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Georgian Mathematical Journal Volume 16 (2009), Number 3, 553–559

NECESSARY CONDITIONS FOR INTEGRABILITY OF THE FOURIER TRANSFORM ELIJAH LIFLYAND

Dedicated to the memory of L. Zhizhiashvili Abstract. We prove the necessary conditions for the integrability of the Fourier transform. The result is a generalization, on one hand, of the well known necessary condition for absolutely convergent Fourier series and, on the other hand, of an earlier multidimensional result of Trigub. 2000 Mathematics Subject Classification: Primary 42B10; Secondary 42B05. Key words and phrases: Integrable Fourier transform, Bessel function, radial part, fractional integral/derivative.

1. Introduction It is well known that the only necessary condition for the Fourier transform of an integrable function is that the Fourier transform must be uniformly continuous and vanishing at infinity. No a priori condition on the rate of vanishing at infinity can be imposed. However, if the Fourier transform is, in turn, integrable, more can be said. To get a flavor of functions with the integrable Fourier transform, we note that R ∞ such a function necessarily possesses a certain smoothness. Let F (x) = f (t) cos xt dt be the cosine Fourier transform of f. If F (x) is integrable on 0 R+ , then the integrals Zx/2

f (x + t) − f (x − t) dt t

δ

are uniformly bounded. For the well known prototype for Fourier series see [5, Ch.II, §10], while in [10, 3.5.5] even a subtler result of this type is given. Indeed, expressing, say, f (x + t) and f (x − t) via the Fourier inversion, we obtain ¯ ¯ x/2 ¯ ¯Z ¯ f (x + t) − f (x − t) ¯ ¯ dt¯¯ ¯ t ¯ ¯ δ ¯ ¯ x/2 ¯ ¯Z Z∞ ¯ ¯ −1 −1 ¯ F (u)[cos u(x + t) − cos u(x − t)] du dt¯¯ = π ¯ t ¯ ¯ δ

0

c Heldermann Verlag www.heldermann.de ISSN 1072-947X / $8.00 / °

554

E. LIFLYAND

Z∞ ≤ 2π −1 0

¯ ¯ x/2 ¯ ¯Z ¯ sin ut ¯ dt¯¯ du. |F (u)| ¯¯ t ¯ ¯ δ

The last integral on the right is uniformly bounded. The result and the proof for the sine and general Fourier transform are the same. The main goal of this paper is to obtain a multidimensional generalization of this necessary condition. Though there exists such a generalization in [9, Th.3], in our version we deal with all the derivatives, not only usual ones of integer order. This suggests a unified approach regardless of dimension and treats certain cases not covered in [9]. We will denote by γ absolute constants while by γ with subscripts constants depending on parameters in the subscript. We are not interested in their explicit form and hence sometimes denote different constants by the same symbols. 2. Preliminaries First of all we need a version of fractional integration/derivation. For 0 < δ < 1 and a locally integrable function g on (0, ∞) define the fractional (Weyl type) integral of order δ by ( RA 1 g(r)(r − t)δ−1 dr, 0 < t < A, WAδ (g)(t) = Γ(δ) t 0, t ≥ A, and, following Cossar [3], a fractional Weyl derivative of order α by µ ¶ d 1−α (α) g (t) = lim − WA (g)(t) A→∞ dt when 0 < α < 1 and by

µ g

(α)

(t) =

d dt

¶p g (δ) (t)

when α = p + δ with p = 1, 2, . . . , and 0 < δ < 1. In the problems of integrability of the Fourier transform the following T transform of a function h(u) defined on (0, ∞) is of importance Zt/2 T h(t) =

h(t + u) − h(t − u) du, u

(1)

0

where the integral is understood in improper sense, i.e., as lim

R

δ→0+ δ

. In [4] it

is called the Telyakovskii transform. The reason is that in an important asymptotic result for the Fourier transform [6] (cf. [4]) it is used to generalize Telyakovskii’s result for trigonometric series. It is clear that the T -transform should be related to the Hilbert transform (see, e.g., [12]); this is obtained and discussed in [6] and later on in, e.g., [4], [7], etc.

INTEGRABILITY OF THE FOURIER TRANSFORM

555

We denote by f0 (t) the radial part of a function f, more precisely, the average of f over the sphere of radius t > 0 and center at the origin Z 1 f0 (t) = f (tx) dθ, (2) σn Sn−1

where Sn−1 is the unit sphere in the n-dimensional Euclidean space of area 2π n/2 2π n/2 σn = Γ(n/2) , the surface of the unit ball B n of volume ωn = nΓ(n/2) . 3. Main Result We are now in a position to formulate and prove the main result. Theorem 1. Let f ∈ A(Rn ), i.e., Z f (x) = g(u) eixu du,

(3)

Rn

with g ∈ L1 (Rn ). Then the radial part f0 satisfies the following conditions: ((n−1)/2)

f0

∈ C(0, ∞);

(4)

lim ts f0 (t) = 0,

(s)

0 ≤ s ≤ (n − 1)/2;

(5)

(s)

0 < s ≤ (n − 1)/2;

