J Fourier Anal ApplDOI 10.1007/s00041-014-9339-0

On Fourier Re-Expansions

Elijah Liflyand

Received: 2 February 2014© Springer Science+Business Media New York 2014

Abstract An old problem on absolute convergence of the re-expansion in the sine(cosine) Fourier series of an absolutely convergent cosine (sine) Fourier series isextended to the non-periodic case (Fourier transforms). Necessary and sufficient con-ditions are given as relations between the Fourier transforms and their Hilbert trans-forms. Sufficient conditions for integrability of the Hilbert transform are obtained.

Keywords Fourier transform · Integrability · Hilbert transform · Hardy space

Mathematics Subject Classification Primary 42A38 · Secondary 42A50

1 Introduction

In 50-s (see, e.g., [5] or in more detail [6, Chapters II and VI]), the following problemin Fourier Analysis attracted much attention:

Let {ak}∞k=0 be the sequence of the Fourier coefficients of the absolutely convergentsine (cosine) Fourier series of a function f : T = [−π, π) → C, that is

∑ |ak | <

∞. Under which conditions on {ak} will the re-expansion of f (t) ( f (t) − f (0),respectively) in the cosine (sine) Fourier series also be absolutely convergent?

The obtained condition is quite simple and is the same in both cases:

Communicated by Hans G. Feichtinger.

E. Liflyand (B)Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israele-mail: liflyand@gmail.com

J Fourier Anal Appl

∞∑

k=1

|ak | ln(k + 1) < ∞. (1)

In this paper we study a similar problem of the integrability of the re-expansionfor Fourier transforms of functions defined on R+ = [0,∞). We give necessary andsufficient conditions in terms of belonging of the sine or cosine Fourier transformto a certain Hardy space. The proof differs from that for Fourier series. The reasonis that an attempt to repeat that proof comes across a relation equivalent to Lusin’sconjecture on the almost everywhere convergence of the Fourier series. Long beforeCarleson’s solution, it was known that the almost everywhere convergence fails tohold for all L1 functions (see a nice discussion of these issues in [2]). Because of thesecircumstances, we had to find a different proof. A simple observation of old proofs ofthe problem in question shows that similar necessary and sufficient conditions holdtrue for series as well, while (1) is just a sufficient condition for belonging to discreteHardy type spaces. On account of this, we give a continuous analog of such condition.The difference between two cases can partially be seen from the fact that a directanalog of (1) is not enough and is accompanied by an additional condition.

2 Main Results

Let

Fc(x) =∞∫

0

f (t) cos xt dt

be the cosine Fourier transform of f and

Fs(x) =∞∫

0

f (x) sin xt dt

be the sine Fourier transform of f , each understood in certain sense.Let

∞∫

0

|Fc(x)| dx < ∞, (2)

and hence

f (t) = 1

π

∞∫

0

Fc(x) cos t x dx, (3)

J Fourier Anal Appl

or, alternatively,

∞∫

0

|Fs(x)| dx < ∞ (4)

and hence

f (t) = 1

π

∞∫

0

Fs(x) sin t x dx . (5)

1) Under which (additional) conditions on Fc we get (4), or, in the alternative case,2) under which (additional) conditions on Fs we get (2)?

The answer is given by the following theorem. To formulate it we turn to the Hilberttransform of an integrable function g

Hg(x) = 1

π

∫

R

g(t)

x − tdt, (6)

where the integral is understood in the improper (principal value) sense, aslim

δ→0+∫|t−x |>δ

. It is not necessarily integrable, and when it is, we say that g is in

the (real) Hardy space H1(R). If g ∈ H1(R), then

∫

R

g(t) dt = 0. (7)

It was apparently first mentioned in [8].When we start with the sine transform, (7) for it holds automatically. On the same

occasion for the cosine transform, we need

∞∫

0

Fc(t) dt = 0 (8)

to ensure (7).

Theorem 1 In order than the re-expansion Fs of f with the integrable cosine Fouriertransform Fc be integrable, it is necessary and sufficient that its Hilbert transformHFc(x) be integrable and (8) hold.

Similarly, in order than the re-expansion Fc of f with the integrable sine Fouriertransform Fs be integrable, it is necessary and sufficient that its Hilbert transformHFs(x) be integrable.

J Fourier Anal Appl

3 Proof of Theorem 1

Let (2) hold true. Then we can rewrite

Fs(x) =∞∫

0

[1

π

∞∫

0

Fc(u) cos tu du

]

sin xt dt. (9)

If one expects the integrability of Fs , then the inner integral should vanish at the origin.This follows from (8). The right-hand side of (9) can be understood in the (C, 1) senseas

1

πlim

N→∞

N∫

0

(

1 − t

N

) ∞∫

0

Fc(u) cos tu du sin xt dt.

