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Volume 2008, Article ID 451815, 11 pagesdoi:10.1155/2008/451815 Research Article Some Classes of Convolution Functions Feng Guo 1, 2 1 School of Mathematical Sciences, Beijing Normal Uni

Volume 2008, Article ID 451815, 11 pages

doi:10.1155/2008/451815

Research Article

Some Classes of Convolution Functions

Feng Guo 1, 2

1 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

2 Department of Mathematics, Taizhou University, Taizhou, Zhejiang 317000, China

Correspondence should be addressed to Feng Guo, gfeng@tzc.edu.cn

Received 10 December 2007; Revised 7 April 2008; Accepted 16 June 2008

Recommended by Vijay Gupta

We consider some classes of 2π-periodic convolution functions  B p, and K p, which include the classical Sobolev class as a special case With the help of the spectra of nonlinear integral equations,

we determine the exact values of Bernstein n-width of the classes  B p, K p in the space L p for

1 < p <∞.

Copyright q 2008 Feng Guo This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction and main results

Let X be a normed linear space and let A be a subset of X Assume that A is closed, convex, and

centrally symmetrici.e., x ∈ A implies −x ∈ A The Bernstein n-width, which was originally

introduced by Tikhomirov1, of A in X is given by

b n A; X  sup

X n1

sup

X n1

⊆ A, 1.1

where SX n1  {x : x ∈ X n1, x ≤ 1} and X n1is taken over all subspaces of X of dimension

at least n  1 Let T : 0, 2π be the torus, and as usual, let L q : Lq 0, 2π be the classical Lebesgue integral space of 2π-periodic real-valued functions with the usual norm·q , 1 ≤ q ≤

∞

Denote by W r

p the classical Sobolev class of real functions f whose r − 1th derivative is absolutely continuous and whose rth derivative satisfies the condition f rq ≤ 1 The concept

of Bernstein n-width for the Sobolev classes W r

p was originally introduced by Tikhomirov1

He considered b n W r

p ; L q , 1 ≤ p, q ≤ ∞, and found the exact value of b 2n−1 W r

∞; L∞ Pinkus

2 obtained the exact value of b 2n−1 W r

1; L1 Later, Magaril-Il’yaev 3 obtained the exact value

of b 2n−1 W r

p ; L p , 1 < p < ∞ The latest contribution to this field is due to Buslaev et al 4 who

found the exact values of b 2n−1 W r

p ; L q  for all 1 < p ≤ q < ∞.



j1

b j G

· − t j

 :

m



j1

1.2

is of dimension m, and is a weak Tchebycheff- WT- system see 2, page 39 for all m odd A

real, 2π-periodic, continuous function G is said to be B-kernel if G satisfies property B.

One says that K is a cyclic variation diminishing kernel of order 2m−1 CVD2m−1 if there exist

σ n ∈ {−1, 1}, n  1, , m, such that

x i − y j

2n−1

i,j1 ≥ 0, 1.3

for all x1 < · · · < x 2n−1 < x1 2π and y1 < · · · < y 2n−1 < y1 2π One will drop the subscript 2m− 1 from the acronyms CVD, if one assumes that these properties hold for all orders One

says that K is nondegenerate cyclic variation diminishing NCVD if K is nonnegative CVD

and

dim span

x1− ·, , K

x n− · n, 1.4 for every choice of 0≤ x1< · · · < x n < 2π and all n∈ N

Now, we introduce the classes of functions to be studied Let K be a NCVD kernel2

and let G be a B-kernel The 2π-periodic convolution function classes  K pand B pare defined as follows:

B p:f : f

x   G∗hx  a, a ∈ R, h ⊥ 1, h p ≤ 1,



where

g∗hx :

Tg x − yhydy, 1.6

and h⊥ 1 meansTh ydy  0.