(6)

t→∞

and lim ts f0 (t) = 0,

t→0+

where s is any number in the given interval (not only integer). Besides that, ((n−1)/2) the T -transform of f0 exists for any t > 0. Proof. We rewrite the radial part as Z Z 1 f0 (t) = g(u) eitxu dθ du. σn Rn

Sn−1

It is well known (see, e.g., [8, Ch.4]) that Z eixu dθ = (2π)n/2 (|u|)1−n/2 Jn/2−1 (|u|), Sn−1

where Jν is the Bessel function of order ν that admits the integral representation Z1 tν Jν (t) = ν √ (1 − u2 )ν−1/2 eitu du. (7) 2 πΓ(ν + 1/2) −1

In particular, Z1 J0 (t) = π −1

(1 − u2 )−1/2 eitu du. −1

(8)

556

E. LIFLYAND

We will make use of the following properties of the Bessel functions: the asymptotic formula (see, e.g., [2, §7.13.1(3)]) for t → ∞ r 2 cos(t − πν/2 − π/4) + O(t−3/2 ) (9) Jν (t) = πt and the formulas for derivatives ¤ d £ ±ν t Jν (t) = ±t±ν Jν∓1 (t). dt

(10)

Indeed, (9) holds for all t > 0 but is (asymptotically) meaningful for large t. Thus we have Z n/2−1 f0 (t) = Γ(n/2)2 g(u) jn/2−1 (t|u|) du, (11) Rn

where jν (t) = t−ν Jν (t). To find the fractional derivative of f0 we have to know RA such a derivative for eiBt . We cannot differentiate t (r−t)−δ eiBr dr immediately and should first perform integration by parts to get (r − t) in a positive power. Routine calculations yield d − dt

ZA

ZA (r − t)−δ eiBr dr = eiBA (A − t)−δ − iB

t

R iBt ∞

(r − t)−δ eiBr dr, t

and therefore (eiBt )(δ) = |B|e r−δ eiBr dr times a constant. Since the 0 −δ Fourier transform of u at B is |B|δ−1 times a constant (see, e.g., [1]), we obtain (eiBt )(s) = γs |B|s eiBt , which yields Z1 jν(s) (t)

(1 − v 2 )ν−1/2 |v|s eitv dv.

= γν,s −1

It follows from the definition of jn/2−1 (t) and from (10) and (9) that for any s (s)

jn/2−1 (t) = O(t−(n−1)/2 ).

(12)

Therefore, for each s ≤ (n − 1)/2 and t ∈ (0, ∞) there exists the s-th derivative of f0 equal to Z1

Z (s) f0 (t)

= γn,s

g(u) |u| Rn

(1 − v 2 )(n−3)/2 v s eit|u|v dv du.

s

(13)

−1

Because of the integrability of g and (12), we have (4). If s < (n − 1)/2, then (5) holds by (12) and the Lebesgue dominated convergence theorem. For s = (n − 1)/2 the proof of (5) is twofold. For n odd we just integrate by parts (n − 1)/2 times in the inner integral of (13) and use the Lebesgue-Riemann theorem. For n even, we first integrate by parts (n − 2)/2

INTEGRABILITY OF THE FOURIER TRANSFORM

557

times in the same integral. The only summand in the resulting formula which is questionable is Z1

Z 1/2

(1 − v 2 )−1/2 |v|β eit|u|v dv du,

g(u) (t|u|) Rn

−1

where β > 1/2. We can rewrite it as Z g(u) (t|u|)1/2 Rn



Z1

×



Z1

(1 − v 2 )−1/2 (1 − v β )eit|u|v dv  du. (14)

(1 − v 2 )−1/2 eit|u|v dv du −

−1

−1

In the second integral in the square brackets we can integrate by parts at least once more, and then all becomes clear. The rest is, by (8), Z g(u) (t|u|)1/2 J0 (t|u|) du Rn

times a constant. Applying now (9), we are again in a position to use the Lebesgue-Riemann theorem which completes the proof of (5). For t → 0, the estimate (12) allows us to apply the Lebesgue dominated (s) convergence theorem as above. Since the integral representation for jn/2−1 (t) is always bounded, the presence of t in positive power leads to (6), i.e., for 0 < s ≤ (n − 1)/2. Let us go to the last assertion of the theorem. We will explicitly give the (n−1)/2) proof of the existence of the T -transform of f0 , i.e., the existence of the ((n−1)/2) integral in (1) for h = f0 , only for n even. For n odd the proof is carried our along the same lines but is even easier since no fractional derivatives are involved. For any δ ∈ (0, t/2) we have Zt/2

((n−1)/2)

f0

δ

dz

|u|(n−1)/2 (1 − v 2 )(n−3)/2 |v|(n−1)/2

g(u) Rn

(t − z)

" Z1

Z = γn

((n−1)/2)

(t + z) − f0 z

−1

Zt/2 ×

# eiv(t+z)|u| − eiv(t−z)|u| dz dv du. z

δ

To apply the Lebesgue dominated convergence theorem as δ → 0+, it suffices to check the boundedness in δ and u of the expression in the square brackets.