In virtue of (2) we can change the order of integration:

1

πlim

N→∞

∞∫

0

Fc(u)

N∫

0

(

1 − t

N

)

cos tu sin xt dt du

= 1

πlim

N→∞

∞∫

0

Fc(u)1

2

N∫

0

(

1 − t

N

) [

sin(u + x)t − sin(u − x)t

]

dt du.

We now need the next simple formula

N∫

0

(

1 − t

N

)

sin At dt = 1

A− sin N A

N A2 . (10)

Applying it yields

Fs(x) = 1

πlim

N→∞

∞∫

0

Fc(u)

[1

u + x− sin(u + x)N

N (u + x)2

]

du

− 1

πlim

N→∞

∞∫

0

Fc(u)

[1

u − x− sin(u − x)N

N (u − x)2

]

du = I1 + I2. (11)

Let us begin with I2. Substituting u − x = t, we obtain

I2 = − 1

πlim

N→∞

∞∫

−x

Fc(x + t)

[1

t− sin Nt

Nt2

]

dt.

J Fourier Anal Appl

For I1, we first substitute u = −v. Thus

I1 = 1

πlim

N→∞

0∫

−∞Fc(−v)

[1

−v + x− sin(−v + x)N

N (x − v)2

]

dv

= − 1

πlim

N→∞

0∫

−∞Fc(−v)

[1

v − x− sin(v − x)N

N (v − x)2

]

dv

= − 1

πlim

N→∞

0∫

−∞Fc(v)

[1

v − x− sin(v − x)N

N (v − x)2

]

dv.

The last equality follows from the evenness of Fc. Substituting v − x = t, we obtain

I1 = − 1

πlim

N→∞

−x∫

−∞Fc(x + t)

[1

t− sin Nt

Nt2

]

dt.

Therefore,

Fs(x) = − 1

πlim

N→∞

∞∫

−∞Fc(x + t)

[1

t− sin Nt

Nt2

]

dt.

We are now in a position to apply the following result (see [20, Vol.II, Ch.XVI,Th. 1.22]; even more general result can be found in [18, Th.107]).

Theorem A If| f (t)|1 + |t | is integrable on R, then the (C, 1) means

− 1

π

∞∫

−∞f (x + t)

[1

t− sin Nt

Nt2

]

dt

converge to the Hilbert transform H f (x) almost everywhere as N → ∞.

It follows from Theorem A that for almost all x

Fs(x) = HFc(x). (12)

We remark that any integrable function satisfies the assumption of Theorem A and (8)is necessary for the integrability of the Hilbert transform.

Now, let (4) holds true. Then we can rewrite

Fc(x) =∞∫

0

[1

π

∞∫

0

Fs(u) sin tu du

]

cos xt dt. (13)

J Fourier Anal Appl

The right-hand side can be understood in the (C, 1) sense as

1

πlim

N→∞

N∫

0

(

1 − t

N

) ∞∫

0

Fs(u) sin tu du cos xt dt.

In virtue of (4) we can change the order of integration:

1

πlim

N→∞

∞∫

0

Fs(u)

N∫

0

(

1 − t

N

)

sin tu cos xt dt du

= 1

πlim

N→∞

∞∫

0

Fs(u)1

2

N∫

0

(

1 − t

N

)

[sin(u + x)t + sin(u − x)t] dt du.

Applying (10), we get

Fs(x) = 1

πlim

N→∞

∞∫

0

Fs(u)

[1

u + x− sin(u + x)N

N (u + x)2

]

du

+ 1

πlim

N→∞

∞∫

0

Fs(u)

[1

u − x− sin(u − x)N

N (u − x)2

]

du = J1 + J2. (14)

Let us begin with J2. Substituting u − x = t, we obtain

J2 = 1

πlim

N→∞

∞∫

−x

Fs(x + t)

[1

t− sin Nt

Nt2

]

dt.

Treating J1 as I1 above, we get

J1 = − 1

πlim

N→∞

0∫

−∞Fs(−v)

[1

v − x− sin(v − x)N

N (v − x)2

]

dv

= 1

πlim

N→∞

0∫

−∞Fs(v)

[1

v − x− sin(v − x)N

N (v − x)2

]

dv.

The last equality follows from the oddness of Fs . Substituting v − x = t, we obtain

J1 = 1

πlim

N→∞

−x∫

−∞Fs(x + t)

[1

t− sin Nt

Nt2

]

dt.

