dt  ∞  q dt  1q −θ(α0 −α1 ) (α0 −α1 ) −θ(α −α ) (α −α ) 0 1 0 1 α1 t t  t f W Bp,τ K(f, t ) + t t 0 1 ∞ 1 q q  (α1 −α0 )k (α0 −α1 )k α  f W Bp,τ1 + K(f, 2 ) . 2 

∞

k=0

α1  f W B α for α1 < α and estimate (5.7), we arrive at Using f W Bp,τ p,q α0 f (W Bp,q

0

α

1 )) ,W Bp,q 1 θq

+

α  f W Bp,q

∞  r=0

2(α−α0 )rq

r  

2α0 kτ Δτk

1/τ

+ 2(α0 −α1 )r

k=0

∞  

2α1 kτ Δτk

1/τ q  1 q

.

k=r+1

Finally, taking into account α1 < α < α0 and using Hardy’s inequalities (1.2) and (4.1), we derive α0 f (W Bp,q

0

α

1 ) ,W Bp,q 1 θ,q

α .  f W Bp,q

12

ERLAN NURSULTANOV AND SERGEY TIKHONOV

To obtain (5.6), we first apply the power theorem for interpolation spaces (see [BL, Th. 3.11.6]), we obtain q    (5.8) (W Bpα00,q0 , W Bpα11,q1 )θ,q = (W Bpα00,q0 )q0 , (W Bpα11,q1 )q1 η,1 , where

qθ q = (1 − η)q0 + ηq1 , 0 < η < 1. η= , q1  Observe that for f (x) = m∈Z cm eimx we have   f0 qW0 B α0 + tf1 qW1 B α1 inf K(t, f ; (W Bpα00,q0 )q0 , (W Bpα11,q1 )q1 ) = =

f =f0 +f1 ∞  k=0

k+1



p0 ,q0

inf

bk =b0,k +b1,k

k+1

p1 ,q1

2α0 q0 k b0,k qlp0 + t2α1 q1 k b1,k qlp1 0

1

 ,

k+1

2 −1 −1 −1 0 2 1 2 0 1 where bk = {cm }m=2 k , b0,k = {cm }m=2k , b1,k = {cm }m=2k , cm = cm + cm , m ∈ Z. This together with (5.8) implies

f q(W B α0

α1 p0 ,q0 ,W Bp1 ,q1 )θ,q

 f ((W Bpα0,q )q0 ,(W Bpα1,q )q1 ) 0 0 1 1 η,1  ∞ dt = t−η K(t, f ; (W Bpα00,q0 )q0 , (W Bpα11,q1 )q1 ) t 0  ∞ ∞  dt 2α0 q0 k t−η K(t2(α1 q1 −α0 q0 )k , bk ; (lp0 )q0 , (lp1 )q1 ) . = t 0 k=0

Changing variables (note that α0 q0 +(α1 q1 −α0 q0 )η = αq) and applying again the power theorem, we obtain ∞ ∞   f q(W B α0 ,W B α1 ) = 2αqk bk ((lp )q0 ,(lp )q1 )  2αqk bk q l ,l ( p0 p1 )θ,q p0 ,q0 p1 ,q1 θ,q 0 1 η,1 k=0



∞  k=0

k=0

2αqk bk qlq ,

1 θ 1−θ + . = q p0 p1 

Theorem 5.1 immediately implies the following Corollary 5.1. Let 0 < q  ∞, β > 0. We have   α1 β α , (i) W Bβ,q , A = W Bβ,q α = (1 − θ)α1 , 0 < θ < 1, α1 = 0. 1 θ,q  ∗β  α1 α , (ii) A , W B∞,q = W B∞,q α = 1−θ 0 < θ < 1, α1 = 1/β. β + θα1 , θ,q 1   ∗β β 1 1−θ θ , 0 < θ < 1, 0 < β0 < β1 < ∞. (iii) A 0 , A∗β1 θ,q = W B∞,q β = β0 + β1 , In particular,   1 θ 1−θ = A∗β , + , 0 < θ < 1. = A∗β0 , A∗β1 β β0 β1 θ,β 1  −1 q cn einx ∈ W B1,r if (iv) Let 0 < q0 , q1 < ∞, q0 = q1 , and 0 < r  ∞. Then f (x) = and only if 1 θ 1−θ {cn } ∈ (cop(q0 ), cop(q1 ))θ,r , (5.9) + . = q q0 q1

WIENER–BEURLING SPACES

(v)

13

In particular, taking r = q, we arrive at (5.2). Similarly, for 1 < q0 , q1 < ∞, q0 = q1 , and 0 < r  ∞, (5.9) can be equivalently replaced by 1 θ 1−θ + . = q q0 q1

{cn } ∈ (ces(q0 ), ces(q1 ))θ,r , In particular, taking r = q, we arrive at (5.1).