558

E. LIFLYAND

Denoting it by Φt (δ, u) and ϕ(v) = (1 − v 2 )(n−3)/2 |v|(n−1)/2 , we then integrate in v by parts n/2 − 1 times (we remind that n is even). Since the integrated terms vanish there, we obtain Zt/2·

Z1 Φt (δ, u) = γn |u|

1/2

ϕ

(n/2−1)

(v)

−1

eiv(t+z)|u| eiv(t−z)|u| − (t + z)n/2−1 (t − z)n/2−1

¸

dz dv. z

δ

Like in (14), the main problem is reduced to estimating Zt/2·

Z1 |u|

1/2

2 −1/2

(1 − v ) −1

eiv(t+z)|u| eiv(t−z)|u| − (t + z)n/2−1 (t − z)n/2−1

¸

dz dv. z

δ

Keeping in mind that (t ± z)1−n/2 − t1−n/2 = O(zt−n/2 ), as in [9, Th. 3], and taking into account (8) and (9), we arrive at proving the boundedness in δ and u of the integral Z1 |u|

1/2

Zt/2 2 −1/2

(1 − v ) −1

eiv(t+z)|u| − eiv(t−z)|u| dz dv z

δ

Z1 = 2i|u|1/2

tv|u|/2 Z

(1 − v 2 )−1/2 eitv|u| −1

z −1 sin z dz dv. δv|u|

But it is shown in [9] that in this case eit|u|v is multiplied by a function from Lip 12 in the L1 norm. The required boundedness is obvious then, which completes the proof. ¤ 4. Concluding Remarks One can see from the proof that ¯ t/2 ¯ ¯Z ((n−1)/2) ¯ Z ((n−1)/2) ¯ f0 ¯ (t + v) − f (t − v) ((n−1)/2) 0 )(t)| = ¯¯ |(T f0 dv ¯¯ ≤ γn |g(u)| du. v ¯ ¯ Rn

δ

Moreover, considering a periodic variable function f with an absoP inˆ each ikx lutely convergent Fourier series f (k)e and repeating the proof, one can k∈Zn

obtain a direct generalization of the one-dimensional necessary conditions mentioned in the Introduction, with an estimate ¯ ¯ t/2 ¯ ¯Z ((n−1)/2) ((n−1)/2) X ¯ ¯ f0 (t + v) − f (t − v) 0 ¯ ≤ γn ¯ |fˆ(k)|. dv ¯ ¯ v ¯ ¯ n k∈Z δ

Many sufficient conditions for the integrability of the Fourier transform and the absolute convergence of Fourier series in the multidimensional case are known; the reader can consult, e.g., [10] and the references therein.

INTEGRABILITY OF THE FOURIER TRANSFORM

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Acknowledgements To my regret I never met Prof. Zhizhiashvili personally, but I was well familiar with his works. In my young years my research was in particular based on his survey paper [11]. I am also honored that Prof. Zhizhiashvili was aware of my results and referred to them. The author thanks the referee for valuable remarks that definitely improved the presentation. References ´lyi, Tables of integral transforms. Vol. I. McGraw Hill Book 1. H. Bateman and A. Erde Company, New York, 1953. ´lyi, Higher transcendental functions. Vol. II. McGraw Hill 2. H. Bateman and A. Erde Book Company, New York, 1953. 3. J. Cossar, A theorem on Ces`aro summability. J. London Math. Soc. 16(1941), 56–68. 4. S. Fridli, Hardy spaces generated by an integrability condition. J. Approx. Theory 113(2001), No. 1, 91–109. 5. J.-P. Kahane, S´eries de Fourier absolument convergentes. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50. Springer-Verlag, Berlin–New York, 1970. 6. E. R. Liflyand, On asymptotics of Fourier transform for functions of certain classes. Anal. Math. 19(1993), No. 2, 151–168. 7. E. Liflyand, Lebesgue constants of multiple Fourier series. Online J. Anal. Comb. No. 1 (2006), Art. 5, 112 pp. (electronic). 8. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971. 9. R. M. Trigub, Absolute convergence of Fourier integrals, summability of Fourier series and approximation by polynomials of functions on a torus. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 44(1980), No. 6, 1378–1409, 1439; English trans.: Math. USSR Izv. 17(1981), 567–593. 10. R. M. Trigub and E. S. Bellinsky, Fourier analysis and approximation of functions. Kluwer Academic Publishers, Dordrecht, 2004. 11. L. V. Zhizhiashvili, Some problems in the theory of simple and multiple trigonometric and orthogonal series. (Russian) Uspehi Mat. Nauk 28(1973), No. 2(170), 65–119; English transl.: Russian Math. Surveys 28(1973), No. 2, 65–127. 12. L. V. Zhizhiashvili, Trigonometric Fourier series and their conjugates. Revised and updated translation of Some problems of the theory of trigonometric Fourier series and their conjugate series (Russian) [Tbilis. Gos. Univ., Tbilisi, 1993]. Mathematics and its Applications, 372. Kluwer Academic Publishers Group, Dordrecht, 1996.

(Received 12.12.2008) Author’s address: Department of Mathematics Bar-Ilan University 52900 Ramat-Gan Israel E-mail: [email protected]