A new approach to Lipschitz spaces of periodic integrable functions

SPACES OF PERIODIC

INTEGRABLE FUNCTIONS

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Miguel A. Jimenez

Benemerita Universidad Autonoma de Puebla Abstract:

The usual de nition of Lipschitz subspaces ofL^p2 ,1 p <¹ is modi ed in order to obtain homogeneous Banach spaces and a Hilbert space for p = 2: In the latter case it is shown that the trigonometric system is an orthogonal basis.

1. Introduction

To introduce the Lipschitz spaces we have restricted ourselves to the linear space ^F2 , of all real Lebesgue measurable 2^;periodic functions dened on the real space IR with the usual identication of points modulo 2:

The continuous functions in ^F2 , form a particular space denoted by C² , which becomes a Banach space under the sup-norm^kk1: This is also the space of all continuous real functions on the interval 02 equipped with the metric (1:1) ⁸xy²02 d(xy) := min^fjx^;y^j2^;jx^;y^jg:

The other well known Banach spaces L^p2 , 1 p < ¹, consist of all functions f for which

Partially supported by CONACyT project 3749P-E9608 and SNI, Mexico and DAAD grant A-98-0265, Germany

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Key words: Homogeneous Banach space, Lipschitz (or Holder) space, Fourier series, best approximation.

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(1:2) ^kf^k_p := ^{1 2R}

0

jf (x)^jpdx

! 1p

<¹:

Here, as usual, two functions that are equal a.e. (i.e. equally Lebesgue almost everywhere) are identied.

For a function f, we denote

(1:3) ⁴t(fx) := f (x + t)^;f (x) t > 0

and for 0 < 1 1 p ¹ , the Lipschitz space Lipp is the class of all functions f ²L^p2 , if 1 p <¹ or f ²C² if p = ¹ such that

(1:4) 'p(f) := supⁿt^;k4t(fx)^k_p :t > 0^o<¹:

For > 1 and each p1, the only functions that (1:4) holds for are constant.

Since Lipp is a linear space and 'p is a semi-norm, a natural norm on Lipp

is usually given by

(1:5) ^kf^kp :=^kf^kp+ 'p(f):

Then one proves that Lipp is a Banach space.

Now, let us denote by ^Tn the nite dimensional linear space of all trigono- metric polynomials of degree n

(1:6) Tn(x) := ^a20 +^P_nk⁼¹(akcoskx + bksinkx):

For any Banach space B such that n^Tn B ^F2 we denote the best approximation off ²B to ^Tn in the norm ofB, by

(1:7) En(f) := En(fB) := inf ^fkf^;Tn^kB :Tn^2Tn^g:

In the above frame, with^k:^k_B =^k:^k_p 1 p ¹ a series of typical problems in Approximation Theory have been well studied for functions in Lipp and, at present, they form an important part of the basis of Approximation Theory. Here we only quote the representative advanced books 2], 4], 6], 7].

When ^k:^kB=^k:^kp , the main trouble is that translations are not continuous operators with respect to the parameter. To explain this situation and for further use let us recall (c. f. 10] and 14] for instance) that a Banach space B L¹² is homogeneous if there exits a constant C > 0 such that ^kf^k1 C^kf^k_B for every f ²B and if the following two conditions concerning translations are satised

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(H¹) f ²B and h²R imply f (x + h)²B and ^kf(x + h)^kB =^kf(x)^kB (H²) f ²B, hh^{0 2}R and h ^!h⁰ imply ^kf(x + h)^;f(x + h⁰)^kB ^;!0.

Many good properties of approximation by Fourier series have been proved for homogeneous Banach spaces. However, Lipp are not homogeneous because they do not satisfy (H²) and this is no good news. In particular, the sequence En(fLip_p) does not always converge to zero.

With this bad property at hands, the researches have been organized follow- ing several directions. We only quote here a few representative papers which together with the already mentioned books, give an idea of the State-of-the-Art in a neighbourhood of our subject (c.f. 1], 3], 5], 8], 9], 11], 12], 13]).

However, we will see in this paper that an appropiate modication of the denition of Lipschitz spaces for 1 p < ¹ , provides us homogeneous Banach spaces. Moreover, for p = 2 the corresponding version leads to a Hilbert space where the trigonometric system

(1:8) ¹² cos(x)sin(x)cos(kx)sin(kx) is orthogonal and complete.