J Fourier Anal Appl

Therefore,

Fc(x) = 1

πlim

N→∞

∞∫

−∞Fs(x + t)

[1

t− sin Nt

Nt2

]

dt.

Finally, it follows from Theorem A that for almost all x ,

Fc(x) = −HFs(x). (15)

This completes the proof. ��Let us comment on the obtained results. In fact, the proof of Theorem 1 shows

that more general results than stated are obtained. Indeed, formulas (12) and (15) aremore informative than the assertion of Theorem 1. To be precise, such formulas areknown, see [7, (5.42) and (5.43)]. However, the situation is much more delicate. Theseformulas are proved in [7] for square integrable functions by applying the Riemann-Lebesgue lemma in an appropriate place (5.44). But in [7, §6.19] more details aregiven (see also [2]) and it is shown that the possibility to apply the Riemann-Lebesguelemma in that argument is equivalent to (Carleson’s solution of) Lusin’s conjecture. Inour L1 setting this is by no means applicable. And, indeed, our proof is different andrests on less restrictive Theorem A. This well agrees with what E.M. Dyn’kin wrotein his noted survey on singular integrals [2]: “In fact, the theory of singular integralsis a technical subject where ideas cannot be separated from the techniques.”

4 Sufficient Conditions

Analyzing the proof in [5], one can see that in fact their results are similar to ours,that is, can also be given in terms of the (discrete) Hilbert transform. In that case (1)is merely a sufficient condition for the summability of the discrete Hilbert transform.

A direct analog of (1) for functions cannot be the only sufficient condition forthe integrability of the Hilbert transform. Indeed, a known counter-example of theindicator function of an interval works here as well: of course, it stands up to themultiplication by logarithm.

Thus, we are going to give sufficient conditions for the integrability of the Hilberttransforms in the spirit of those in [5] for sequences supplied by some additionalproperties.

4.1 General Conditions

First of all, examining integrability of the Hilbert transform, one can test the integralover, say, |t | ≥ 3

2 |x |. Indeed, for x > 0, we have

∞∫

0

∣∣∣∣

∞∫

32 x

g(t)

x − tdt

∣∣∣∣ dx

J Fourier Anal Appl

≤∞∫

0

|g(t)|2t3∫

0

dx

x − tdt = ln 3

∞∫

0

|g(t)| dt.

The rest is estimated in a similar manner.Further, since

a∫

−a

dt

x − t= 0 (16)

for any a > 0 when the integral is understood in the principal value sense, we canalways consider

a∫

−a

g(t) − g(x)

x − tdt

instead of the Hilbert transform truncated to [−a, a].When in the definition of the Hilbert transform (6) the function g is odd, we will

denote this transform by Ho, and it is equal to

Hog(x) = 2

π

∞∫

0

tg(t)

x2 − t2 dt. (17)

When g is even its Hilbert transform He can be rewritten as

Heg(x) = 2

π

∞∫

0

xg(t)

x2 − t2 dt. (18)

Of course, both integrals should be understood in the principal value sense (see, e.g.,[7, Ch.4, §4.2]).

Since the functions Fc and Fs are even and odd, respectively, their Hilbert transformsin (12) and (15) can be represented as

He Fc(x) = 2

π

∞∫

0

x Fc(t)

x2 − t2 dt. (19)

and

Ho Fs(x) = 2

π

∞∫

0

t Fs(t)

x2 − t2 dt. (20)

J Fourier Anal Appl

These may be useful in some applications.In fact, the most known condition is the following. If g is of compact support,

a classical Zygmund L log L condition (see, e.g., [20]) ensures the integrability ofthe Hilbert transform. More precisely, the condition is the integrability of g log+ |g|,where the log+ |g| notation means log |g| when |g| > 1 and 0 otherwise. As E.M.Stein has shown in [15], this condition is necessary on the intervals where the functionis positive.

Let us mention a simple condition given in [16, Ch.2, Ex.21]: if a function satisfies

(7) and behaves as1

(1 + |x |)2 , then its Hilbert transform is integrable.

We will prove the following result, less restrictive in certain respects than thoseabove. A different result of such type is recently given in [11].

Theorem A Let g be an integrable function on R which satisfies conditions (7),

∫

|x |≥ 12

|g(x)| log 3|x | dx (21)

and

∫

R

12 min(|x |,1)∫

− 12 min(|x |,1)

∣∣∣∣g(x + t) − g(x)

t

∣∣∣∣ dt dx . (22)

Then g ∈ H1(R).

Since each function can be represented as the sum of its even and odd parts, we willprove Theorem A separately for odd and even functions. Thus, from now on we canconsider g to be defined on R+ and analyze either (17) or (18) rather than the generalHilbert transform.