The exact values of b n  B p ; L q  and b n K p ; L q  are known for the cases p  q  1, p 

q  ∞, and n is odd see 2 for more details Chen 5 is the one who found the lower

estimate of b 2n−1  B p , L p  and b 2n−1 K p , L p  for 1 < p < ∞ In this paper, we will determine the

exact constants of some classes of periodic convolution functions B p with B-kernelor NCVD-kernel for p ∈ 1, ∞, which include the classical Sobolev class as its special case

Now, we are in a position to state our main results of this paper.

Theorem 1.3 Let G be a B-kernel, and n  1, 2, Then

b 2n−1 B p ; L p

 λ n p, p, G, 1 < p < ∞, 1.7

s 2n B p ; L p

 b 2n−1 B p ; L p

 λ n p, p, G, 1.8

where

n



 −hx, hx{sin} nx ≥ 0, h p≤ 1



,

λ n: λn p, q, G  {sup}{G∗h q : h ∈ D n , }, 1 < q ≤ p < ∞,

1.9

Theorem 1.4 Let K be a {NCVD} kernel and n  1, 2, Then

b 2n−1 K p ; L p

 λ n p, p, K, 1 < p < ∞,

s 2n K p ; L p

 b 2n−1 K p ; L p

 λ n p, p, K,

λ n p, q, K  {sup}{K∗h q : h ∈ D n , }, 1 < q ≤ p < ∞.

1.10

We will only give the proof for the case of a B-kernel As for the case of a NCVD kernel,

the proof is similar and even more simple

2 Nonlinear integral equation and its spectral couple

Before we proveTheorem 1.3, we need some results about nonlinear integral equations and their spectral couple First, we introduce some definitions and notations

i S−x indicates the number of sign changes in the sequence x1, , x nwith zero terms

discarded The number S−c x of cyclic variations of sign of x is given by

S−c x : max

i S−x i , x i1, , x n , x1, , x i   S−x k , , x n , x1, , x k



, 2.1

where k is some integer for which x k /  0 Obviously, S−

c x is invariant under cyclic permutations, and S−c x is always an even number.

ii Sx counts the maximum number of sign changes in the sequence x1, , x nwhere zero terms are arbitrarily assigned values1 or −1 The number S

c x of maximum cyclic variations of sign of x is defined by

Sc x : max

i S

x i , x i1, , x n , x1, , x i



Let f be a piecewise continuous, 2π-periodic, real-valued function onR One assumes

that f x  fx  fx−/2 for all x and

S c f : sup S−

c



x1



, , f



where the supremum is taken over all x1< · · · < x m < x1 2π and all m ∈ N.

Moreover, one needs further counts of zeros of a function Suppose that f is a continuous, 2π-periodic, real-valued function onR One defines



Z c f : sup S

c



x1



, , f



where the supremum runs over all x1 < · · · < x m < x1 2π and all m ∈ N Assume that f is a 2π-periodic, real-valued function on R for which f is sufficiently smooth The number of zeros

of f on a period, counting multiplicities, is denoted by Z 

c f.

Clearly, S c f denotes the number of sign changes of f on a period, and  Z c f denotes the number of zeros of f on a period, where the zeros which are sign changes are counted once

and zeros which are not sign changes are counted twice Moreover, we have

S c f ≤  Z c f ≤ Z 

We define Q pto be the nonlinear transformation:



Q p f

t :f tp−1signft, 1 < p < ∞. 2.6

Since the function F y : |y| p−1sign y is continuous and strictly increasing, Q p f is continuous

if and only if f is Moreover, since Fy is uniformly continuous on every compact interval,

Q p f ∈ L p , p  p/p − 1, and Q p Q p f  f for every f For 1 ≤ q, p < ∞, f, λ q is called a

spectral couple, and f is called a spectral function if

h p  1, fx  G∗hx  β,



Q p h

y  λ −q

TG x − yQ q f

where β satisfies the condition

inf

c∈RG∗h  c q  G∗h  β q , 2.8 when

TG xdx  0 It is well known that if 1 < q < ∞, then β is unique The set of all spectral

couples is denoted byΓp, q, G, and the spectral class Γ 2n p, q, G is given by

Γ2n p, q, G :f, λ q

∈ Γp, q, G : S c f  2n. 2.9

Lemma 2.2 see 2, page 177 Let φ be a real piecewise continuous 2π-periodic function satisfying

φ ⊥ 1 and set ψx : a  G∗φx If G satisfies property B, then



Z c ψ ≤ S c φ. 2.10

Lemma 2.3 For 1 < p, q < ∞, if f, λ q  ∈ Γp, q, G with S c h < ∞ Then, f has a finite number of

zeros, and all its zeros are simple.



