Volume 2008, Article ID 894529, 8 pagesdoi:10.1155/2008/894529 Research Article Some Classes of Periodic Functions with Formal Self-Adjoint Linear Differential Operators Feng Guo Departm
Trang 1Volume 2008, Article ID 894529, 8 pages
doi:10.1155/2008/894529
Research Article
Some Classes of Periodic Functions with Formal
Self-Adjoint Linear Differential Operators
Feng Guo
Department of Mathematics, Taizhou University, Zhejiang, Taizhou 317000, China
Correspondence should be addressed to Feng Guo, guofeng0576@163.com
Received 10 December 2007; Accepted 18 June 2008
Recommended by Vijay Gupta
We consider the classes of periodic functions with formal self-adjoint linear differential operators
W pLr, which include the classical Sobolev class as its special case With the help of the spectral of linear differential equations, we find the exact values of Bernstein n-width of the classes WpLr in
the 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 result
LetC, R, Z, N, and N be the sets of all complex numbers, real numbers, integers, nonnegative integers, and positive integers, respectively Let T be the unit circle realized as the interval
0, 2π with the points 0 and 2π identified, 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 Sobolev space of functions x· on T such that the r − 1st derivative
x r−1 · is absolutely continuous on T and x r · ∈ L p , r ∈ N The corresponding Sobolev class
is the set
W p r :W r
Tikhomirov1 introduced the notion of Bernstein width of a centrally symmetric set C
in a normed space X It is defined by the following formula:
b n C, X : sup
L
Trang 2where BX is the unit ball of X and the outer supremum is taken over all subspaces L ⊂ X such that dim L ≥ n 1, n ∈ N.
In particular, Tikhomirov posed the problem of finding the exact value of b n C; X, where
C W r
p and X L q , 1 ≤ p, q ≤ ∞ He also obtained the first results 1 for p q ∞ and
n 2k − 1 Pinkus 2 found b 2n−1 W r
p ; L q , where p q 1 Later, Magaril-Il’yaev 3 obtained
the exact value of b 2n−1 W r
p ; L p , for 1 < p < ∞ The latest contribution to this fields 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 < ∞.
Let
Lr D D r a r−1D r−1 · · · a1D a0, D d
be an arbitrary linear differential operator of order r with constant real coefficients
a0, a1, , a r−1 Denote by p r the characteristic polynomial of Lr D The linear differential
operatorLr D will be called formal self-adjoint if p r −t −1 r p r t, for each t ∈ C.
We define the function classes W pLr as follows:
W p
Lr
x· : x r−1∈ AC 2π ,Lr Dx· p ≤ 1, 1.4
where 1≤ p ≤ ∞.
In this paper, we will determine the exact values of Bernstein n-width of some classes of
periodic functions with formal self-adjoint linear differential operators WpLr, which include the classical Sobolev class as its special case
We define Q pto be the nonlinear transformation
Q p f
t :ftp−1
The maim result of this paper is the following
Theorem 1.1 Assume that 1 < p < ∞ Let L r D be an arbitrary formal self-adjoint linear differential operators given by1.3 Then, there exists a number N ∈ Nsuch that for every n ≥ N:
b 2n−1
W pLr
; L p λ 2n: λ2n
p, p,Lr
where λ 2n is that eigenvalue λ of the boundary value problem
Lr Dyt −1 r λ −p
Q p x
t, yt Q pLr Dxt,
x j 0 x j 2π, y j 0 y j 2π, j 0, 1, , n − 1,
1.7
for which the corresponding eigenfunction x· x 2n · has only 2n simple zeros on T and is normalized
by the conditionLr Dx· p 1.
Trang 32 Proof of the theorem
First we introduce some notations and formulate auxiliary statements
LetLr D be an arbitrary linear differential operator 1.3 Denote the 2π-periodic kernel
ofLr D by
KerLr D x· ∈ C rT : Lr Dxt ≡ 0. 2.1
Let μ 0 ≤ μ ≤ r be the dimension of KerL r D and {ϕ i , , ϕ μ} an arbitrary basis in KerLr D.
