approximation by the nonlinear fourier basis

SCIENCE CHINAMathematics

. ARTICLES . June 2011 Vol. 54 No. 6: 1207–1214

doi: 10.1007/s11425-011-4211-z

c© Science China Press and Springer-Verlag Berlin Heidelberg 2011 math.scichina.com www.springerlink.com

Approximation by the nonlinear Fourier basis

HUANG Chao1 & YANG LiHua1,2,∗

1School of Mathematics and Computing Science, Sun Yat-sen University, Guangzhou 510275, China;2Guangdong Province Key Laboratory of Computational Science, Guangzhou 510275, China

Email: boss [email protected], [email protected]

Received October 17, 2010; accepted December 8, 2010

Abstract As a typical family of mono-component signals, the nonlinear Fourier basis {eikθa(t)}k∈Z, defined

by the nontangential boundary value of the Mobius transformation, has attracted much attention in the field

of nonlinear and nonstationary signal processing in recent years. In this paper, we establish the Jackson’s and

Bernstein’s theorems for the approximation of functions in Xp(T), 1 � p � ∞, by the nonlinear Fourier basis.

Furthermore, the analogous theorems for the approximation of functions in Hardy spaces by the finite Blaschke

products are established.

Keywords nonlinear Fourier basis, Jackson’s theorem, Bernstein’s theorem, finite Blaschke products

MSC(2000): 41A17, 41A25, 32A35, 42A50

Citation: Huang C, Yang L H. Approximation by the nonlinear Fourier basis. Sci China Math, 2011, 54(6): 1207–

1214, doi: 10.1007/s11425-011-4211-z

1 Introduction

Let R denote the set of all real numbers and Z the set of all integers, set T := R/2πZ. Let N denote theset of all positive integers, and Z+ := N ∪ {0}. Let C(T) denote the set of all the continuous functionson T, and the norm on C(T) is defined by ‖f‖C(T) := supt∈T

|f(t)|. For 1 � p < ∞, Lp(T) stands for thespace of all the p-power integrable functions on T endowed with the following norm

‖f‖Lp(T) :=(

12π

∫T

|f(t)|pdt

)1/p

. (1.1)

For simplicity, we denote

Xp(T) :=

{Lp(T), 1 � p < ∞,

C(T), p = ∞.

It is a classical approximation problem to estimate the approximation error of f ∈ Xp(T) by functionsin an approximation space — a given subspace of Xp(T). A typical approximation space is the space ofall the trigonometric polynomials of degree less than or equal to n, that is, Tn := span{eikt : |k| � n}.The set of functions {eikt : k ∈ Z} is called the Fourier basis. For the subspace Tn, the approximationerror of f ∈ Xp(T) by Tn is denoted by

En(f)p := infT∈Tn

‖f − T ‖Xp(T).

∗Corresponding author

1208 Huang C et al. Sci China Math June 2011 Vol. 54 No. 6

Another concept of the same importance is the modulus of smoothness defined by

ω(f, t)p := sup0�h�t

‖f(· + h) − f‖Xp(T)

for any f ∈ Xp(T), 1 � p � ∞. Accordingly, for 0 < α � 1, the Lipschitz class of order α is defined by

(Lipα)p := {f : |f |(Lipα)p< ∞}, where |f |(Lipα)p

:= supt>0

{t−αω(f, t)p}.

In approximation theory, we need to discuss the relation between the decreasing velocity of En(f)p asn → ∞ and the smoothness of f , such as f ∈ (Lipα)p for some α > 0. For any two sequences of numbers{fn} and {gn}, we use the notation fn = O(gn) to mean that there exists some constant C > 0 suchthat |fn| � C|gn|, fn � gn means that fn = O(gn) and gn = O(fn) hold simultaneously. In the theory ofapproximation, we have the following classical theorems [9]: Let f ∈ Xp(T), 1 � p � ∞. Then,