Q p h

≤ S c



Q q f

 S c f Obviously, S c f  S c h   Z c f Therefore, f has a finite number of zeros, and all its zeros

are simple

Lemma 2.4 a If 1 < q < p < ∞, and f1and f2are two spectral functions, then

S c



f1 f2



≤ maxS c



f1



, S c



f2



< ∞. 2.11

b If 1 < q ≤ p < ∞, and f1and f2correspond to the same spectral value and f1/  f2, then all

σ ε : S c f1εf2 For all sufficiently small ε, we have σε  S c f1  Z c f1 : N Indeed, let

for all small ε, so that f1 εf2has exactly one zero in each V t i On the other hand, f1 εf2/ 0 if

t ∈ T \i V t i  and ε > 0 is sufficiently small By using 2.5–2.7,Lemma 2.2, and the identity signa  b  sign|a|p−1sign a  |b| p−1sign b, we have

σ ε  S c



f1 εf2



≤ Z c



f1 εf2



≤ S c



h1 εh2



 S c



Q p h1 Q p



εh2



 S c



Q p h1 ε p−1

Q p h2



≤ S c



λ −q1 Q q f1 ε p−1λ −q2 Q q f2



 S c



Q q f1 Q q



ε p−1/q−1 λ1/λ2q/ q−1 f2



 S c



f1 ε p−1/q−1 λ1/λ2q/ q−1 f2



 σε p−1/q−1 λ1/λ2q/ q−1

.

2.12

Iterating this inequality for 0 < ε < 1, we obtain σε ≤ σε0, where ε0 can be made arbitrarily close to zero due to 1 < q < p < ∞, so that we may assume that σε0  N Consequently, σε ≤ N for 0 < ε < 1 But then also σ1  S c f1 f2 ≤ N for otherwise one can choose ε < 1 so close to 1 that σε > N.

Now, we turn to prove part b Taking λ1  λ2, ε  1 in 2.12, we get S c f1  f2 



Z c f1 f2.Lemma 2.4is proved

For a spectral function f, let t1 < t2 < · · · < t m be all its zeros onT, and let s k : tk 

t k1/2, k  1, , m, t m1 t1 2π be the midpoints of the intervals between them.

Lemma 2.5 For 1 < q ≤ p < ∞, a spectral function f is odd with respect to each of its zeros t k , that

m  2n, and the points t k are equidistant on T The f is periodic with period 2π/n.

function with the same λ Therefore, Ft  ft k − t  ft k  t has a zero at t  0 without sign

change Byb ofLemma 2.4, this function Ft must be zero.

The proof ofLemma 2.5is complete

Lemma 2.6 see 6 Let G be a B-kernel, n ∈ N, 1 < p, q < ∞ Then, Γ 2n p, q, G / Ø Moreover, if

f, λ q ∈ Γ2n p, q, G, then the function f : G∗h  β satisfies the following conditions:

n



 −fx, ∀x ∈ 0, 2π, 2.13

n



 −ht, ∀t ∈ 0, 2π. 2.14

Lemma 2.7 Let G be a B-kernel For n ∈ N, 1 < q ≤ p < ∞, if f, λ q ∈ Γ2n p, q, G Then, there

h x0 ≥ 0, x0 ∈ 0, π/n, then hx0 sin nx0 ≥ 0, x0 ∈ 0, π/n For x ∈ T, there exists a

i, i  1, , 2n, such that x ∈ i − 1π/n, iπ/n Since hx  π/n  −hx Thus

h x sin nx  h x0i − 1π

n

 sin n x0i − 1π

n



 hx0 sin nx0≥ 0. 2.15 Combining2.14, we get h ∈ D n , and λ  f q  G∗h q The proof ofLemma 2.7is complete

3 Upper estimate of Bernsteinn-width

Following some ideas of Buslaev4, Tikhomirov 1, Chen and Li 7, and Chen 5, the proofs

of our main results are based on some iteration process which starts with an arbitrary function

h0∈ L p with mean value zero and produces a sequence of functions h k, and then a subsequence

of their integrals f k converges to a spectral function f.

