AN ANALOG OF SOME TITCHMARSH THEOREM FOR THE …gauss40.pdmi.ras.ru/ma26/presentations/Platonov.pdf · on locally compact Vilenkin groups. Let us present necessary de nitions from

General de�nitions

Proofs of Theorems 2 and 3

AN ANALOG OF SOME TITCHMARSH

THEOREM FOR THE FOURIER TRANSFORM

ON LOCALLY COMPACT VILENKIN GROUPS

26th St.Petersburg Summer Meeting in Mathematical AnalysisJune 25th, 2017

S. S. Platonov

Petrozavodsk State University

3 èþëÿ 2017 ã.

S. S. Platonov AN ANALOG OF SOME TITCHMARSH THEOREM FOR THE FOURIER TRANSFORM ON LOCALLY COMPACT VILENKIN GROUPS



Contents

1 General de�nitions

2 Proofs of Theorems 2 and 3




Contents

1 General de�nitions

2 Proofs of Theorems 2 and 3




The main themes of the talk are some analogues of one classicalTitchmarsh theorem on description of the image under the Fouriertransform of a class of functions satisfying the Lipschitz conditionin L2 and the sharp estimate (with exact constant) for decreasingof certain Fourier transforms of L2 functions in mean.




Suppose that f (x) is a function in the L2(R) space (all functionsbelow are complex-valued), ‖ · ‖L2(R) is the norm of L2(R), and α isan arbitrary number in the interval (0, 1).

De�nition 1

A function f (x) belongs to the Lipschitz class Lip(α, 2) if

‖f (x − t)− f (x)‖L2(R) = O(tα)

as t → 0.




Theorem 1 ([T], Theorem 85)

If f (x) ∈ L2(R) and f̂ (λ) is its Fourier transform, then the

conditions

f ∈ Lip(α, 2), 0 < α < 1,

and ∫|λ|≥r

|f̂ (λ)|2 dλ = O(r−2α)

as r →∞ are equivalent.

[T] Titchmarsh E. C., Introduction to the theory of Fourier

integrals, Oxford: Clarendon Press, 1937.




There are many analogues of Theorem 1: for the Fourier transformon noncompact Riemannian rank 1 symmetric spaces, in particularfor the Fourier transform on the Lobachevski�i plane; for the Fourier� Jacobi transform; for the Fourier � Dunkl transform and etc. See,for example:[Pl1] Platonov S. S., The Fourier transform of functions satisfying

the Lipschitz condition on rank 1 symmetric spaces 1, Sib. Math.J., 46:6 (2005), 1108�1118.[Y] Younis M. S., Fourier transform of Lipschitz functions on the

hyperbolic plane, Internat. J. Math.& Math. Sci., 21:2 (1998),397-401.[DH] Daher R., Hamma M., An analog of Titchmarsh's theorem of

Jacobi transform, Int. J. of Math. Anal., 6:17-20 (2012), 975-981.[M] Maslouhi M., An analog of Titchmarsh's theorem for the

Dunkl transform, Integral Transforms Spec. Funct., 21:10 (2010),771-778.




We obtain some analogue of Theorem 1 for the Fourier transformon locally compact Vilenkin groups.Let us present necessary de�nitions from harmonic analysis onlocally compact Abelian groups (see, for example, [Õ-Ð], [Rud]).[H-R] E. Hewitt and K. A. Ross, Abstract harmonic analysis, vol. I:Structure of topological groups. Integration theory, grouprepresentations, Grundlehren Math. Wiss., vol. 115, AcademicPress, Inc., Publishers, New York; Springer-Verlag,Berlin-G�ottingen-Heidelberg 1963, viii+519 pp[Rud] Rudin W., Fourier analysis on groups. New York and London:Interscience Publishers, 1962.




Let G be a locally compact Abelian group. A character of G is acontinuous complex-valued function χ(x) on G such that|χ(x)| = 1 and χ(x + y) = χ(x)χ(y) for any x , y ∈ G . Let Γ bethe set of all characters of G . The set Γ equipped with thecompact-open topology and the operation of point-wisemultiplication of characters becomes an LCA-group which is said tobe the dual group of G . We note that the group operation in thegroup G is always written additively and the operation in the dualgroup Γ is written multiplicatively.




De�nition 2

A locally compact Abelian group G is said to be a locally compact

Vilenkin group if there exists a strictly decreasing sequence of

compact open subgroups {Gn}n∈Z (that is Gn+1 $ Gn) such that⋃n∈Z Gn = G and

⋂n∈Z Gn = {0}.

The factor group Gn/Gn+1 is a �nite Abelian group. Let dn be theorder of the group Gn/Gn+1, then dn ≥ 2. Examples of locallycompact Vilenkin groups are the group p-adic numbers and, moregenerally, the additive group of a local �eld, see [Taibl].[Taibl] Taibleson M. H., Fourier analysis on local �elds, Math.Notes, vol. 15, Prinston Univ. Press, 1975.