(i) Jackson’ theorem: If f ∈ (Lipα)p, 0 < α � 1, then En(f)p = O(n−α).(ii) Bernstein’ theorem: If En(f)p = O(n−α), 0 < α < 1, then f ∈ (Lipα)p.Fourier basis is a powerful tool in stationary signal processing. In recent years, the nonlinear and

nonstationary signal processing has achieved a rapid progress. A family of nonlinear Fourier bases, as theextension of the classical Fourier basis, have been constructed and applied to signal processing [2,3,10–12].For any complex number a = |a|eita , |a| < 1, the nonlinear phase function θa(t) is defined by the radicalboundary value of the Mobuis transformation

τa(z) =z − a

1 − az,

that is,

eiθa(t) := τa(eit) =eit − a

1 − aeit.

It is easily seen that θa(t + 2π) = θa(t) + 2π and its derivative is the Poisson kernel

θ′a(t) =1 − |a|2

1 − 2|a| cos(t − ta) + |a|2 , (1.2)

which satisfies

0 <1 − |a|1 + |a| � θ′a(t) � 1 + |a|

1 − |a| . (1.3)

Hence, θa(t) is a strictly monotonic increasing function, which makes cos θa(t) be a special mono-component signal [10, 12]. It has been shown that for any sequence {ck}k∈Z of finite nonzero terms,there holds ∑

k∈Z

|ck|2 =12π

∫T

∣∣∣∣∑k∈Z

ckeikx

∣∣∣∣2

dx =12π

∫T

∣∣∣∣∑k∈Z

ckeikθa(t)

∣∣∣∣2

θ′a(t)dt,

which combining with (1.3) implies that the so-called nonlinear Fourier basis {einθa(t)}n∈Z forms a Rieszbasis for L2(T) with the upper bound

√(1 + |a|)/(1 − |a|) and the lower bound

√(1 − |a|)/(1 + |a|).

When a = 0, {einθa(t)}n∈Z is simply the Fourier basis {eint}n∈Z.Let T a

n be the space of all the nonlinear trigonometric polynomials of degree less than or equal to n,that is,

T an := span{eikθa(t) : |k| � n}.

This paper aims at establishing the Jackson’s and Bernstein’s theorems for the approximation of functionsin Xp(T) by functions in T a

n . The approximation error of f ∈ Xp(T), 1 � p � ∞, by the nonlinear Fourierbasis {einθa(t)}n∈Z is denoted by

Ean(f)p := inf

T∈T an

‖f − T ‖Xp(T).

The rest of the paper is organized as follows: In Sections 2 and 3 we establish the Jackson’s andBernstein’s theorems for the approximation of functions in Xp(T), 1 � p � ∞, by the nonlinear Fourierbasis. In Section 4, the approximation of holomorphic functions in Hardy spaces by the finite Blaschkeproducts is discussed.

Huang C et al. Sci China Math June 2011 Vol. 54 No. 6 1209

2 Jackson’s theorem

2.1 Jackson’s theorem in C(T)

Lemma 2.1. Let f ∈ C(T). Then Ean(f)∞ = En(f ◦ θ−1

a )∞.

Proof. Suppose ‖f ◦ θ−1a −

∑|k|�n ckeik·‖C(T) = En(f ◦ θ−1

a )∞. Note that θa(t) is a strictly monotonicincreasing function satisfying θa(t + 2π) = θa(t) + 2π, we have

∥∥∥∥f −∑|k|�n

ckeikθa(·)∥∥∥∥

C(T)

=∥∥∥∥f ◦ θ−1

a −∑|k|�n

ckeik·∥∥∥∥

C(T)

= En(f ◦ θ−1a )∞.

It follows that Ean(f)∞ � En(f ◦ θ−1

a )∞.

Similarly, we can prove En(f ◦ θ−1a )∞ � Ea

n(f)∞. The proof is complete. �

Lemma 2.2. Let f ∈ C(T). Then we have

1 − |a|2

ω(f, t)∞ � ω(f ◦ θ−1a , t)∞ � 2

1 − |a|ω(f, t)∞.