6. Final remarks (α)

1. Note that the classical Bernstein inequality TN p  N α TN p for trigonometric poly nomials TN (x) = |k|N ck eikx has the usual form in the Wiener-Beurling spaces, namely, (α)

α α  N TN W B α . TN W Bp,q p,q

2. Let us also discuss the Nikolskii inequality for trigonometric polynomials. Its analogue for the Wiener-Beurling spaces reads as follows: If 0 < r  p  ∞ and α + 1/r − β − 1/p  0, then (α+1/r)−(β+1/p) α  N TN W Br,q TN W B β .

(6.1)

p,q

Indeed, let

2m−1

α TN W Br,q

N <

2m

and γ = α + 1/r − 1/p. Taking into account (2.1), we derive that

γ  TN W Bp,q



 m−1 

2γqk



k=0



|cn |p

q/p

+ 2γqm





|cn |p

q/p 1/q

[2m−1 ]|n|N

[2k−1 ]|n|<2k

 N (α+1/r)−(β+1/p) TN W B β . p,q

Note that if the first integrability parameters are the same but the second ones are different, then the Nikolskii inequality has a different form. Namely, if α ∈ R, 0 < p < ∞, 0 < q1 < q  ∞, then (6.2)

1

α TN W Bp,q  ([ln N ] + 1) q 1

− q1

1

α . TN W Bp,q

This follows from H¨ older’s inequality. Thus, in view of (6.1) and (6.2), the Nikolskii inequalities in the Wiener–Beurling spaces are similar to those in the Lorentz spaces, cf. [NT]. 3. Now  we study the Fourier series with averages of Fourier coefficients. Given a series f (x) = k∈Z ck eikx , define Hf (x) =



chk eikx ,

chk

k∈Z

and Bf (x) =

 k∈Z

cbk eikx ,

|k|  1 = |cm | 2|k| + 1 m=−|k|

cbk =

∞  |cn | + |c−n | < ∞. 2n

n=|k|

This topic was originated by Hardy [Ha], who proved that if f ∈ Lp , 1 < p < ∞, then Hf ∈ Lp . See also [An, Be]; for the recent results see [DNT].

14

ERLAN NURSULTANOV AND SERGEY TIKHONOV

We will see that Wiener–Beurling spaces have the remarkable property that series with Hardy-Ces´ aro and Copson averages belong to Wiener–Beurling spaces if and only if the original series belong to similar spaces with different values of parameters. Theorem 6.1. If 0 < p, q  ∞, then (6.3)

α+ p1 −1

α ⇐⇒ f ∈ W B1,q Hf ∈ W Bp,q

if and only if

α<1−

if and only if

α+

1 p

and (6.4)

α+ p1 −1

α Bf ∈ W Bp,q ⇐⇒ f ∈ W B1,q

1 > 0. p

Remark 6.1. In a particular case p = 1 and α = 1q − 1 < 0 we arrive at the known results for Ces´aro and Copson sequence spaces by Bennett [Ben, Theorem 20.31]:

and

{cm } ∈ ces(q) ⇐⇒ {chm } ∈ ces(q) ⇐⇒ {cbm } ∈ ces(q),

1
{cm } ∈ cop(q) ⇐⇒ {chm } ∈ cop(q) ⇐⇒ {cbm } ∈ cop(q),

1 < q < ∞.

Proof. To show (6.3), we observe that α Hf W Bp,q



∞ 

kαq

2

k=0



∞ 

k+1 m  2 1  p q/p 1/q |cs | m k

|s|=0

m=2

kq(α+ p1 −1)



2

k=0

The latter is equivalent to f 

1 −1 α+ p

W B1,q

this condition, assume that β = α +

1 p

ck =

for α +

2k 

|cs |

q 1/q

.

|s|=0 1 p

− 1 < 0, cf. (1.3). To verify the sharpness of

− 1  0 and define 1 , (|k| + 1)1+β+ε

ε > 0. 

ck eikx .  The proof of (6.4) is similar. The sharpness can be seen considering f = ck eikx with 1 1 1 α +  0, 0<ε , ck = 1+ε 1 , α+ p q (ln(|k| + 1)) q (|k| + 1) p

α = ∞ for f = It is routine to show that f W B β < ∞ and Hf W Bp,q p,q

for which one has f 

1 −1 α+ p

W B1,q

α = ∞. < ∞ but Bf W Bp,q



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