ISSN 1068-3623, Journal of Contemporary Mathematical Analysis, 2013, Vol. 48, No. 4, pp. 139–144. c© Allerton Press, Inc., 2013.Original Russian Text c© R. H. Barkhudaryan, 2013, published in Izvestiya NAN Armenii. Matematika, 2013, No. 4, pp. 3-12.

DIFFERENTIAL EQUATIONS

On the Gibbs Phenomenon for Expansions by Eigenfunctionsof the Boundary Problem for Dirac System

R. H. Barkhudaryan*

1Institute of Mathematics of National Academy of Sciences of ArmeniaReceived December 14, 2011

Abstract—The paper considers expansions by eigenfunctions of the boundary problem for Diracsystem. The Gibbs phenomenon for such expansions is revealed.

MSC2010 numbers : 34L10, 34L40DOI: 10.3103/S1068362313040018

Keywords: Expansions by eigenfunctions; Dirac system; Gibbs phenomenon.

1. INTRODUCTION

It is well known that expansions by eigenfunctions of the regular boundary problems for ordinarydifferential equations with smooth coefficients on a finite interval converge uniformly, provided thatthe underlying function belongs to the domain of definition of the corresponding operator. Otherwisea phenomenon, similar to the Gibbs phenomenon for classical Fourier series, can occur. Such aphenomenon for some special cases were considered in a number of papers (see L. Mishoe [1, 2],L. Brandolini and L. Colzani [3], M. Taylor [4], K. Coletta et al. [5], and S. Kaber [6]).

In this paper the Gibbs phenomenon for the components of the vector function of the followingboundary problem for Dirac system is revealed:⎛

⎝ 0 1

−1 0

⎞⎠ dy

dx−

⎛⎝p(x) 0

0 r(x)

⎞⎠ y = λy, (1.1)

y2(−1) cos α + y1(−1) sin α = 0, (1.2)

y2(1) cos β + y1(1) sin β = 0, (1.3)

where p and r are real-valued functions defined on [−1, 1].Based on the physical reasons R. Szmytkowski (see [7, 8]) showed that the expansion by eigenfunc-

tions of Dirac system does not converge to the underlying function at the endpoints of the interval evenif each component of the function belongs to the class C∞[−1, 1], but does not satisfy the boundaryconditions.

We state some known facts and formulas related to the problem (1.1)-(1.3), which will be used below(see [9], p. 71). We denote by {λn}∞n=−∞ and

{vn = (vn,1, vn,2)T

}∞n=−∞ the sets of eigenvalues and the

normed eigenvector-functions of this problem, respectively. Notice that, without loss of generality, wecan assume that the number λ = 0 is not an eigenvalue. For conciseness, the series by eigenfunctions{vn} we will refer as a Fourier series, and the corresponding coefficients as Fourier coefficients.

For a given vector-function f(x) = (f1(x), f2(x))T ∈ L22[−1, 1] := L2[−1, 1] × L2[−1, 1] we denote

SN (f) =N∑

n=−N

cnvn(x), (1.4)

*E-mail: rafayel@instmath.sci.am

139

140 BARKHUDARYAN

cn =

1∫

−1

vTn (x)f(x)dx, (1.5)

RN (f) = f(x) − SN (f).

It is known (see, e.g., [9], p. 82) that the eigenvector-functions of the Dirac problem form a completeorthogonal system in the Hilbert space L2

2[−1, 1], implying that SN (f) converges to f in the norm of L22,

defined by

‖f‖2 =

⎛⎝

1∫

−1

(f21 (x) + f2

2 (x))dx

⎞⎠

1/2

.

For the following asymptotic formulas we refer to [9], p.75 (note that in [9] the formulas (1.7), (1.8) werestated with some typos, here we state the improved versions):

λn = n − θ

π+ O

(n−1

), n → ∞, (1.6)

vn,1(x) = cos(ξn − α) + O(n−1

), n → ∞, (1.7)

vn,2(x) = sin(ξn − α) + O(n−1

), n → ∞, (1.8)

where

θ = β − α − 12

∫ 1

−1(p(t) + r(t))dt, (1.9)

ξn = ξ(x, λn) = λn(x + 1) − 12

x∫

−1

(p(τ) + r(τ))dτ .

The main result of this paper is the following theorem.

Theorem 1.1. Let p, r ∈ C1[−1, 1], f ∈ C12 [−1, 1] and the function f does not satisfy the boundary

conditions. The following assertions hold:(a) if B(f,−1, α) �= 0, then

lim supN→∞x→−1

|B(SN (f), x, α)||B(f,−1, α)| =

2π

∫ π

0

sin t

tdt,

(b) if B(f, 1, β) �= 0, then

lim supN→∞x→1

|B(SN (f), x, β)||B(f, 1, β)| =

2π

∫ π

0

sin t

tdt,

where B(f, x, γ) = f1(x) sin γ + f2(x) cos γ.