Proof. Using the following inequality

|θ−1a (x) − θ−1

a (y)| = |(θ−1a )′(ξ)| · |x − y| =

1θ′a(θ−1

a (ξ))|x − y| � 1 + |a|

1 − |a| |x − y|,

where ξ ∈ (x, y) or ξ ∈ (y, x), we conclude that

ω(f ◦ θ−1a , t)∞ = sup

|x−y|�t

|f(θ−1a (x)) − f(θ−1

a (y))| � ω

(f,

1 + |a|1 − |a| t

)∞

� 21 − |a| ω(f, t)∞.

Similarly, we have ω(f, t)∞ � 21−|a| ω(f ◦ θ−1

a , t)∞. The proof ends. �

Below is the Jackson’s theorem for the approximation of functions in C(T) by the nonlinear Fourierbasis {einθa(t)}n∈Z.

Theorem 2.3. Let f ∈ C(T). Then we have

Ean(f)∞ � 24

1 − |a| ω

(f,

1n

)∞

.

Consequently, if f ∈ (Lipα)∞, 0 < α � 1, then Ean(f)∞ = O(n−α).

Proof. Since f ∈ C(T), then f ◦ θ−1a ∈ C(T). The result in [9, p. 64] implies that En(f ◦ θ−1

a )∞ �12 ω(f ◦ θ−1

a , 1/n)∞. Using Lemmas 2.1 and 2.2, we conclude that

Ean(f)∞ = En(f ◦ θ−1

a )∞ � 12 ω

(f ◦ θ−1

a ,1n

)∞

� 241 − |a| ω

(f,

1n

)∞

. �

In what follows we discuss the conditions such that Ean(f)∞ = O(n−(r+α)) holds for a given r ∈ N.

For any r ∈ N, denote by Cr(T) the set of all the r-times continuously differentiable functions on T.

Theorem 2.4. Suppose that f ∈ Cr(T), r ∈ N, satisfies f (r) ∈ (Lipα)∞, 0 < α < 1. ThenEa

n(f)∞ = O(n−r−α).

Proof. The fact that f ∈ Cr(T) implies that f ◦ θ−1a ∈ Cr(T). By Lemma 2.1 and the result in

[9, p. 66] we have

Ean(f)∞ = En(f ◦ θ−1

a )∞ � 12nr

ω

((f ◦ θ−1

a )(r),1n

)∞

.

It is easy to verify that

(f ◦ θ−1a )(r) =

f (r) ◦ θ−1a

(θ′a ◦ θ−1a )r

+r−1∑j=1

(f (j) ◦ θ−1a )uj , (2.1)


where uj , j = 1, 2, . . . , r − 1, are some infinitely differentiable functions on T.According to Lemma 2.2 we have f (r)◦θ−1

a ∈ (Lipα)∞. On the other hand, since (θ′a ◦θ−1a )−r ∈ C1(T),

we have (θ′a ◦θ−1a )−r ∈ (Lip1)∞ ⊂ (Lipα)∞. Therefore their product (f (r) ◦θ−1

a ) · (θ′a ◦θ−1a )−r ∈ (Lipα)∞.

Similarly,∑r−1

j=1(f(j) ◦ θ−1

a )uj is also in (Lipα)∞ since it is continuously differentiable on T. Thus,(f ◦ θ−1

a )(r) ∈ (Lipα)∞ and consequently Ean(f)∞ = O(n−r−α) holds. �

2.2 Jackson’s theorem in Lp(T)

In this subsection we discuss the Jackson’ theorem in Lp(T), 1 � p < ∞.

Lemma 2.5. Let f ∈ Lp(T), 1 � p < ∞. Then Ean(f)p � En(f ◦ θ−1

a )p.

Proof. Suppose ‖f ◦ θ−1a −

∑|k|�n ckeik·‖Lp(T) = En(f ◦ θ−1

a )p. Using the change of variable u = θa(t)in the integral we deduce that

(12π

∫T

∣∣∣∣f(t) −∑|k|�n

ckeikθa(t)

∣∣∣∣p

θ′a(t) dt

)1/p

= En(f ◦ θ−1a )p,

which implies

Ean(f)p �

(1 + |a|1 − |a|

)1/p

En(f ◦ θ−1a )p.

