##### Document Text Contents

Page 1

Certain relations between the Fourier and Hilbert

transforms

Elijah Liflyand

Bar-Ilan University

March, 2012

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 1 / 24

Page 2

The Fourier transform of a function of bounded variation

Let us start with the next result:

L 1993; Fridli, 2001

We define the following T -transform of a function

g : R = [0,∞)→ C:

Tg(t) =

∫ t/2

0

g(t+ s)− g(t− s)

s

ds,

where the integral is understood in the improper (principal value)

sense, that is, as lim

δ→0+

∫

δ

.

Here and in what follows we use the notation “. ” and “& ” as

abbreviations for “≤ C ” and “≥ C ”, with C being an absolute

positive constant.

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 2 / 24

Page 51

The Fourier transform of a function of bounded variation

At first sight, Theorem 2 does not seem to be a result at all, at most

a technical reformulation of the definition.

This could be so but not after the appearance of the analysis of Q by

Johnson-Warner.

Indeed, the well-known extension of Hardy’s inequality

∫

R

|ĝ(x)|

|x|

dx . ‖g‖H1(R)

implies

H1(R) ⊆ Q ⊆ L10(R),

where the latter is the subspace of g in L1(R) which satisfy the

cancelation property.

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 13 / 24

Page 52

The Fourier transform of a function of bounded variation

At first sight, Theorem 2 does not seem to be a result at all, at most

a technical reformulation of the definition.

This could be so but not after the appearance of the analysis of Q by

Johnson-Warner.

Indeed, the well-known extension of Hardy’s inequality

∫

R

|ĝ(x)|

|x|

dx . ‖g‖H1(R)

implies

H1(R) ⊆ Q ⊆ L10(R),

where the latter is the subspace of g in L1(R) which satisfy the

cancelation property.

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 13 / 24

Page 102

Integrability of the Hilbert transform

More is known about the odd case.

Paley-Wiener’s theorem asserts that if g ∈ L1(R) is an odd and

monotone decreasing on R+ function, then Hg ∈ L1, i.e., g is in

H1(R).

Recently, this theorem has been extended to a class of functions more

general than monotone ones by L-Tikhonov.

Also, sufficient conditions in the odd case were obtained by Trigub,

Giang-Moricz, L.

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 24 / 24

Page 103

Integrability of the Hilbert transform

More is known about the odd case.

Paley-Wiener’s theorem asserts that if g ∈ L1(R) is an odd and

monotone decreasing on R+ function, then Hg ∈ L1, i.e., g is in

H1(R).

Recently, this theorem has been extended to a class of functions more

general than monotone ones by L-Tikhonov.

Also, sufficient conditions in the odd case were obtained by Trigub,

Giang-Moricz, L.

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 24 / 24

Certain relations between the Fourier and Hilbert

transforms

Elijah Liflyand

Bar-Ilan University

March, 2012

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 1 / 24

Page 2

The Fourier transform of a function of bounded variation

Let us start with the next result:

L 1993; Fridli, 2001

We define the following T -transform of a function

g : R = [0,∞)→ C:

Tg(t) =

∫ t/2

0

g(t+ s)− g(t− s)

s

ds,

where the integral is understood in the improper (principal value)

sense, that is, as lim

δ→0+

∫

δ

.

Here and in what follows we use the notation “. ” and “& ” as

abbreviations for “≤ C ” and “≥ C ”, with C being an absolute

positive constant.

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 2 / 24

Page 51

The Fourier transform of a function of bounded variation

At first sight, Theorem 2 does not seem to be a result at all, at most

a technical reformulation of the definition.

This could be so but not after the appearance of the analysis of Q by

Johnson-Warner.

Indeed, the well-known extension of Hardy’s inequality

∫

R

|ĝ(x)|

|x|

dx . ‖g‖H1(R)

implies

H1(R) ⊆ Q ⊆ L10(R),

where the latter is the subspace of g in L1(R) which satisfy the

cancelation property.

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 13 / 24

Page 52

The Fourier transform of a function of bounded variation

At first sight, Theorem 2 does not seem to be a result at all, at most

a technical reformulation of the definition.

This could be so but not after the appearance of the analysis of Q by

Johnson-Warner.

Indeed, the well-known extension of Hardy’s inequality

∫

R

|ĝ(x)|

|x|

dx . ‖g‖H1(R)

implies

H1(R) ⊆ Q ⊆ L10(R),

where the latter is the subspace of g in L1(R) which satisfy the

cancelation property.

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 13 / 24

Page 102

Integrability of the Hilbert transform

More is known about the odd case.

Paley-Wiener’s theorem asserts that if g ∈ L1(R) is an odd and

monotone decreasing on R+ function, then Hg ∈ L1, i.e., g is in

H1(R).

Recently, this theorem has been extended to a class of functions more

general than monotone ones by L-Tikhonov.

Also, sufficient conditions in the odd case were obtained by Trigub,

Giang-Moricz, L.

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 24 / 24

Page 103

Integrability of the Hilbert transform

More is known about the odd case.

Paley-Wiener’s theorem asserts that if g ∈ L1(R) is an odd and

monotone decreasing on R+ function, then Hg ∈ L1, i.e., g is in

H1(R).

Recently, this theorem has been extended to a class of functions more

general than monotone ones by L-Tikhonov.

Also, sufficient conditions in the odd case were obtained by Trigub,

Giang-Moricz, L.

Elijah Liflyand (Bar-Ilan University) Fourier and Hilbert transforms March, 2012 24 / 24