In what follows G is a locally compact Vilenkin group, Γ its dualgroup. For any n ∈ Z let Γn be the annihilator of Gn, that is

Γn = {χ ∈ Γ : χ(x) = 1 for any x ∈ Gn}.

It follows from the properties of dual groups and the annihilators ofsubgroups (see [H-R, (23.24), (23.29)]) that Γn is a compact opensubgroup of Γ, the sequence of subgroups {Γn}n∈Z is strictlyincreasing,

⋂n∈Z Γn = {1} and

⋃n∈Z Γn = Γ.




We choose Haar measures dx on G and dχ on Γ so that∫G0

dx =

∫Γ0

dχ = 1. (1)

We denote by µ(A) the Haar measure of a subset A ⊂ G , and byλ(B) the Haar measure of a subset B ⊂ Γ.For every s ∈ Z we de�ne the the number mn by

mn :=

d1d2 . . . dn if n > 0,

1 if n = 0,

d−10 d−1

−1 . . . d−1−n+1 if n < 0.

(2)

Then

µ(Gn) =1

mn, λ(Γn) = mn.




Let Lp(G ), 1 ≤ p <∞, be a Banach space of all measurableC-valued functions f (x) on G with �nite norm

‖f ‖p = ‖f ‖Lp(G) :=

∫G

|f (x)|p dx

1/p

.

Similarly, let Lp(Γ) be a Banach space of all measurable C-valuedfunctions g(χ) on Γ with �nite norm

‖g‖p = ‖g‖Lp(Γ) :=

∫Γ

|g(χ)|p dχ

1/p

.

As usual, functions from the spaces Lp are considered up to theirvalues on a set of measure 0.




For any function f (x) ∈ L1(G ), by the Fourier transform of f wemean the function f̂ (ξ) on Γ de�ned by the formula

f̂ (χ) :=

∫G

f (x)χ(x) dx , χ ∈ Γ.

If f ∈ L2(G ), then its Fourier transform f̂ (ξ) can be de�ned as thelimit in L2(G ) of a sequence of the functions

f̂n(χ) :=

∫Gn

f (x)χ(x) dx

as n→∞. The Fourier transform F : f (x) 7→ f̂ (χ) is a linearisomorphism of the space L2(G ) into the space L2(Γ), and for anyfunction f ∈ L2(G ) we have the Parseval's identity

‖F (f )‖L2(Γ) = ‖f ‖L2(G).




For a function f (x) on G and for any h ∈ G let

(τhf )(x) := f (x − h).

The operator τh is called the translation operator. If f ∈ L2(G ) andF (f )(χ) = f̂ (χ) is its Fourier transform, then we have:

F (τhf )(χ) = χ(h) f̂ (χ). (3)




For f ∈ L2(G ) and n ∈ N let

ω2(f ; n) := sup{‖f − τhf ‖2 : h ∈ Gn}.

The sequence of numbers {ω2(f ; n)}n∈N is called the moduluscontinuity of f in the space L2(G ).Let be ω = {ωn}n∈N a sequence of real numbers monotonouslydecreasing to zero (that is (i) ωn ≥ 0; (ii) ωn ≥ ωn+1 ∀n ∈ N; (iii)ωn → 0 as n→∞).

De�nition 3

A function f (x) belongs to the space Hω2 (G ), if f ∈ L2(G ) and for

some constant c = c(f ) > 0 we have

ω2(f ; n) ≤ c ωn, n ∈ N.




Let ω = {ωn}n∈N and ω′ = {ω′n}n∈N be sequences of real numbersmonotonously decreasing to zero. The sequences ω and ω′ will becalled equivalent if we have

c1 ωn ≤ ω′n ≤ c2 ωn, n ∈ N

for some positive constants c1 and c2. It can be proved that everyspace Hω

2 (G ) is nonzero, and Hω2 (Gp) = Hω′

2 (G ) if and only if thesequences ω and ω′ are equivalent.




The main results of the talk are the next theorems.

Theorem 2

For every f ∈ L2(G ) we have the inequality( ∫Γ\Γn

|f̂ (χ)|2 dχ)1/2

≤ 1√2ω2(f ; n), n ∈ N, (4)

where constant 1√2in (4) is exact.




The following theorem is an analogue of the Tichmarsh theorem.

Theorem 3

Let ω = {ωn}n∈N be any sequence of real numbers monotonously

decreasing to zero. Then the next conditions are equivalent:

f ∈ Hω2 (G ) (5)

and ( ∫Γ\Γn

|f̂ (χ)|2 dχ)1/2

≤ c ωn, n ∈ N, (6)

where c = c(f ) is some positive constant.




For the special case when G = Qp is the group of p-adic numbers,the Theorems 2 and 3 was proved in[Pl2] Platonov S. S., An analogue of the Titchmarsh theorem for

the Fourier transform on the group of p-adic numbers, p-AdicNumbers, Ultrametric Analysis and Appl., 9:2 (2017), 158�164.




Lemma 1

Let χ be a character of ãðóïïû G , n ∈ Z. Then∫Gn

χ(x) dx =

{µ(Gn), if χ ∈ Γn,

0, if χ /∈ Γn.