Similarly, we have

En(f ◦ θ−1a )p �

(1 + |a|1 − |a|

)1/p

Ean(f)p.

The proof is complete. �

In the previous subsection, we obtained the equivalence between ω(f ◦ θ−1a , t)∞ and ω(f, t)∞ for any

f ∈ C(T). Unfortunately, it is nearly impossible to deduce the equivalence between ω(f ◦ θ−1a , t)p and

ω(f, t)p directly for any f ∈ Lp(T), 1 � p < ∞. However, we can prove that f ∈ (Lipα)p if and only iff ◦ θ−1

a ∈ (Lipα)p by using a classical approximation result as follows.Let D : t0 < t1 < · · · < tN = t0 + 2π be a partition of T, which divides T into finite disjoint intervals

Ik := [tk−1, tk), 1 � k � N . Denote δD := max1�k�N |tk − tk−1|, S(D) := span{χIk: 1 � k � N}

the space of all the piecewise constant functions with respect to this partition, where χIkstands for the

characteristic function, and denote

s(f, D)p := infS∈S(D)

‖f − S‖Lp(T)

for any f ∈ Lp(T), 1 � p < ∞. It is proved in [4, 5] that for any 0 < α � 1, f ∈ (Lipα)p if and onlyif s(f, D)p = O(δα

D) holds for any partition D. By this equivalence, we can easily deduce the followinglemma.

Lemma 2.6. Let f ∈ Lp(T), 1 � p < ∞. Then for any 0 < α � 1, f ∈ (Lipα)p if and only iff ◦ θ−1

a ∈ (Lipα)p.

Proof. Let D : t0 < t1 < · · · < tn = t0 + 2π be a partition of T and S ∈ S(D). For any f ∈ Lp(T),1 � p < ∞, it is easy to see that

∫T

|(f ◦ θ−1a )(x) − S(x)|pdx =

∫T

|f(t) − (S ◦ θa)(t)|p θ′a(t)dt,

which implies that ‖f ◦ θ−1a − S‖Lp(T) � ‖f − S‖Lp(T), where S := S ◦ θa ∈ S(D) and D : θ−1

a (t0) <

θ−1a (t1) < · · · < θ−1

a (tn) = θ−1a (t0) + 2π is another partition of T depending on D and θa. It is then

followed that s(f ◦ θ−1a , D) � s(f, D).

It is easy to show that D generates D and vice versa. Meanwhile, there holds that δD � δD. Therefores(f ◦ θ−1

a , D) = O(δαD) if and only if s(f, D) = O(δα

D). Accordingly, f ∈ (Lipα)p if and only if f ◦ θ−1

a ∈(Lipα)p. �


Based on the above lemmas, the following Jackson’s theorem for the nonlinear Fourier basis holds inLp(T), 1 � p < ∞.

Theorem 2.7. Let f ∈ (Lipα)p, 1 � p < ∞, 0 < α � 1. Then Ean(f)p = O(n−α).

Proof. By Lemma 2.6 we have f ◦ θ−1a ∈ (Lipα)p. Lemma 2.5 and the Jackson’s theorem for the linear

Fourier basis [1, Subsection 2.2] yield that Ean(f)p � En(f ◦ θ−1

a )p = O(n−α). The proof ends. �

To estimate the convergence rate of differentiable functions, we denote the space of all the absolutelycontinuous functions on T by AC(T) and define

W rLp(T) := {f ∈ Lp(T) : ∃φ ∈ Cr−1(T) such that φ(r−1) ∈ AC(T), φ(r) ∈ Lp(T)

and f(x) = φ(x) a.e. x ∈ T}

for r ∈ N and 1 � p < ∞.For any f ∈ W rLp(T), the function φ ∈ Cr−1(T) such that φ(r−1) ∈ AC(T), φ(r) ∈ Lp(T) and

φ(x) = f(x) a.e. x ∈ T exists uniquely. For this reason, we assume without losing generality that f itselfis in Cr−1(T) and satisfies f (r−1) ∈ AC(T) and f (r) ∈ Lp(T).