Z c f denotes the number of zeros of f in a period, counting multiplicity, and S c f is the cyclic sign change count for a piecewise continuous, 2π-periodic function f2 Following,
x·, λ is called the spectral pair of 1.7 if the function x· is normalized by the condition
Lr Dx· p 1 The set of all spectral pairs is denoted by SPp, p, L r Define the spectral classes SP2k p, p, L r as
SP2k p, p, L r x·, λ∈ SPp, p,Lr : S c
x· 2k. 2.2 Let x 2n· denotes the solution of the extremal problem as follows:
π/2n
0
|Xt| p dt −→ sup,
π/2n
0
|Lr DXt| p dt ≤ 1,
x k π
2n −1k1π
2n
/2
0, k 0, 1, , n − 1,
2.3
and the function x 2n · is such that x 2n t −x 2n t − π/n for all t ∈ T:
x 2n t :
⎧
⎪
⎪
x 2n t, 0≤ t ≤ π
2n ,
x 2n π
n − t
, π 2n < t≤ π
n .
2.4
Let us extend periodically the function x 2n t onto R, and normalize the obtained function as
it is required in the definition of spectral pairs From what has been done above, we get a
function x 2n t belongs to SP 2n p, p, L r Furthermore, by 5, which any other function from
SP2n p, p, L r differs from x 2n· only in the sign and in a shift of its argument, and there exists a
number N ∈ Nsuch that for every n ≥ N, all zeros of x 2n· are simple, equidistant with a step
equal to π/n, and S c x 2n S cLr Dx 2n 2n We denote the set of zeros sign variations
ofLr Dx 2n on the period by Q 2n τ1, , τ 2n Let
G r t 1
2π
k/∈Λ
e ikt
whereΛ {k ∈ Z : p r ik 0} and i is the imaginary unit.
Trang 4The 2π-periodic G-splines are defined as elements of the linear space
S
Q 2n , G r
spanϕ1t, , ϕ μ t, G r
t − τ1
, , G r
t − τ 2n
As was proved in6, if n ≥ N, then dim SQ 2n , G r 2n.
We assumeshifting x· if necessary that L r Dx 2n · is positive on −π, π π/n Let
L 2n: L2n r, p, p denote the space of functions of the form
xt
μ
j1
a j ϕ j t 1
π
TG r t − τ
2n
i1
b i y i τ
where a1, , a μ , b1, , b 2n ∈ R, 2n
i1b i 0, y i · χ i·Lr Dx 2n · − i − 1π/n, and χ i·
is the characteristic function of the intervalΔi : −π i − 1π/n, −π iπ/n, 1 ≤ i ≤ 2n
Obviously, dim L 2n 2n and L 2n ⊂ W pLr
Let us now consider exact estimate of Bernstein n-width This was introduced in1 We
reformulate the definition for a linear operator P mapping X to Y
Definition 2.1see 2, page 149 Let P ∈ LX, Y Then the Bernstein n-width is defined by
b n PX, Y sup
X n1
inf
P x ∈X n1
P x / 0
Px Y
where X n1is any subspace of span{Px : x ∈ X} of dimension ≥ n 1.
2.1 Lower estimate of Bernstein n-width
Consider the extremal problem
x· p p
Lr Dx· p
p
and denote the value of this problem by α p Let us show that α ≥ λ n, this will imply the desired
lower bound for b 2n−1 Let x· ∈ L 2n, then
Lr Dx· p p2n
i1
Δi
2n
i1
b i y i t
p
dt2n
i1
Δi
|b i|p|Lr Dx n t| p dt 1
2n
2n
i1
|b i|p , 2.10 and by setting
z i· : 1
π
TG r · − τy i τdτ, i 1, 2, , 2n, 2.11
we reduce problem2.9 to the form
μ
j1a j ϕ j· 2n
i1b i z i·p
p
1/2n2n
i1|b i|p −→ inf, a1, , a μ , b1, , b 2n ∈ R. 2.12
Trang 5This is a smooth finite-dimensional problem It has a solutiona1, a μ , b1, , b 2n, and, moreover, b1, , b 2n / 0 According to the Lagrange multiplier rule, there exists a η ∈ R
such that the derivatives of the function a1, , a μ , b1, , b 2n → ga1, , a μ , b1, , b 2n
ηb1 b2 · · · b 2n where g· is the function being minimized in2.12 with respect to
a1, , a μ , b1, , b 2n at the point a1, a μ , b1, , b 2n are equal to zero This leads to the relations
Tϕ j tQ p x
tdt 0, j 1, , μ, 2.13
Tz i tQ p x
tdt α p
2n Q p b i , i 1, , 2n, 2.14
where x· μ
j1a j ϕ j t 2n
i1b i z i·
We remark that ga1, , a μ , b1, , b 2n gda1, , da μ , db1, , db 2n for any d / 0,
and hence the vectord a1, , d a μ , d b1, , d b 2n is also a solution of 2.12 Thus, it can be assumed that|b i | ≤ 1, i 1, , 2n, and b i0 −1i0 1for some i0, 1 ≤ i0≤ 2n.
