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BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS

FALKO BAUSTIAN AND VLADIMIR BOBKOV

Abstract. We establish suﬃcient assumptions on sequences of Fučík eigenvalues of the one-

dimensional Laplacian which guarantee that the corresponding Fučík eigenfunctions form a

Riesz basis in 𝐿2(0, 𝜋).

1. Introduction

The classical spectral theorem for compact self-adjoint (linear) operators on a Hilbert space

𝑋asserts the existence of a sequence of eigenfunctions of such an operator that forms an

orthogonal basis of 𝑋, see, e.g., [30]. One of the simplest examples is given by the sine

functions sin(𝑛𝑥),𝑛∈N, which are eigenfunctions of the one-dimensional Dirichlet Laplacian,

i.e.,

−𝑢′′ =𝜆𝑢 in (0, 𝜋), 𝑢(0) = 𝑢(𝜋)=0,

and hence they form an orthogonal basis in 𝐿2(0, 𝜋). Although the assumptions of the spectral

theorem can not be weakened in general, eigenfunctions of certain operators which do not

satisfy the imposed requirements might still form a basis of a space 𝑋. Classes of such

operators have been an active topic of research, and a great amount of signiﬁcant results have

been obtained. We refer to [1,13,22,24,26,32] and the extensive bibliographies therein for

a deeper discussion.

The references indicated above are concerned mainly with linear operators. At the same

time, the conclusion of the spectral theorem can be valid even for some nonlinear operators.

As an example, there exists 𝑝0>1such that the generalized trigonometric functions sin𝑝(𝑛𝑥)

deﬁned as eigenfunctions of the one-dimensional Dirichlet 𝑝-Laplacian with 𝑝>𝑝0, i.e.,

−(|𝑢′|𝑝−2𝑢′)′=𝜆|𝑢|𝑝−2𝑢in (0, 𝜋𝑝), 𝑢(0) = 𝑢(𝜋𝑝)=0,

form a Riesz basis in 𝐿2(0, 𝜋𝑝), see [4], and [6,7,16] for further developments. Here,

𝜋𝑝=2𝜋

𝑝sin(𝜋/𝑝). While this basis is non-orthogonal except for 𝑝= 2, it has certain advan-

tages, in particular, in a numerical study of nonlinear equations, see [5]. Let us remark that

it remains unknown whether the same basisness result holds true for all 𝑝 > 1. Notice that

the generalized trigonometric functions sin𝑝(𝑛𝑥)share with sin(𝑛𝑥)the anti-periodic struc-

ture which is signiﬁcantly used in the proof of [4]. In general, there exist several results on

assumptions for a function 𝑓to guarantee that the corresponding system of dilated functions,

i.e., the sequence of the form {𝑓(𝑛𝑥)}, is a basis, see, e.g., [3,6,18,19,35], to mention a few.

2010 Mathematics Subject Classiﬁcation. 34L10, 34B25, 34B08, 47A70.

Key words and phrases. Fucik specturm, Fucik eigenfunctions, Riesz basis, Paley-Wiener stability.

V. Bobkov was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports,

and by the grant 18-03253S of the Grant Agency of the Czech Republic.

1

arXiv:2012.10368v1 [math.CA] 18 Dec 2020

2 F. BAUSTIAN AND V. BOBKOV

In the present work, we investigate the basisness of eigenfunctions for another type of

eigenvalue problems for the classical Dirichlet Laplacian in one dimension which does not ﬁt

into the framework of the spectral theorem, namely,

−𝑢′′ =𝛼𝑢+(𝑥)−𝛽𝑢−(𝑥)in (0, 𝜋), 𝑢(0) = 𝑢(𝜋)=0,(1.1)

where both 𝛼, 𝛽 are spectral parameters. This so-called Fučík (or Dancer-Fučík)eigenvalue

problem was originated in the works [11] and [17] in the context of studies of elliptic equations

with “jumping” nonlinearities. Various aspects of the Fučík eigenvalue problem for second-

order elliptic operators where intensively studied afterwards. We refer to [8,33] for the local

existence of curves in the Fučík spectrum emanated from classical eigenvalues 𝜆𝑘, to [9,14]

for variational characterizations of such curves, to [29,31] for their asymptotic behaviour, to

[10,23] for nodal properties of the corresponding Fučík eigenfunctions, and to [12,28] for

algebro-topological properties of the associated energy functional. Although this list is far

from being complete, many other relevant references can be found in the indicated papers.