Theorem 2.8. Suppose that f ∈ W rLp(T), r ∈ N, 1 � p < ∞, satisfies f (r) ∈ (Lipα)p, 0 < α < 1.Then Ea

n(f)p = O(n−r−α).

Proof. Assume that f (r−1) ∈ AC(T) and f (r) ∈ Lp(T). It is easy to see that (f ◦ θ−1a )(r−1) ∈ AC(T)

and (f ◦ θ−1a )(r) ∈ Lp(T). Since f (r) ∈ (Lipα)p, using (2.1) and Lemma 2.6, we obtain that (f ◦ θ−1

a )(r) ∈(Lipα)p. Hence, according to the Jackson’s theorem for the linear Fourier basis [1, Subsection 2.2], thereholds that En(f ◦ θ−1

a )p = O(n−r)ω((f ◦ θ−1a )(r), n−1)p = O(n−r−α). Finally, by Lemma 2.5 we have

Ean(f)p = O(n−r−α). �

3 Bernstein’s theorem

3.1 Bernstein’s theorem in C(T)

Theorem 3.1. Let f ∈ C(T) satisfy Ean(f)∞ = O(n−(r+α)) for some r ∈ Z+ and 0 < α � 1. Then

f ∈ Cr(T). Moreover, f (r) ∈ (Lipα)∞ when 0 < α < 1 and ω(f (r), t)∞ = O(t| ln t|) when α = 1.

Proof. We only consider the case that r � 1. The case that r = 0 can be handled similarly and moreeasily.

From Lemma 2.1, we know that En(f ◦ θ−1a )∞ = Ea

n(f)∞ = O(n−(r+α)). According to the classicalBernstein’s theorem for the linear Fourier basis [1, Subsection 2.3], we have f ◦θ−1

a ∈ Cr(T), (f ◦θ−1a )(r) ∈

(Lipα)∞ when 0 < α < 1 and ω((f ◦ θ−1a )(r), t)∞ = O(t| ln t|) when α = 1. It is easy to verify that

f = (f ◦ θ−1a ) ◦ θa ∈ Cr(T) and ω(f (r), t)∞ = ω([(f ◦ θ−1

a ) ◦ θa](r), t)∞ = O(ω((f ◦ θ−1a )(r), t)∞ + t), which

implies that f (r) ∈ (Lipα)∞ when 0 < α < 1 and ω(f (r), t)∞ = O(t| ln t|) when α = 1. The proof ends.�

3.2 Bernstein’s theorem in Lp(T)

Theorem 3.2. Let f ∈ Lp(T), 1 � p < ∞. If Ean(f)p = O(n−(r+α)), where r ∈ Z+ and 0 < α < 1,

then f ∈ W rLp(T) and f (r) ∈ (Lipα)p.

Proof. According to Lemma 2.5, we have En(f ◦ θ−1a )p = O(Ea

n(f)p) = O(n−r−α). By the classicalBernstein’s theorem for the linear Fourier basis [1, Subsection 2.3], we deduce that there is a function g ∈Cr−1(T) such that g(r−1) ∈ AC(T), g(r) ∈ Lp(T), f ◦ θ−1

a (x) = g(x) a.e. x ∈ T, and ω(g(r), t)p = O(tα),i.e., g(r) ∈ (Lipα)p.