Then several questions on Fourier series and on best approximation by trigono- metric polynomials in these spaces could be viewed in a frame similar to this one in L^p2 spaces. This is the goal of the paper.

I am indebted to my colleague Jorge Bustamante, who has supported me with valuable advice.

2. The spaces B

p1 p <¹

Our rst objective is to extend the functiond on 01², given by (1:1), to the whole plane. We dene

(2:1) ⁸xy²02, ⁸jk ²ZZ, d(x + 2jy + 2k) = d(xy).

It is easy to prove the following result which is a corner stone for our purposes:

Proposition 1

The function d is a pseudometric that is 2^;periodic in each of its two variables and translation invariant, i.e.

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(2:2) ⁸xyh²IR, d(x + hy + h) = d(xy):

In the following we assume that > 0 is xed and denote by^{F(2 )2} the space of all real Lebesgue measurable functions on IR² that are 2^;periodic in each variable. We dene the translation operator on ^{F(2 )m} m = 12 by

(2:3) (Thf)(x) = f(x+h) and (Thf)(xy) = f(x+hy +h), h ²IR respectively. Then Th denotes two dierent linear operators.

We introduce the operator F : ^{F2 ;!F(2 )2} by

(2:4) (Ff)(xy) = ^f(_d^x();_xy^f)(y) x⁶=y mod (2) (or 0 ifx = y mod (2)) A simple but important remark is that F is linear and antisymmetric. This property means that

(2:5) ⁸xy²IR, ⁸f ^2F2 , (Ff)(xy) =^;(Ff)(yx)

Proposition 2

The operators Th and F commute in the following sense:

(2:6) ⁸f ^2F2 , ⁸xyh²IR, F(Thf)(xy) = Th(Ff)(xy).

Proof

We use proposition 1. For x⁶=y mod (2)

F(Thf)(xy) = ^Thf(_d^x();_xy^T)hf(y) = ^f(_d^x(+_x^+h);_hy^f+(_h^y+)h) =Th(Ff)(xy) LetL^{p(2 )2} be the Banach spaces of functions f^{2F(2 )2} for which (2:7) ^kf^kp := ^{12 2R}

0 2

R

0

jf (xy)^jpdxdy

!1

p <¹ 1 p < ¹

Denition 1

^{Fix 1} p <^1and > 0:^{The space}Bp is the class of all functions f ²L^{p2 for which} Ff ²L^{p(2 )2}:

Clearly,^kF(:)^k_p is a semi-norm onBp: Then Bp is a normed space with 4

(2:8) ^kf^kp :=^kf^k_pp+^kF(f)^k_pp^1=p:

We will prove in Remark 1 after formula (2:23) that ^\>¹Bp is reduced to constant functions for every 1 p < ¹. This is the only reason for which we restrict ourselves to the bound 1.

Theorem 1

For every 1 p <¹ and0< 1the spaceBp is a homogeneous Banach Space.

Proof

We begin with the proof that the space is complete.

Let (fn) be a Cauchy sequence inBp: In particular, (fn) is a Cauchy sequence in L^p2 : Then there exists f ²L^p2 such that

(2:9) ^kfn^;f^kp ^;!0, if n ^;!1. We need to prove that f ²Bp and that

(2:10) ^kF(fn^;f)^kp ^;!0 if n ^;!1.

Since f ² L^p2 and ^kFf^k_p ^kF(fn^;f)^k_p+ ^kFfn^kp the assertion that f ²Bp automatically follows from (2:10):

To prove this last property, observe that (Ffn) is also a Cauchy sequence in L^{p(2 )2}: Then there is a g ²L^{p(2 )2} such that

(2:11) ^kFfn^;g^kp ^;!0, if n^;!1:

On the other hand^kF(fn^;f)^k_p =^kFfn^;Ff^k_p: So we only have to prove that

(2:12) Ff = g a.e.

By (2:9), there exists a subsequence (fnj) converging to f a.e. on 02 and by (2:11), another sub-sequence (Ffn_jk) converges to g a.e. on 02²: Then (2:12) holds.