4.2 Odd Functions

Surprisingly, there exist much more conditions for the integrability of the Hilberttransform of an odd function than for an even one. One of the explanations is that justthe odd Hilbert transform is strongly involved in the problems of the integrability ofthe Fourier transform (see, e.g., [9]) and has been singled out for deeper study.

Though an odd function always satisfies (7), not every odd integrable functionbelongs to H1(R), for counterexamples, see [12] and [10]. Paley-Wiener’s theorem(see [14]; for an alternative proof and discussion, see Zygmund’s paper [19]) assertsthat if g ∈ L1(R) is an odd and monotone decreasing on R+ function, then Hg ∈ L1,i.e., g is in H1(R). Recently, in [13, Thm.6.1], this theorem has been extended to aclass of functions more general than monotone ones. However, it is doubtful that theseresults are really practical in our situation.

J Fourier Anal Appl

Back to Theorem A, we can consider

2

π

3x2∫

x2

tg(t)

x2 − t2 dt

instead of (17). Indeed, the possibility of restricting to that upper limit has been justifiedabove. Similarly,

∞∫

0

∣∣∣∣

x2∫

0

tg(t)

x2 − t2 dt

∣∣∣∣ dx ≤

∞∫

0

|g(t)|t∞∫

2t

dx

x2 − t2 dt

≤ 2

3

∞∫

0

|g(t)| dt. (23)

Now, like in (16) and further, we have

1∫

0

∣∣∣∣

3x2∫

x2

tg(t)

x2 − t2 dt

∣∣∣∣ dx ≤

1∫

0

∣∣∣∣

3x2∫

x2

t[g(t) − g(x)]x2 − t2 dt

∣∣∣∣ dx + O

⎛

⎝

∞∫

0

|g(t)| dt

⎞

⎠

≤1∫

0

x2∫

− x2

|g(x + t) − g(x)||t | dt dx + O

⎛

⎝

∞∫

0

|g(t)| dt

⎞

⎠ . (24)

When x ≥ 1 we first estimate

∞∫

1

∣∣∣∣

x− 12∫

x2

tg(t)

x2 − t2 dt

∣∣∣∣ dx ≤

∞∫

12

t |g(t)|2t∫

t+ 12

dx

x2 − t2 dt

≤ C

∞∫

12

|g(t)| ln 3t dt

and

∞∫

1

∣∣∣∣

3x2∫

x+ 12

tg(t)

x2 − t2 dt

∣∣∣∣ dx ≤

∞∫

32

t |g(t)|t− 1

2∫

2t3

dx

t2 − x2 dt

J Fourier Anal Appl

≤ C

∞∫

12

|g(t)| ln 3t dt.

These two bounds lead to the logarithmic condition (21). The remained integral

∞∫

1

∣∣∣∣

x− 12∫

x+ 12

tg(t)

x2 − t2 dt

∣∣∣∣ dx

is estimated exactly like that in (24). Applying (16), we obtain

∞∫

1

∣∣∣∣

x+ 12∫

x− 12

tg(t)

x2 − t2 dt

∣∣∣∣ dx ≤

∞∫

1

∣∣∣∣

x+ 12∫

x− 12

t[g(t) − g(x)]x2 − t2 dt

∣∣∣∣ dx + O

⎛

⎝

∞∫

0

|g(t)| dt

⎞

⎠

≤∞∫

1

12∫

− 12

|g(x + t) − g(x)||t | dt dx + O

⎛

⎝

∞∫

0

|g(t)| dt

⎞

⎠ .

(25)

Combining all the obtained estimates, we arrive at the required result. ��

4.3 Even Functions

While an odd function always satisfies (7), in the case of even functions the situationis more delicate: the function must satisfy (7) already on the half-axis, like (8). Withthis in hand, the proof goes along the same lines as that for odd functions. The onlyproblem is that an estimate like (23) does not follow immediately from the formula(18). However, using the above remark on the cancelation property for g on the half-axis, we can rewrite (18) as

Heg(x) = 2

π

∞∫

0

g(t)

[x

x2 − t2 − 2

x

]

dt. (26)

Now,

∞∫

0

∣∣∣∣

x2∫

0

t2g(t)

x(x2 − t2)dt

∣∣∣∣ dx ≤

∞∫

0

|g(t)|t2

∞∫

2t

dx

x(x2 − t2)dt

J Fourier Anal Appl

≤ 1

6

∞∫

0

|g(t)| dt. (27)

This additional term 2x does not affect the other estimates of the previous subsection.