Let φ = g ◦ θa. It is easy to verify that φ ∈ Cr−1(T), φ(r−1) ∈ AC(T) and φ(r) ∈ Lp(T). ByLemma 2.6 and (2.1), we conclude that φ(r) ∈ (Lipα)p from g(r) ∈ (Lipα)p. Let E be a subset of T

whose Lebesgue measure |E| is 0. It can be easily shown that |θ−1a (E)| is also 0. Thus the fact that

f ◦ θ−1a (x) = g(x) a.e. x ∈ T implies that f(x) = g ◦ θa(x) = φ(x) a.e. x ∈ T. Then it follows that

f ∈ W rLp(T) and f (r) ∈ (Lipα)p. The proof ends. �


4 Jackson’s and Bernstein’s theorems in Hardy spaces

Approximation in Hardy spaces is a classical and still interesting topic. Since the nontangential boundaryvalue of a function in the Hardy space Hp(U) is a function in Lp(T), the corresponding approximationtheorems in Lp(T) can be established in the Hardy space. As mentioned in Section 1, for every k ∈ Z+,eikθa(t) is the nontangential boundary value of

(τa(z))k =(

z − a

1 − az

)k

,

which is a sequence of functions in the Hardy space Hp(U) for 1 � p � ∞.The definition for the classical Hardy space can be found in many books on complex analysis such

as [6, 7, 13]. Let U := {z : |z| < 1}, we denote by H(U) the set of all the holomorphic functions definedon U. For 1 � p � ∞, the Hardy space is defined by

Hp(U) := {f ∈ H(U) : ‖f‖Hp(U) < ∞}, where ‖f‖Hp(U) := sup0�r<1

‖f(rei·)‖Lp(T).

It is a well-known fact that the mapping f �→ f(ei·), where f(ei·) stands for the nontangential boundaryvalue of f , is an isometric and linear mapping from Hp(U) to

ALp(T) := {f ∈ Lp(T) : c−k(f) = 0 for all k ∈ N} ,

where ck(f) is the k-th Fourier coefficient of f defined by

ck(f) :=12π

∫T

f(t)e−iktdt, k ∈ Z,

and (ref. [6, p. 21])‖f‖Hp(U) = ‖f(ei·)‖Lp(T), ∀f ∈ Hp(U). (4.1)

Based on this fact, the analogous Jackson’s and Bernstein’s theorems can be established in Hp(U) forthe sequence of functions {τk

a }k∈Z+ , which are called the finite Blaschke products.Each function f in ALp(T), which is called an analytic signal, satisfies Hf = −i(f − c0(f)), where H

stands for the classical Hilbert transform defined by

Hf(t) := p.v.12π

∫T

f(t − s) cots

2ds, (4.2)

where p.v. expresses the Cauchy principal value of the singular integral. It is well-known that H is alinear and bounded operator on Lp(T), 1 < p < ∞, thus, ‖Hf‖Lp(T) = O(‖f‖Lp(T)) for any f ∈ Lp(T).

To set up the approximation theorems in the Hardy space Hp(U), we introduce the modulus ofsmoothness and the approximation error as follows: The modulus of smoothness in the Hardy spaceHp(U), 1 � p � ∞, is defined by

ω(f, t)∗p := sup0�h�t

‖f(zeih) − f(z)‖Hp(U), ∀f ∈ Hp(U).

Accordingly, for 0 < α � 1, the Lipschitz class of order α in Hp(U) is defined by

(Lipα)∗p := {f : |f |(Lipα)∗p < ∞}, where |f |(Lipα)∗p := supt>0

{t−αω(f, t)∗p}.

We denote by Ban the linear space generated by the Blaschke product sequence {τk

a }k∈Z+ , that is,

Ban := span{τk

a : 0 � k � n}.

Then the approximation error of f ∈ Hp(U), 1 � p � ∞, by Ban is defined by

Ean(f)∗p := inf

B∈Ban

‖f − B‖Hp(U).


According to (4.1), we easily conclude that ω(f, t)∗p = ω(f(ei·), t)p holds for any f ∈ Hp(U), 1 � p � ∞.Consequently, for any 0 < α � 1, f ∈ (Lipα)∗p if and only if f(ei·) ∈ (Lipα)p. In the following, we givethe equivalence between Ea

n(f)∗p and Ean(f(ei·))p.