Remark 1.1. Since all the eigenfunctions satisfy the boundary conditions, the limit of the truncatedseries also should satisfy these conditions (provided that the convergence is uniformly). In the casewhere the underlying function does not satisfy the boundary conditions, that is, either B(f,−1, α) �= 0or B(f, 1, β) �= 0, we have an analog of the Gibbs phenomenon in terms of violation of the boundaryconditions.

Remark 1.2. If we consider the expansion (1.4) component-wise, then it can easily be shown (followsfrom the proof of the theorem) that if either α = 0 or α = π/2, then one component of the series SN (f)converges uniformly, while the other obeys Gibbs phenomenon at the point −1 (see Section 3). Thisassertion remains valid also at the point 1 provided that either β = 0 or β = π/2.

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ON THE GIBBS PHENOMENON FOR EXPANSIONS BY EIGENFUNCTIONS 141

Remark 1.3. The quantity 2π

∫ π0

sin tt dt ≈ 1.17898 is exactly the constant appearing in the classical

Gibbs phenomenon for Fourier series.

Remark 1.4. To overcome Gibbs phenomenon, in [10] was suggested a convergence accelerationmethod of expansions by eigenvector-functions for the problem (1.1)-(1.3), similar to the Krylov-Eckhoff convergence acceleration method for the classical Fourier series (see [11, 12]).

2. PROOF OF THE THEOREMDenote

Lf =

⎛⎝ 0 1

−1 0

⎞⎠ df

dx−

⎛⎝p(x) 0

0 r(x)

⎞⎠ f, Bf = B

⎛⎝f1(x)

f2(x)

⎞⎠ =

⎛⎝−f2(x)

f1(x)

⎞⎠ ,

fk(x) = BLkf(x), k � 0,

where L0 stands for the identity operator.

Lemma 2.1. Let p, r ∈ Cq−1[−1, 1], p(q−1), r(q−1) ∈ AC[−1, 1], f ∈ Cq2 [−1, 1] and f (q) ∈ AC2[−1, 1]

with q � 1. Then the coefficients cn defined by (1.5) can be represented cn = Pn + Fn, where

Pn = vTn (1)

q∑k=0

λ−k−1n fk(1) − vT

n (−1)q∑

k=0

λ−k−1n fk(−1), (2.1)

Fn = λ−q−1n

∫ 1

−1vTn (x)Lq+1(f(x))dx.

Proof. We have

cn =∫ 1

−1vTn (x)f(x)dx = λ−1

n

∫ 1

−1(f1(x)(v′n,2(x) − p(x)vn,1(x)) − f2(x)(v′n,1(x) + r(x)vn,2(x)))dx.

Integration by parts yields

cn = λ−1n (f1(x)vn,2(x) − f2(x)vn,1(x))|1−1

+λ−1n

∫ 1

−1vn,1(x)(f ′

2(x) − p(x)f1(x)) + vn,2(x)(−f ′1(x) − r(x)f2(x))dx

= λ−1n vT

n (x)f0(x)∣∣∣1

−1+ λ−1

n

∫ 1

−1vTn (x)L(f(x))dx.

Repeating integration by parts q times, we get the result.

Now we consider the function

κ(x) =

⎛⎝ x − 1

−x + 1

⎞⎠, (2.2)

which does not satisfy the boundary conditions at the point x = 0 for α �= π4 . Let

SN (κ) =N∑

n=−N

cn(κ)vn (2.3)

be the expansion of κ(x) by the system {vn(x)}, where

cn(κ) =

1∫

−1

vTn (x)κ(x)dx. (2.4)

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 48 No. 4 2013

142 BARKHUDARYAN

Lemma 2.2. If p, r ∈ C1[−1, 1], then for the coefficients cn(κ) defined by (2.4) the followingasymptotic formula is fulfilled, cn(κ) =

√2λ−1

n (cos α − sinα) + αn, where∑∞

n=1 |αn| < +∞.

Proof. Using Lemma 2.1 applied to the function (2.2), we obtain

cn = vTn (1)λ−1

n κ0(1) − vTn (−1)λ−1

n κ0(−1) + λ−1n

∫ 1

−1vTn (x)L1(κ(x))dx

=(

22

)vTn (−1)λ−1

n κ0(−1) + λ−1n

1∫

−1

vTn (x)L1(κ(x))dx

=√

2λ−1n (cos α − sin α) + O

(1n2

)+ λ−1

n

1∫

−1

vTn (x)L1(κ(x))dx.

Taking into account that L1(κ(x)) ∈ L22[−1, 1], we obtain the desired result.

Now we show that if p, r ∈ C1[−1, 1] and α �= π4 , then the function κ defined by (2.2) obeys Gibbs

phenomenon. To this end we consider the error of approximation of function (2.2) by the truncated series:

RN (κ) = κ(x) − SN (κ) =∑

|n|>N

cnvn(x) =∑

|n|>N

cn

(vn,1(x)vn,2(x)

).