The proof is complete. ��

5 Concluding Remarks

First of all, the relations (12) and (15) are of interest by their own.The assertions of Theorem 1 can be reformulated in terms of Hardy spaces: belong-

ing of Fc (Fs) to the real Hardy space H1(R) ensures the integrability of Fs (Fc).The problem of sharpness is simple in this case: any known counterexample of an

integrable function with non-integrable Hilbert transform works perfectly. For exam-ple, let Fc(x) = 1

1+x2 , the Fourier transform of f (t) = e−|t |. Surely, Fc ∈ L1(R).

However, f cannot be re-expanded in the integrable sine Fourier transform, sinceHFc(x) = x

1+x2 ∈ L1(R). That this is true, one can see from the fact that the oddextension of this Fc from the right half-axis to the whole R is not continuous at zero,or, equivalently, this Fc does not satisfy (8).

More can be said about odd functions. Certain convenient conditions for belongingof such functions to H1(R) are known for quite a long time. They are function (Fouriertransform) analogs of important sufficient sequence conditions for the integrability oftrigonometric series (see, e.g., [17] and [3]) and can be found, for example, in [9] andin [4]. In fact, many of these subspaces first appeared in [1]. For 1 < q ≤ ∞, set

‖g‖Aq =∞∫

0

⎛

⎜⎝

1

u

∫

u≤|t |≤2u

|g(t)|qdt

⎞

⎟⎠

1q

du,

with a standard modification when q = ∞. In other words, belonging of g to one ofthe spaces Aq ensures the integrability of the odd Hilbert transform of g.

Acknowledgments The author’s attention to this problem was brought by R.M. Trigub. Thanks to himas well as to A. Iosevich and J.-P. Kahane for stimulating discussions.

References

1. Borwein, D.: Linear functionals connected with strong Cesáro summability. J. Lond. Math. Soc. 40,628–634 (1965)

2. Dyn’kin, E.M.: Dyn’kin, Methods of the Theory of Singular Integrals: Hilbert Transform and Calderón-Zygmund Theory. In: Khavin, V.P., Nikol’skij, N.K. (eds.) Commutative Harmonic Analysis I, pp.167–259. Springer, Berlin (1991)

3. Fomin, G.A.: A class of trigonometric series, Mat. Zametki 23 (1978), 213–222 (Russian). - Englishtransl. Math. Notes 23, 117–123 (1978)

4. Giang, D.V., Móricz, F.: Lebesgue integrability of double Fourier transforms. Acta Sci. Math. (Szeged)58, 299–328 (1993)

J Fourier Anal Appl

5. Izumi, S.I., Tsuchikura, T.: Absolute convergence of trigonometric expansions. Tôhoku Math. J. 7,243–251 (1955)

6. Kahane, J.-P.: Séries de Fourier absolument convergentes. Springer, Berlin (1970)7. F.W. King, Hilbert transforms, Vol. 1, Enc. Math/ Appl., Cambridge Univ. Press, Cambridge (2009).8. Kober, H.: A note on Hilberts operator. Bull. Amer. Math. Soc. 48(1), 421–426 (1942)9. Liflyand, E.: Fourier transform of functions from certain classes. Anal. Math. 19, 151–168 (1993)

10. Liflyand, E.: Fourier transform versus Hilbert transform. Ukr. Math. Bull. 9, 209–218 (2012)11. E. Liflyand, Interaction between the Fourier transform and the Hilbert transform, Acta et Commenta-

tiones Universitatis Tartuensis de Mathematica, 2014.12. Liflyand, E., Tikhonov, S.: C.R. Acad. Sci. Paris Ser. I. Weighted Paley-Wiener theorem on the Hilbert

transform 348, 1253–1258 (2010)13. Liflyand, E., Tikhonov, S.: A concept of general monotonicity and applications. Math. Nachrichten

284, 1083–1098 (2011)14. Paley, R.E.A.C., Wiener, N.: Notes on the theory and application of Fourier transform, Note II. Trans.

Amer. Math. Soc. 35, 354–355 (1933)15. Stein, E.M.: Note on the class L log L . Studia Math. XXXII, 305–310 (1969)16. Stein, E.M., Shakarchi, R.: Functional Analysis: Introduction to Further Topics in Analysis. Princeton

Univ. Press, Princeton and Oxford (2011)17. Telyakovskii, S.A.: On a Sufficient Condition of Sidon for the Integrability of Trigonometric Series,

(Mat. Zametki 14 (1973), 317–328 (Russian)). - English transl. Math. Notes 14, 742–748 (1973)18. Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford (1937)19. Zygmund, A.: Some points in the theory of trigonometric and power series. Trans. Amer. Math. Soc.

36, 586–617 (1934)20. A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some

additions, Cambridge University Press, London, 1968.