Let
x 2n t
μ
j1
a j ϕ j t 2n
i1
and x 2nsatisfies1.7 Let a a
1, , a 2n and b 1, −1, , 1, −1 ∈ R 2n It follows from the definitions of x 2n · and x· that
Lr Dx 2n t − L r Dxt 2n
i1
i / i0
−1i1− b i
χ i tL r Dx 2n t−i − 1π
n
and hence S cLr Dx 2n ·, L r Dx· has at most 2n−2 sign changes Then, by Rolle’s theorem,
S cLr Dx 2n· − Lr Dx· ≤ 2n − 2 For any a, b ∈ R, signa b signQ p a Q p b, therefore
S c
Q p x 2n
· −Q p x·≤ 2n − 2. 2.17
In addition, since x 2n is 2π-periodic solution of the linear differential equation
Lr Dyt −1 r λ −p Q p xt, and ϕ j t ∈ KerL r D Then, by 7, page 94, we have
Tϕ j tQ p xtdt 0, j 1, , μ. 2.18
If we now multiply both sides of 2.15 by Q p x 2n t, and integrate over the interval
Δi , 1 ≤ i ≤ 2n, we get
Δi
z i tQ p x 2n
tdt −1 i1
Δi
|x 2n t| p dt −1i1λ p 2n
Due to
Tz i tQ p x 2n
tdt Δi z i tQ p x 2n
tdt Therefore, we have
Tz i tQ p x 2n
tdt −1 i1λ p 2n
2n , i 1, , 2n. 2.20
Trang 6Changing the order of integration and using2.14 and 2.20, we get that
Δi
Lr Dx 2n t− i − 1π
n
1
π
TG r t − τQ p x 2n
τ −Q p x
τdτ
dt
Tz i tQ p x 2n
t −Q p xtdt 1
2n
−1i1λ p 2n − α p Q p b i
.
2.21
Denote by f· the factor multiply Lr Dx 2n t − i − 1π/n in the integral in the left-hand side
of this equality If we assume that λ 2n > α, then we arrive at the relations
sign
Δi
Lr Dx 2n t−i − 1π
n
f·dt −1 i1, i 1, , 2n. 2.22
Suppose for definiteness thatLr Dx 2n t − i − 1π/n > 0 interior to Δ i , i 1, , 2n.
Then it follows from 2.22 that there are points t i ∈ Δi such that signf t i −1i1, i
1, , 2n, that is, S c f· ≥ 2n − 1 But f· is periodic, and hence S c f· ≥ 2n, therefore,
S cLr Df· ≥ 2n Further, L r Df· Q p x 2n
t − Q p xt, that is, S c Q p x 2n t −
Q p xt≥ 2n.
We have arrived at a contradiction to2.17, and hence λ 2n ≤ α Thus b 2n−1 W pLr ; L p ≥
λ 2n
2.2 Upper estimate of Bernstein n-width
Assume the contrary: b 2n−1 W pLr ; L p > λ 2n , 1 < p < ∞ Then, by definition, there exists a linearly independent system of 2n functions L 2n : spanf1, , f 2n
⊂ L p and number γ > λ 2n such that L 2n ∩ γSL p ⊆ Lr D, or equivalently,
min
x ·∈L 2n
x· p
Lr Dx· p ≥ γ > λ 2n 2.23
Let us assign a vector c∈ R2n to each function x· ∈ L 2nby the following rule:
x· −→ c c1, , c 2n ∈ R2n , where x· 2n
j1
c j f j ·. 2.24
Then2.23 acquires the form
min
c∈R2n\{0}
2n
j1c j f j·p
2n
j1c jLr Df j·p
Let c0 0 Consider the sphere S 2n−1in the spaceR2n with radius 2π, that is,
S 2n−1:
c : c c1, , c 2n ∈ R2n , c 2n
j1
|c j | 2π
Trang 7
To every vector c∈ R2n we assign function ut, c defined by
ut, c
⎧
⎨
⎩
2π −1/p sign c j , for t∈t k−1, t k
, k 1, , 2n,
0, for t t k , k 1, , 2n − 1, 2.27 where t0 0, t k k
i1|c i |, k 1, , 2n, and the extended 2π-periodically onto R.