Nevertheless, the structure and properties of the Fučík spectrum and Fučík eigenfunctions are

understood only to a rather limited extent, and their investigation can be a hard task even

in the case of operators acting in ﬁnite-dimensional spaces, see the discussion in [20,25]. In

particular, we are not aware of previous results on the basis properties of Fučík eigenfunctions

of the problem (1.1) for the one-dimensional Laplacian.

Fučík eigenfunctions associated with Fučík eigenvalues (𝛼, 𝛽)of the problem (1.1) can be

interpreted as speciﬁc alterations of the sine functions that, in general, do not lead to a system

of dilated or orthogonal functions when 𝛼̸=𝛽, although they preserve a certain periodic

structure. By establishing explicit formulas for the distances between Fučík eigenfunctions and

corresponding sine functions, and making use of the periodic structure, we obtain sequences

of Fučík eigenvalues (𝛼(𝑛), 𝛽(𝑛)) whose corresponding Fučík eigenfunctions form a Riesz basis

in 𝐿2(0, 𝜋). To this end, we employ several classical results from the stability theory of Paley

and Wiener [27], which provides basic information on the basisness of Fučík eigenfunctions,

and we leave more developed approaches for further investigation.

Let us outline the structure of this article. In the remainder of the introduction, we rig-

orously introduce normalized Fučík eigenfunctions and Fučík systems as sequences of such

functions in Section 1.1, and we state our main results in Section 1.2. We derive explicit

formulas for the norms of the normalized Fučík eigenfunctions, their distances to the cor-

responding sine functions, and some scalar products between Fučík eigenfunctions and sine

functions in Section 2. In Section 3, we establish upper bounds on the distances which are

essential for the proof of Theorem 1.4. The proof of Theorem 1.7 is given in Section 4, and

the proof of Theorem 1.9 is given in Section 5. We conclude the article with ﬁnal remarks in

Section 6. Appendices Aand Bcontain several auxiliary technical details.

1.1 Fučík eigenvalues and eigenfunctions. The Fučík spectrum Σ(0, 𝜋)⊂R2of the

linear Dirichlet Laplacian in one dimension is deﬁned as the set of all pairs of parameters

(𝛼, 𝛽)∈R2for which the problem

−𝑢′′(𝑥) = 𝛼𝑢+(𝑥)−𝛽𝑢−(𝑥)in (0, 𝜋),

𝑢(0) = 𝑢(𝜋) = 0,(1.1)

has a non-trivial solution. Here, 𝑢+= max(𝑢, 0) and 𝑢−= max(−𝑢, 0) denote the positive and

negative parts of 𝑢, respectively. Clearly, for 𝛼=𝛽=: 𝜆we obtain the standard eigenvalue

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 3

problem −𝑢′′ =𝜆𝑢 with zero Dirichlet boundary conditions, which possesses the complete

sequence of eigenvalues 𝜆𝑛=𝑛2,𝑛∈N, and the sine functions 𝜑𝑛= sin(𝑛𝑥)as corresponding

eigenfunctions. Since 𝜑1has deﬁnite sign, the lines {1} × Rand R× {1}are the trivial part

of Σ(0, 𝜋)and we have

Σ(0, 𝜋)∖(({1} × R)∪(R× {1})) ⊂ {(𝛼, 𝛽)∈R2:𝛼 > 1, 𝛽 > 1}

by the variational characterization of the ﬁrst eigenvalue 𝜆1= 1. Any (𝛼, 𝛽)∈Σ(0, 𝜋)is called

Fučík eigenvalue and any corresponding non-zero solution of (1.1) is called Fučík eigenfunction.