Lemma 4.1. Let f ∈ Hp(U), 1 < p < ∞. Then Ean(f)∗p � Ea

n(f(ei·))p.

Proof. Choose coefficients c0, c1, . . . , cn such that

Ean(f)∗p =

∥∥∥∥f −n∑

k=0

ckτka

∥∥∥∥Hp(U)

.

By (4.1) we have

Ean(f)∗p =

∥∥∥∥f(ei·) −n∑

k=0

ckeikθa

∥∥∥∥Lp(T)

� Ean(f(ei·)). (4.3)

To get the inverse inequality, we choose coefficients c−n, . . . , c0, . . . , cn such that

Ean(f(ei·)) =

∥∥∥∥f(ei·) −n∑

k=−n

ckeikθa

∥∥∥∥Lp(T)

= ‖g + h‖Lp(T), (4.4)

where g := f(ei·) −∑n

k=0 ckeikθa and h := −∑n

k=1 c−keikθa . Since g and h are respectively the non-tangential boundary values of the Hp(U) functions f(z) −

∑nk=0 ckτk

a (z) and −∑n

k=1 c−kτka (z), we have

that Hg = −i(g − c0(g)) and Hh = −i(h − c0(h)), which yield H(g + h) = −i(g − h + 2L0), whereL0 := (−c0(g) + c0(h))/2 and consequently

‖g − h + 2L0‖Lp(T) = ‖H(g + h)‖Lp(T) = O(‖g + h‖Lp(T)) = O(Ean(f(ei·))). (4.5)

Combining (4.4) and (4.5), we obtain that

‖g + L0‖Lp(T) � 12(‖g + h‖Lp(T) + ‖g − h + 2L0‖Lp(T)) = O(Ea

n(f(ei·))).

Using (4.1) we have

Ean(f)∗p �

∥∥∥∥f + L0 −n∑

k=0

ckτka

∥∥∥∥Hp(U)

= ‖g + L0‖Lp(T) = O(Ean(f(ei·))). (4.6)

Finally, by (4.3) and (4.6) we obtain that Ean(f)∗p � Ea

n(f(ei·))p. �

Based on the norm equivalence relation (4.1) and Jackson’s theorem 2.7 and Bernstein’s theorem 3.2and Lemma 4.1, we easily obtain the following approximation theorem in the Hardy space Hp(U), 1 <

p < ∞.

Theorem 4.2. Let f ∈ Hp(U), 1 < p < ∞. Then for any given 0 < α < 1, Ean(f)∗p = O(n−α) if and

only if f ∈ (Lipα)∗p.

To discuss the high order approximation theorems in Hardy spaces, we denote

W rHp(U) := {f ∈ H(U) : f, f ′, . . . , f (r) ∈ Hp(U)}, r ∈ N.

The following lemma gives the relation between the smoothness of a function in W rHp(U) and that ofits nontangential boundary value in W rLp(T).

Lemma 4.3. Let f ∈ Hp(U), 1 < p < ∞. Then for any given r ∈ N, f ∈ W rHp(U) if and only iff(ei·) ∈ W rLp(T).

Proof. Let f ∈ W rHp(U). Then for any 0 � k � r − 1 and t0, t ∈ T there holds that

f (k)(ρeit) − f (k)(ρeit0) =∫ t

t0

iρeixf (k+1)(ρeix)dx, 0 < ρ < 1.


Assume that t0 satisfies limρ→1− f (k)(ρeit0) = f (k)(eit0). The fact that f (k+1) ∈ Hp(U) implies thatFk(x) := supρ<1 |f (k+1)(ρeix)| ∈ Lp(T) (see [6, p. 12]). Then, setting ρ → 1− we obtain that

f (k)(eit) − f (k)(eit0) =∫ t

t0

ieixf (k+1)(eix)dx a.e. t ∈ T,

which concludes that f (k)(eit) is absolutely continuous on T and

[f (k)(ei·)]′(t) = ieitf (k+1)(eit) a.e. t ∈ T,

in which the terms f (k)(eit) and f (k+1)(eit) are respectively the nontangential boundary values of f (k)(z)and f (k+1)(z). Accordingly,

[f(ei·)](r)(t) = ireirtf (r)(eit) +r−1∑k=1

ckeiktf (k)(eit) a.e. t ∈ T, (4.7)

where {ck}r−1k=1 are some complex constants. Hence [f(ei·)](r) ∈ Lp(T) and consequently f(ei·) ∈ W rLp(T).