First, we take some h0∈ L psuch thath0p  1, h0⊥ 1 Let

f0x G ∗h0



x  β0, 3.1

where β0satisfies the condition:

inf

c∈RG ∗h0



 c q  G ∗h0



 β0q , 1 < q < ∞. 3.2 Next, we construct the sequences of functions{h k } and {f k} as follows:

f k x G ∗h k



x  β k , k  1, 2, , 3.3



Q p h k1

y  μ −q k1

TG x − yQ q f k



xdx, k  0, 1, 2, , 3.4

where β k is uniquely determined by the condition

f k1q inf

c∈RG ∗h k1

 c q G ∗h k1

 β k1q , 1 < q < ∞, 3.5

and μ k1> 0 is determined by the condition h k1p  1, 1 < p < ∞.

Lemma 3.1 Let 1 < p, q < ∞ Then

f kq ≤ μ k1≤ f k1q , k  1, 2, 3.6

1 h k1p−1

p ·h kp ≥ Q p h k1, h k −q

k1f kq

which proves the first inequality in3.6 We now use this first inequality and similarly prove the second inequality:

1 h k1p p  Q p h k1, h k1 −q k1G ∗ Q q f k , h k1

≤ μ −q k1f k1q · Q q f kq  μ −q k1f k1q · f kq−1

q ≤ μ−1

k1f k1q

3.8

The proof ofLemma 3.1is complete

It follows fromLemma 3.1that the construction of the sequence{f k}∞k1is unambiguous Moreover, it follows from3.6 that {μ k1}∞k1is monotonic nondecreasing sequence and tends

to some number μ It is clear that

μ : lim

k→∞μ k  lim

Lemma 3.2 For each starting function h0/  0, h0 ⊥ 1, the sequence {h k}∞k1 of 3.4 contains

choose a subsequence {h k i}∞i1 converging weakly to some h with h p  1, with {f k i}∞i1

converging uniformly to f : G ∗ h  β It follows from 3.4 that {Q p h k i1}∞i1 converges

uniformly because the operator Q p , 1 < p <∞, preserves uniform convergence Consequently,

{Q p Q p h k i1 h k i1}∞i1converges uniformly to some v with v p  1, where 1/p  1/p  1 Let

k→∞ in 3.4 and with μ in 3.9 Then, we can obtain



Q p v

y  μ −q

TG x − yQ q f

xdx. 3.10

Now, we turn to prove thatf, μ is a spectral couple Since in the following inequality,

Q p h k i1, h k i

−q

k i1Q q f k i , h k i

−q

k i1f k iq

q −→ μ −q · μ q  1, 3.11 which impliesQ p v, h p  h p  1,

we get

Q p v, h p p−1· h p  1. 3.12 Therefore, the case of equality can occur only if|Q p v|p  |h| p , sign Q p v sign h almost every,

or, equivalently, if v  h Comparing 3.10 with 2.7, we get μ  λ.

The proof ofLemma 3.2is complete

For convenience, we denote byG, λ n  all the function h n , where h nis sufficiently

i

G ∗ h nq  λ n: λp, q, G  λn h np , 3.13

ii

2π

0

x − yQ q G ∗ h n



xdx  λ q n





ydy, y ∈ T. 3.14

In what follows, we need to convolute G with periodic kernel for

φ σ  φσ, t : √1

2π

∞



n−∞

exp



− 1

2σ2t − 2nπ2



i Z 

c φ σ ∗f ≤ S c f,

ii limσ→0 φ σ ∗f  f uniformly holds for every continuous function f with 2π-period Let G be a B-kernel G σ : φσ ∗G is said to be the mollification of G by φ σ It is easily

verified that G σ is a B-kernel.