An analog of the Buslaev iteration process8 is constructed in the following way: the
function xt, c is found as a periodic solution of the linear differential equation L r Dx0
u, then the periodic functions {x k t, c} k∈N are successively determined from the differential equations
Lr Dx k t Q p y k
t,
Lr Dy k t −1 r μ −p k−1
Q p x k−1
where p p/p − 1, and the constants {μ k : k 0, , } are uniquely determined by the
conditions
Lr Dx kp 1, Q p x k
t ⊥ KerL r D, Q p y k
t ⊥ KerL r D. 2.29
By analogy with the reasoning in8, we can prove the following assertions:
i the iteration procedure 2.28-2.29 is well de fined, the sequences {μ k}k∈N is
monotone nondecreasing and converge to an eigenvalue λc > 0 of the problem 1.7,
ii the sequence {x k ·, c} k∈N has a subsequence that is convergent to an eigenfunction
x·, c of the problem 1.7, with λc x·, c p,
iii for any k ∈ N there exists a c ∈ S 2n−1 such that x k ·, c has at least 2n zeros
Z c x k ·, c ≥ 2n on T,
iv in the set of spectral pairs λc, x·, c, there exists a pair λc, x·, c such that
S c x·, c 2N ≥ 2n.
Items i and ii can be proved in the same way as 8, Sections 6 and 10 Item iii follows from the Borsuk theorem 9, which states that there exists a c ∈ S 2n−1 such that
Z c x k ·, c ≥ 2n−1, but since the function x k ·, c is periodic, we actually have Z c x k ·, c ≥ 2n.
Finally, itemiv, by ii and iii, which Z c x·, c ≥ 2n In view of x·, c zeros are simple, therefore, S c x·, c ≥ 2n.
Since spectral pairs of 1.7 are unique and the Kolmogorov width d 2n W pLr ; L q
λ 2n p, q, L r for p ≥ q 5, when n ≥ N, it follows that
λc λ 2N d 2N
W p
Lr
; L p
≤ d 2n
W p
Lr
; L p
Therefore, by virtue of itemsi, ii, and 2.30, we obtain
min
c∈R2n\{0}
2n
j1c j f j·p
2n
j1c jLr Df j·p
2n
j1c j f j·p
2n
j1c jLr Df j·p
≤ Lx k ·, c p
r Dx k ·, c p ≤ λc λ 2N ≤ λ 2n ,
2.31 which contradicts 2.25 Hence b 2n−1 W pLr ; L p ≤ λ 2n Thus, the upper bound is proved This completes the proof of the theorem
Trang 8Project was supported by the Natural Science Foundation of ChinaGrant no 10671019 and Scientific Research Fund of Zhejiang Provincial Education DepartmentGrant no 20070509
References
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Moscow, Russia, 1976.
2 A Pinkus, n-Widths in Approximation Theory, vol 7 of Results in Mathematics and Related Areas 3, Springer,
Berlin, Germany, 1985.
3 G G Magaril-Il’yaev, “Mean dimension, widths and optimal recovery of Sobolev classes of functions
on a straight line,” Mathematics of the USSR Sbornik, vol 74, no 2, pp 381–403, 1993.
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classes of periodic functions,” Matematicheskie Zametki, vol 58, no 1, pp 139–143, 1995Russian.
5 S I Novikov, “Exact values of widths for some classes of periodic functions,” East Journal on Approximations, vol 4, no 1, pp 35–54, 1998.
6 N T T Hoa, Optimal quadrature formulae and methods for recovery on function Classds defined by variation diminishing convolutions, Candidate’s dissertation, Moscow State University, Moscow, Russia, 1985.
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8 A P Buslaev and V M Tikhomirov, “Spectra of nonlinear differential equations and widths of Sobolev
classes,” Mathematics of the USSR Sbornik, vol 71, no 2, pp 427–446, 1992.
9 K Borsuk, “Drei S¨atze ¨uber die n-dimensionale euklidische Sph¨are,” Fundamenta Mathematicae, vol 20,
pp 177–190, 1933.
... ? ?Exact values of widths for some classes of periodic functions, ” East Journal on Approximations, vol 4, no 1, pp 35–54, 1998.6 N T T Hoa, Optimal quadrature formulae... A P Buslaev, G G Magaril-Il’yaev, and N T’en Nam, ? ?Exact values of Bernstein widths of Sobolev
classes of periodic functions, ” Matematicheskie Zametki, vol 58, no 1,... completes the proof of the theorem
Trang 8Project was supported by the Natural Science Foundation of ChinaGrant