Hereinafter, under a solution of (1.1) we mean the classical (i.e., 𝐶2-regular) solution of that

problem. Any Fučík eigenfunction consists of positive bumps of length 𝑙1=𝜋

√𝛼alternating

with negative bumps of length 𝑙2=𝜋

√𝛽, and has the form of translations and multiples of

sin(√𝛼𝑥)at positive bumps and sin(√𝛽𝑥)at negative bumps. For each even number 𝑛of

bumps there exists one curve

Γ𝑛=(𝛼, 𝛽)∈R2:𝑛

2

𝜋

√𝛼+𝑛

2

𝜋

√𝛽=𝜋

in the Fučík spectrum Σ(0, 𝜋)that contains (𝜆𝑛, 𝜆𝑛), while for each odd number 𝑛≥3of

bumps there are two curves

Γ𝑛=(𝛼, 𝛽)∈R2:𝑛+ 1

2

𝜋

√𝛼+𝑛−1

2

𝜋

√𝛽=𝜋,

Γ𝑛=(𝛼, 𝛽)∈R2:𝑛−1

2

𝜋

√𝛼+𝑛+ 1

2

𝜋

√𝛽=𝜋

in Σ(0, 𝜋)intersecting at (𝜆𝑛, 𝜆𝑛), see Figure 1. These curves completely describe the non-

trivial part of Σ(0, 𝜋), see [17, Lemma 2.8]. Observe that if 𝑢is a Fučík eigenfunction for the

pair of parameters (𝛼, 𝛽), then so is 𝑡𝑢 for any 𝑡 > 0, while −𝑡𝑢 is a Fučík eigenfunction for

(𝛽, 𝛼). As a consequence, since (𝛼, 𝛽 )∈Γ𝑛for odd 𝑛implies (𝛽, 𝛼)∈

Γ𝑛, we will neglect the

curve

Γ𝑛from our further consideration. Moreover, notice that if 𝑢is a Fučík eigenfunction

for the pair of parameters (𝛼, 𝛽)∈Γ𝑛for even 𝑛, then 𝑣(𝑥) = 𝑢(𝜋−𝑥)is also a Fučík

eigenfunction for (𝛼, 𝛽)and 𝑣̸=𝑢.

Figure 1. Several curves of the Fučík spectrum

In order to uniquely specify a Fučík eigenfunction for each point of Σ(0, 𝜋), let us introduce

the following special choice of Fučík eigenfunctions, see Figure 2.

4 F. BAUSTIAN AND V. BOBKOV

Deﬁnition 1.1. Let 𝑛≥2and (𝛼, 𝛽 )∈Γ𝑛. The normalized Fučík eigenfunction 𝑓𝑛

𝛼,𝛽 is the

𝐶2-solution of the boundary value problem (1.1) with (𝑓𝑛

𝛼,𝛽)′(0) >0which is normalized by

‖𝑓𝑛

𝛼,𝛽‖∞= sup

𝑥∈[0,𝜋]|𝑓𝑛

𝛼,𝛽(𝑥)|= 1.

For 𝑛= 1, we set 𝑓1

𝛼,𝛽 =𝜑1for every (𝛼, 𝛽)∈({1} × R)∪(R× {1}).

The normalized Fučík eigenfunctions can be described more explicitly by the following

piecewise deﬁnition. Let 𝑛≥2. For 𝛼≥𝑛2≥𝛽we have

𝑓𝑛

𝛼,𝛽(𝑥) = √𝛽

√𝛼sin(√𝛼(𝑥−𝑘𝑙)) for 𝑘𝑙 ≤𝑥 < 𝑘𝑙 +𝑙1,

−sin(√𝛽(𝑥−𝑘𝑙 −𝑙1)) for 𝑘𝑙 +𝑙1≤𝑥 < (𝑘+ 1)𝑙, (1.2)

and for 𝛽 > 𝑛2> 𝛼 we have

𝑓𝑛

𝛼,𝛽(𝑥) = sin(√𝛼(𝑥−𝑘𝑙)) for 𝑘𝑙 ≤𝑥 < 𝑘𝑙 +𝑙1,

−√𝛼

√𝛽sin(√𝛽(𝑥−𝑘𝑙 −𝑙1)) for 𝑘𝑙 +𝑙1≤𝑥 < (𝑘+ 1)𝑙, (1.3)

where 𝑙=𝑙1+𝑙2and 𝑘∈N0=N∪ {0}. Notice that (1.2) and (1.3) deﬁne 𝑓𝑛

𝛼,𝛽 on the whole

R+. We also remark that 𝑓𝑛

𝛼,𝛽 ̸∈ 𝐶3[0, 𝜋]provided 𝛼̸=𝛽.

(a) 𝑛= 2 (b) 𝑛= 3

Figure 2. A normalized Fučík eigenfunction 𝑓𝑛

𝛼,𝛽 (solid) and sin(𝑛𝑥)(dotted)

In our purpose to form a basis in 𝐿2(0, 𝜋)that consists solely of normalized Fučík eigen-

functions, we pick one normalized Fučík eigenfunction from each curve Γ𝑛,𝑛≥2, and the sine

function as the normalized Fučík eigenfunction of the trivial part of Σ(0, 𝜋). In this regard,

the following deﬁnition will be useful.