Lemma 3.3 see 5 Suppose h n,σ ∈ G σ , λ n,σ , where λ n,σ : λn p, q, G σ  Then

i limσ→0 λ n,σ  λ n ,

ii there exists a sequence of real number σ k > 0 such that σ k→0and the corresponding sequence

iii denote h n x  lim k→∞h n,σ k x, then h n ∈ G, λ n .

We recall an equivalent definition on the Bernstein n-width of a linear operator P from a linear normed space X to Y

b n



P X, Y sup

X n1

inf

P x ∈X n1

P x / 0

Px Y

x X , 3.16

where X n1is any subspace of span{Px : x ∈ X} of dimension ≥ n  1.

Lemma 3.5 Let G be a B-kernel For each p ∈ 1, ∞ and n  1, 2, , then

b 2n−1 B p ; L p

≤ λ n: λn p, p, G. 3.17

Proof We first prove the theorem under the assumption that G is su fficiently smooth, and Z 

c c

G ∗ h ≤ S c h is true An example of such function is G σ , the mollification of G by φ σ Assume

that b 2n−1  B p ; L p  > λ n From the definition of Bernstein n-width, there exists a 2n-dimensional linear subspace L 2n : lin{g1, g2, , g 2n }, and a number γ > λ n , such that L 2n ∩ γSL p  ⊆ B p,

where SL p  is the unit ball of L p, that is,

min

c G∗h∈L 2n

c G ∗ hp

h p  min

f ∈L 2n

f p

h p ≥ γ > λ n 3.18

For every f ∈ L 2n , f  2n

j1ξ j g j , define a mapping f→ξ  ξ1, ξ2, , ξ 2n



∈ R2n Using the similar method as that in9, pages 214–216, we get hp  2n

j1c j |ξ j|p1/p , where c j  jπ/n

j−1π/n |hx| p dx, j  1, , 2n, and c j π/n

0 |hx| p dx  c1, j  1, , 2n, if h ∈ D n By3.18,

we have

min

ξ∈R2n\{0}

2n

j1ξ j g jp

2n

Let



ξ : ξ ξ1, , ξ 2n



∈ R2n ,

2n



i1

ξ i  0,2n

i1

|ξ i |  2π

. 3.20

For every vector ξ ∈ S 2n−1, we take

h ξ0t 

⎧

⎨

⎩

2π −1/p sign ξ k , for t∈t k−1, t k



, k  1, , 2n,

0, for t  t k , k  1, , 2n − 1, 3.21 where t0 0, t k k

i1|ξ i |, k  1, , 2n, and let

f0ξ x G ∗ h ξ

0



x  β0, 1 < p < ∞, 3.22

where β0satisfies the condition

inf

c∈RG ∗ h0



 c p  G ∗ h0



 β0p 3.23

Next, for p  q, we consider the iterative procedure 3.3-3.4 beginning with h ξ

0 and

f0ξ instead of h0and f0, respectively The analogues of Lemmas3.1and3.2hold Moreover, for

the limit element f ξ, there exists ξ ∈ S 2n−1 such that f ξ has at least 2n simple zeros in 0, 2π

i.e., S c f ξ  ≥ 2n Indeed, let O 2n−1

k  {ξ : ξ ∈ S 2n−1 , Z 

c f ξ

k  ≤ 2n − 2}, where the function f ξ

k

defined by3.3 Clearly, the set O 2n−1

k is open in S 2n−1 Let H k 2n−1  S 2n−1 \O 2n−1

k Then, H k 2n−1is

a nonempty closed set, and that H k 2n−11 ⊂ H 2n−1

k , k ∈ N First, we prove that H 2n−1

k is nonempty

For fixed 0 < x1< x2< · · · < x 2n−1 < 2π, let η ξ  η1ξ, η2ξ, , η 2n ξ, where

η i ξ 

⎧

⎪

⎪ T

h ξ0tdt, for i  1,

f k ξ

x i−1

, for i  2, , 2n.