To prove the properties (H¹) and (H²) inBp, we will utilize the fact that both of them are satised in L^{p(2 )m} m = 12, as well as proposition 2. Let f ²B_p be given and h > 0: Then

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kThf^kp_p= ^kThf^kp_p+^kF(Thf)^kp_p =^kThf^kp_p+^kTh(Ff)^kp_p

=^kf^kp_p+^kFf^kp_p =^kf^kp_p

So (H¹) holds inBp. Further, it is enough to consider the caseh⁰ = 0 in (H²):

kThf^;f^kp_p =^kThf^;f^kp_p+^kF(Thf ^;f)^kp_p =^kThf ^;f^kp_p+^kTh(Ff)^;Ff^kp_p that converges to 0 if h tends to 0.

Corollary 1

For each f ²Bp and each summability kernel of 2^;periodic con- tinuous functions (Kn) in L¹² , one has

(2:13) ^kKnf ^;f^k_p ^;!0, if n^;!1. In particular, the Fejer's sums of the Fourier series (2:14) ^a20 +^P1_n⁼¹(ancosnx + bnsinnx) of f where

(2:15) an:= ^{1 2R}

0

f(t)cosnt dt and bn := ^{1 2R}

0

f(t)sinnt dt n = 012 converge tof in the norm^kk_pand the trigonometric polynomials are everywhere dense in B_p:

Proof

See paragraph 2, Chapter 1, of 10], that also includes the denition of summability kernels.

Now one might wish to simplify the double integrals that appear in our ap- proach. We dene the domains:

D : = 02²

D¹ : = ^f(xy)²D : 0 x and x + y 2^g D² : = ^f(xy)²D : 0 x 2 and x y x + ^g D³ : = ^f(xy)²D : (yx)²D^2g

D⁴ : = ^f(xy)²D : (yx)²D^1g

D⁵ : = ^f(xy)²R² : x 2 and 2 y x + ^g Then, for everyf ²L¹(D) = L^{1(2 )2} we have

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(2:16) ^R_D f =_i^P4

=0 R

Dif

and using (2:5), for g = ^jFf^jp if f ² Bp or g = Fu Fv if u ² Bp and v ²B^q 0 < 1, 1=p + 1=q = 1 :

(2:17) ^R_D²_D³ g = 2^R_Di g, i = 23 (2:18) ^RD¹D⁴ g = 2^RDi g, i = 14

Now, sinceg is 2^;periodic in each variable, it follows from (2:16^;17^;18) that

(2:19) ^RD g = 2^RD²D⁵g.

With techniques of Measure Theory, one can write

(2:20) ^RD²D⁵f =^{R0 hR02} f (xx + t)dxⁱ dt for f ²L¹(D²D⁵).

Then, from (2:19^;20), we rediscover the most familiar formulas (2:21) ^2R

0 2

R

0

jFf (xy)^pjdx dy = 2^{R0 R02 f(x);}_t^f(x+t)pdxdt (2:22) ^2R

0 2

R

0

Fu(xy)Fv (xy)dxdy = 2^R

0 2

R

0

(u⁽x^);u⁽x⁺t⁾⁾⁽v⁽x^);v⁽x⁺t⁾⁾

t ⁺ dxdt

if u²Bp ,v ²B^q 0 < 1 < pq <¹ and 1=p + 1=q = 1 Finally we also has

(2:23) ^hRD =^R_D²_D³ =^R_D¹_D⁴ⁱFf = 0, for f ²B¹

Remark 1

The equation (2:21) above shows that Bp is not necessarily reduced to constant functions when > 1: In fact, from this equality and under the optimal assumption on f that there exists a constant C := C(f) > 0 such that ^j4t(fx)^j C t for every t > 0 we deduce that Ff ² L^{p(2 )2} whenever < (p + 1)=p. However, since (p + 1)=p ^;! 1 if p tends to ¹ , we have that 1 represents the common case for every 1 p < ¹: In other words, if f is not a constant function and > 1 , then there exists p <¹ such that f =²Bp:

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Proposition 3

For every 0 < 1 and 1 p < ¹, the classical Lipschitz spaces Lipp de ned in Section 1 are continuously embedded in Bp by the identity operator.