Using Lemma 2.2 and the asymptotic formula for eigenfunctions of κ, we obtain∑

|n|>N

cnvn,1(x) = (cos α − sinα)∑

|n|>N

(cos(ξn − α)

λn+ αn

)

= (cos α − sin α)∑

|n|>N

(cos(π

2 n(x + 1) + ϕ(x))πn/2

+ αn

), (2.5)

where

ϕ(x) = −α − θ

2(x + 1) − 1

2

x∫

−1

(p(τ) + r(τ))dτ . (2.6)

It follows from (2.5) that

SN (κ) = κ(x) −∑

|n|>N

cos α − sin α

πn/2

⎛⎝cos

(π2 n(x + 1) + ϕ(x)

)

sin(

π2n(x + 1) + ϕ(x)

)

⎞⎠ + o(1)

= κ(x) − 2(cos α − sin α)π

⎛⎝− sin (ϕ(x))

cos (ϕ(x))

⎞⎠ ∑

|n|>N

sin(

π2 n(x + 1)

)n

+ o(1).

Observe that the function ϕ defined by (2.6) is continuous and the sum

∑|n|>N

sin(

π2n(x + 1)

)n

is the remainder of the Fourier series of function π2 − π

2 x, that is,∞∑

n=−∞

sin(

π2n(x + 1)

)n

= −π

2x +

π

2, x ∈ (−1, 1]. (2.7)

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 48 No. 4 2013

ON THE GIBBS PHENOMENON FOR EXPANSIONS BY EIGENFUNCTIONS 143

We split the interval (−1, 1] into two parts (−1, ξ] and (ξ, 1] so that |ϕ(x) − ϕ(−1)| < ε if x ∈ (−1, ξ],and |κ(x) − SN (κ)| < ε if x ∈ (ξ, 1] for sufficiently large N . We have then

lim supN→∞x→−1

|B(SN (κ), x, α)| = |B(κ,−1, α)| − |B(κ,−1, α)|π

lim supN→∞x→−1

∑|n|>N

sin(

π2n(x + 1)

)n

. (2.8)

From formulas (2.8) and (2.7) we obtain the desired result.

A similar result can be obtained for the function

(x) =

⎛⎝ x + 1

−x − 1

⎞⎠, (2.9)

which has a singularity at the point 1.

Namely, if p, r ∈ C1[−1, 1] and β �= π4 , then the function defined by (2.9) obeys Gibbs phenomenon,

that is,

lim supN→∞x→1

|B(SN ( ), x, β)||B( , 1, β)| =

2π

∫ π

0

sin t

tdt.

A combination of the above discussed cases yields the following general result.

Assume that α, β �= π4 (similar arguments can be used to prove the result when either α or β are equal

to π4 ). Consider the function

g(x) = f(x) − hκ(x) − H (x),

where

h =f2(−1) cos α + f1(−1) sin α

κ2(−1) cos α + κ1(−1) sin α, H =

f2(1) cos α + f1(1) sin α

2(1) cos α + 1(1) sin α.

Observe that the function g(x) has a continuous first derivative and satisfies the boundary conditions.Therefore the series SN (f(x) − hκ(x) − H (x)) converges uniformly. This completes the proof ofTheorem.

3. NUMERICAL ILLUSTRATIONS

In this section the above results are illustrated for the function κ. Notice that if α = 0, then oneof the components in the decomposition (2.3) converges uniformly, while the other yields the Gibbsphenomenon. To illustrate this phenomenon we consider a system with vanishing potential, and assumethat α = 0, β = −π

4 . For this problem, it is easy to find the eigenvalues and eigenfunctions. Namely, forthe eigenvalues we have

λn =πk

2+

π

8, n = 0,±1,±2, ...,

and the normed eigenfunctions have the form

vn(x) =12

√1 − (−1)k√

2

⎛⎝cos

((πk2 + π

8

)x)cot

(πk2 + π

8

)− sin

((πk2 + π

8

)x)

cos((

πk2 + π

8

)x)

+ cot(

πk2 + π

8

)sin

((πk2 + π

8

)x)

⎞⎠ .

Acknowledgment. The author would like to thank Knut and Alice Wallenberg Foundation and GoranGustafsson Foundation for supporting to work at Swedish Royal Technical University (Royal Instituteof Technology).

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 48 No. 4 2013

144 BARKHUDARYAN

Fig. 1. The approximation error of the second component of function κ(x) by a finite sum of Fourier series at the vicinityof the point x = −1 using N = 5, 10, 20 Fourier coefficients.

Fig. 2. The approximation error of the first component of function κ(x) by a finite sum of Fourier series at the vicinityof the point x = −1 using N = 5, 10, 20 Fourier coefficients (the uniform error decreases as N increases).

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