3.24

It is easily seen that ηξ is a continuous and odd mapping By Borsuk’s theorem 10, there

exists a ξ ∈ S 2n−1 such that ηξ  0 Then, Z 

c f ξ

k   2n − 1, that is, ξ ∈ H 2n−1

k Thus, H k 2n−1

is a nonempty Next, we prove H k 2n−11 ⊂ H 2n−1

k , k ∈ N Assume, on the contrary, there exists a

ξ ∈ H 2n−1

k1 , but ξ/ ∈H 2n−1

k Thus, S c f k ξ  ≤ Z 

c f k ξ  ≤ 2n − 2 results in S c Q q f k ξ  ≤ 2n − 2 By 3.4,

we get

S c



Q p h k ξ1

≤ 2n − 2, S c



h k ξ

≤ 2n − 2. 3.25 According to3.3, we have Z 

c



f k ξ1

≤ 2n − 2, namely, ξ/∈H 2n−1

k1 A contradiction follows from the above We have constructed a system of nonempty closed nested sets Their intersection is nonempty Let ξ ∈ ∞

k1H 2n−1

k  According toLemma 3.2, there existsf ξ x, λ p  ∈ Γp, p, G

such that limk→∞f k ξ x  f ξ x, x ∈ 0, 2π Thus, Z 

c f ξ  ≥ 2n − 1 In view ofLemma 2.3, zeros

of f ξ x are simple Therefore, S c f ξ  ≥ 2n − 1 But since the function f ξ x is periodic, we actually have S c f ξ  ≥ 2n We write S c f ξ   2N.

For the spectral function f ξ corresponding to spectral value λξ, byLemma 2.7, and the

nonincreasing property of Kolmogorov n-widths in n, and d 2n  B p ; L p   λ n p, p, G 7, we have

λ ξ ≤ λ N  d 2N B p ; L p

≤ d 2n B p ; L p

 λ n 3.26 Therefore, by Lemmas3.1,3.2, and3.26, we have

min

ξ∈R2n\{0}

2n

j1ξ j g jp

2n

j1c j |ξ j|p1/p ≤ 

2n

j1ξ j g jp

c11/p2n

j1|ξ j|p1/p  f ξp  λξ ≤ λ n , 3.27 which is contradicted with3.19

For a general B-kernel G, set G σ  φ σ ∗G, and h σ  φ σ ∗h, λ n,σ  φ σ ∗λ n For f  c  G∗h ∈

B p , we set f σ  c  G σ ∗h From the results obtained in the pervious case, we have

G σ ∗h  c p

h σp  f h σp

σp ≤ λ n,σ 3.28 According to Lemma 3.3, we get G∗h  c p / h p ≤ λ n p, p, G Therefore, we obtain

b 2n−1  B p ; L p  ≤ λ n p, p, G The proof ofLemma 3.5is complete

Proof of theorem

Now, we consider the proof ofTheorem 1.3

b 2n−1  B p ; L p  ≤ λ n p, p, G On the other hand, by 5, for each 1 < p ≤ q < ∞ and n  1, 2, , then b 2n−1  B p ; L q  ≥ λ n p, q, G Thus, we have b 2n−1  B p ; L p   λ n p, p, G for p ∈ 1, ∞ and

n ∈ N The result 1.8 is obvious since s 2n  B p ; L p   λ n p, p, G 5 Theorem 1.3is proved completely

The proof ofLemma 3.2is complete

For convenience, we denote byG,... Ft must be zero.

The proof ofLemma 2.5is complete

Lemma 2.6 see 6 Let... proof ofLemma 2.7is complete

3 Upper estimate of Bernstein< /b>n-width

Following some ideas of Buslaev4, Tikhomirov 1, Chen and Li 7, and Chen 5, the proofs