Conversely, let f(ei·) ∈ W rLp(T). Inductively we assume that, for 0 � k � r−1, the k-th derivative off ∈ Hp(U), f (k), is in Hp(U) and its nontangential boundary value f (k)(ei·)∈W 1Lp(T). By [8, p. 410],we have

f (k)(z) =1

2πi

∮|ξ|=1

f (k)(ξ)ξ − z

dξ =12π

∫T

eit

eit − zf (k)(eit)dt, |z| < 1.

It is followed that

f (k+1)(z) =12π

∫T

eit

(eit − z)2f (k)(eit)dt =

12π

∫T

eit

eit − zFk(t)dt, |z| < 1,

where Fk := (−ie−i·)[f (k)(ei·)]′ ∈ ALp(T). Hence f (k+1) ∈ Hp(U), ‖f (k+1)‖Hp(U) = ‖Fk‖Lp(T) andf (k+1)(eit) = Fk(t) ∈ Lp(T). By induction we have f ∈ W rHp(U). The lemma is proved. �

Theorem 4.4. Let f ∈ Hp(U), 1 < p < ∞. Then for any given r ∈ N and 0 < α < 1, Ean(f)∗p =

O(n−r−α) if and only if f ∈ W rHp(U) and f (r) ∈ (Lipα)∗p.

Proof. By Lemma 4.1 we know that Ean(f)∗p = O(n−r−α) if and only if Ea

n(f(ei·))p = O(n−r−α), which,according to Theorems 2.8 and 3.2, is equivalent to f(ei·) ∈ W rLp(T) and [f(ei·)](r) ∈ (Lipα)p. ByLemma 4.3 and (4.7), it is further equivalent to f ∈ W rHp(U) and ω(f (r), t)∗p = ω(f (r)(ei·), t)p = O(tα),that is, f (r) ∈ (Lipα)∗p.

Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos.

11071261, 60873088, 10911120394).

References

1 Butzer P L, Nessel R J. Fourier Analysis and Approximation, volume 1. New York: Academic Press, 1971

2 Chen Q H, Li L Q, Qian T. Stability of frames generated by nonlinear fourier atoms. Int J Wavelets Multiresolut Inf

Process, 2005, 3: 465–476

3 Chen Q H, Li L Q, Qian T. Two families of unit analytic signals with nonlinear phase. Phys D, 2006, 221: 1–12

4 DeVore R A. Nonlinear approximation. Acta Numer, 1998, 7: 51–150

5 DeVore R A, Lorentz G G. Constructive Approximation. New York: Springer-Verlag, 1993

6 Duren P L. Theory of Hp Spaces. New York: Academic Press, 1970

7 Garnett J B. Bounded Analytic Functions. New York: Academic Press, 1981

8 Greene R E, Krantz S G. Function Theory of One Complex Variable, 3rd ed. Providence, RI: American Mathematical

Society, 2006

9 Mo G D, Liu K D. Methods for Approximation Theory (in Chinese). Beijing: Science Press, 2003

10 Qian T. Analytic signals and harmonic measures. J Math Anal Appl, 2006, 314: 526–536

11 Qian T, Chen Q H. Characterization of analytic phase signals. Comput Math Appl, 2006, 51: 1471–1482

12 Qian T, Chen Q H, Li L Q. Analytic unit quadrature signals with nonlinear phase. Phys D, 2005, 303: 80–87

13 Rudin W. Real and Complex Analysis, 3rd ed. New York: McGraw-Hill, 1987

approximation by the nonlinear fourier basis

Documents