Proof of Theorem 2

1) Let f ∈ L2(G ), h ∈ Gn, n ∈ N. By de�nition of the modulus ofcontinuity we have

ω2(f ; n) := sup{‖f − τhf ‖2 : h ∈ Gn}. (7)

It follows from (3) that

F (f − τhf )(ξ) = (1− χp(ξh)) f̂ (ξ), (8)

then, using the Parseval's identity, we have

‖f − τhf ‖22 =

∫Γ

|1− χ(h)|2 |f̂ (χ)|2 dχ. (9)




If χ ∈ Γn, h ∈ Gn, then χ(h) = 1. Hence, the equality (9) can berewritten in the form

‖f − τhf ‖22 =

∫Γ\Γn

|1− χ(h)|2 |f̂ (χ)|2 dχ. (10)

Integrating the equality (10) with respect to h ∈ Gn, we obtain∫Gn

‖f − τhf ‖22 dh =

∫Γ\Γn

(∫Gn

|1− χ(h)|2 dh)|f̂ (χ)|2 dχ. (11)




It follows from |χ(h)| = 1 that

|1− χ(h)|2 = 2− 2 Reχ(h). (12)

It follows from Lemma 1 that∫Gn

χ(h) dh = 0 for χ ∈ Γ \ Γn, (13)

hence it follows from (12) and (13) that∫Gn

|1− χ(h)|2 dh =

∫Gn

(2− Reχ(h)) dh = 2

∫Gn

dh = 2µ(Gn). (14)

From (11) and (14) it follows that∫Gn

‖f − τhf ‖22 dh = 2µ(Gn)

∫Γ\Γn

|f̂ (χ)|2 dχ. (15)




On the other hand, since ‖f − τhf ‖2 ≤ ω2(f ; n) for h ∈ Gn, then∫Gn

‖f − τhf ‖22 dh ≤ (ω2(f ; n))2

∫Gn

dh = µ(Gn) (ω2(f ; n))2 . (16)

It follows from (15) and (16) that

2µ(Gn)

∫Γ\Γn

|f̂ (χ)|2 dχ ≤ µ(Gn) (ω2(f ; n))2 ,

which implies that the inequality (4) holds.




2) We claim that the constant 1√2in (4) is exact.

For any n ∈ Z and a ∈ G letGn(a) := a + Gn = {x ∈ G : x − a ∈ Gn}. In particular,Gn(0) = Gn. For every s ∈ N let ϕs be the characteristic functionof the subset Gs . Then ‖ϕs‖2

2 = µ(Gs).




It can be proved that∫Γ\Γn

|ϕ̂s(χ)|2 dχ =

{12

(1− mn

ms

)(ω2(ϕs ; n))2 for n < s,

0 for n ≥ s,(17)

where mn and ms is de�ned in (2). Since mnms≤ 2n−s , then it follows

from (17) that, for any n ∈ N and ε > 0, for su�ciently large s wehave the inequality( ∫

Γ\Γn

|ϕ̂s(χ)|2 dχ)1/2

≥ 1√2

(1− ε)ω2(ϕs ; n),

which implies that the constant 1√2in (4) is exact.




Using the results of the paper[Rub] Rubinshtein A. I., Moduli of continuity of functions, de�ned

on a zero-dimensional group, Math. Notes., 23 (1978), 205-211.we can prove the next

Proposition 1

Let {ωn}n∈N be a sequence of real numbers monotonously

decreasing to zero. Then there exist a function f ∈ L2(G ) such that

ω2(f ; n) = ωn ∀n ∈ N. (18)




Corollary 1

For any sequence {ωn}n∈N of real numbers monotonously

decreasing to zero the space Hω2 (G ) is nonzero.

Proposition 2

Let ω = {ωn}n∈N and ω′ = {ω′n}n∈N be sequences of real numbers

monotonously decreasing to zero. Then Hω2 (G ) = Hω′

2 (G ) if and

only if the sequences ω and ω′ are equivalent.




Proof of Theorem 3

It follows from Theorem 2 that (5) entails (6).Let f ∈ L2(G ) and we assume that (6) holds. Arguing as in theproof of Theorem 2, we obtain that for any h ∈ Gn the equality(11) holds. It follows from (11), using the inequalities|1− χ(h)| ≤ 2 and (6), that

‖f −τhf ‖22 =

∫Γ\Γn

|1−χ(h)|2|f̂ (χ)|2 dχ ≤ 4

∫Γ\Γn

|f̂ (χ)|2 dχ ≤ 4c2ω2n

(19)for h ∈ Gn, n ∈ N. Taking in (19) the in�num over all h ∈ Gn, weobtain that

ω2(f ; n) ≤ 2c ωn, n ∈ N,

that is the condition (5) holds.Hence the conditions (5) and (6) are equivalent.


AN ANALOG OF SOME TITCHMARSH THEOREM FOR THE …gauss40.pdmi.ras.ru/ma26/presentations/Platonov.pdf · on locally compact Vilenkin groups. Let us present necessary de nitions from

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