Proof

For any positive nite measure space (X ) and 1 p < ¹ there exits a constant C := Cp > 0 such that ^k:^k_p C^k:^k1: On the other hand, the semi-norm'p in (1:4) could be equivalently dened with 0 < t (see Sec. 4.1 of 2]). Then, the proposition follows from (2:22):

Proposition 4

^{For every} 0< 1 and 1< p < ¹ the spaces B_p are strictly convex.

Proof

Let fg²Bp be such that ^kf^k_p =^kg^k_p = 1. It follows that

kf + g^k_p =^kf + g^kp_p +^kF(f) + F(g)^kp_p^1=p

kf^k_p +^kg^k_p^p+ ^kFf^k_p+^kFg^k_p^p1=p

kf^kp_p +^kF(f)^kp_p^1p +^kg^kp_p+^kF(g)^kp_p^1=p= 2:

Then ^kf + g^k_p = 2 is possible only iff = g a. e.

As a consequence we have the non-trivial result:

Corollary 2

^{For every} f ²B_p , 0< 1 , 1< p < ¹ andn = 12:: there is a unique polynomial of best approximation of f to ^Tn in Bp.

3. The Hilbert Space B

In this section we usually write B instead of B²: We shall prove that the trigonometric system is an orthogonal basis of this space. In fact, one easily proves:

Proposition 5

The bilinear functional

(3:1) (f ^jg) := (f ^j g)_L²2 + (Ff ^jFg)_L²

(2⁾² fg ²B 8

is an inner product whose associated norm ^kf^k = (f ^jg)¹² is equal to ^kf^k2: Here

(f ^jg)_L²2 = ^{1 R02} f(x)g(x)dx (Ff ^jFg)L²

(2⁾² = ^{12 R02 R02} Ff(xy)Fg(xy)dxdy:

Then the general results of Hilbert spaces hold inB:

Theorem 2

The trigonometric system (1:8) is an orthogonal basis of B whose elements have the norms:

(3:2) ¹² = 1 , ^kcos(kx)^k2 =^ksin(kx)^k2 =N(k)² = 1 + ^{4 R0 1;cos(}_t^{2 mt)}dt for k = 12

Proof

The trigonometric system is a set of B whose nite linear combinations (i. e. the trigonometric polynomials) are everywhere dense in B as we stated in Corollary 4. In order to prove that is an orthogonal basis, we only need to check the orthogonality condition. But this, as well as (3.2), are straightforward tasks accomplished by means of (2.21-22) and using that (1.8) is an orthonormal basis in L²² :

I have not found any reference to the following striking result. Then the proof is given here:

Theorem 3

^Let H be any Hilbert space, with inner product (^j)_H and norm

k:^k_H . Let F be a linear subspace of H that becomes a Hilbert space under the inner product (^j)_F and such that ^k:^k_H ^k:^k_F on F. If ^fuj :j = 12::^g is an orthonormal basis of H that simultaneously is an orthogonal basis of F ^{then for} every f ²F, the Fourier series of f are formally equal for both spaces.

Proof

Put Cj := ^kuj^kF. Since the Fourier series ^Pj

f ^j _C^uj_{jF C}^uj_j converges to f in^k:^k_F , it also converges to f in H due to the hypothesis on the norms. Then (f ^juj)_H = _C¹²_j (f ^juj)_F for j = 12:::

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Corollary 3

For every f ²B , the Fourier series of f in this space is given by (3:3) f(x) = ^a20 +^P1_n⁼¹ (an cos(nx) + bnsin (nx))

where an and bn , n = 0,1,2, are calculated in the usual form remembered in (2:15) !

Then a function

f (x) = ^A2o +^P1n⁼¹(Ancos(nx) + Bnsin(nx))²L²² is in B if and only if

P

1n⁼¹N(n)²(An²+B_n²)<¹.

Proof

Combine the last two theorems .

Corollary 4

For every f ²B and n = 12:: the polynomials of best approxi- mation of f ^{to T}n in B are the partial sums of the Fourier series of f ^{given by} (3:3) and

En(fB)² =_k ^P1

=n⁺¹N(k)²(a²k+b²k).

References

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RWTH Aachen, Sharker Verlag, 1996

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Springer-Verlag, 1971

Author's Address:

Apartado Postal J-27 Colonia San Manuel Puebla 72571, Pue., Mexico e. mail: mjimenez@fcfm.buap.mx

fax: (22) 33 24 03

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