Deﬁnition 1.2. We deﬁne a Fučík system 𝐹𝛼,𝛽 ={𝑓𝑛

𝛼(𝑛),𝛽(𝑛)}as a sequence of normal-

ized Fučík eigenfunctions with the mappings 𝛼, 𝛽 :N→Rsatisfying 𝛼(1) = 𝛽(1) = 1 and

(𝛼(𝑛), 𝛽(𝑛)) ∈Γ𝑛for every 𝑛≥2.

1.2 Main results. In this section, we summarize our main results for the basisness of

Fučík systems given by Deﬁnition 1.2. We present growth conditions on the mappings 𝛼, 𝛽

which guarantee that the Fučík system 𝐹𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋). We call a complete

system {𝜓𝑛}in 𝐿2(0, 𝜋)aRiesz basis if there exist positive constants 𝑐, 𝐶 > 0such that the

inequalities

𝑐

𝑁

𝑛=1 |𝛼𝑛|2≤⃦

⃦

𝑁

𝑛=1

𝛼𝑛𝜓𝑛⃦

⃦≤𝐶

𝑁

𝑛=1 |𝛼𝑛|2

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 5

are satisﬁed for arbitrary 𝑁∈Nand any constants 𝛼1, . . . , 𝛼𝑁. In fact, Riesz bases are images

of an orthonormal basis under a linear homeomorphism. Several equivalent characterizations

of a Riesz basis can be found in [36, Theorem 9]. Hereinafter, ‖·‖ denotes the standard norm

in 𝐿2(0, 𝜋).

We make use of several methods from the stability theory of Paley and Wiener to show

that a Fučík system 𝐹𝛼,𝛽 inherits the basis properties from the system of sine functions {𝜑𝑛}

provided that the two sequences are suﬃciently close to each other. There exist various

concepts of nearness between systems of functions, among which we will be interested in the

following two most classical notions, see, e.g., [34].

Deﬁnition 1.3. Let {𝜙𝑛}and {𝜓𝑛}be two sequences of functions. The sequence {𝜓𝑛}is

quadratically near to {𝜙𝑛}if

∞

𝑛=1 ‖𝜙𝑛−𝜓𝑛‖2=𝑟 < ∞(1.4)

for a constant 𝑟≥0, and strongly quadratically near if (1.4) holds for 𝑟 < 1. The sequence

{𝜓𝑛}is Paley-Wiener near to {𝜙𝑛}if there exits a constant 𝜆∈(0,1) such that

⃦

⃦

𝑁

𝑛=1

𝛼𝑛(𝜙𝑛−𝜓𝑛)⃦

⃦< 𝜆⃦

⃦

𝑁

𝑛=1

𝛼𝑛𝜙𝑛⃦

⃦

holds for arbitrary 𝑁∈Nand any constants 𝛼1, . . . , 𝛼𝑁.

We introduce our results, each one connected to one of these nearness concepts. The ﬁrst

general result, Theorem 1.4, allows a rather free choice of the Fučík system 𝐹𝛼,𝛽 and utilizes

the strong quadratic nearness of that system to the sine functions {𝜑𝑛}. In Theorem 1.7,

we signiﬁcantly improve the constants of Theorem 1.4 by means of the quadratic nearness of

special Fučík systems 𝐹𝛼,𝛽 with 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for odd 𝑛to {𝜑𝑛}. For the ﬁnal result the

normalized Fučík eigenfunctions 𝑓𝑛

𝛼,𝛽 for even 𝑛are chosen in such a way that they form a

sequence of dilated functions, while for odd 𝑛we just pick the sine functions as in the previous

case. This speciﬁc choice of the Fučík system allows us to apply the separation of variables

approach of Duﬃn and Eachus [15] in order to establish the Paley-Wiener nearness to the

sine functions. In view of the nature of Riesz bases, both approaches are intrinsically based

on the construction of a bounded invertible operator 𝑇:𝐿2(0, 𝜋)→𝐿2(0, 𝜋)which maps the

trigonometric system {𝜑𝑛}to the Fučík system 𝐹𝛼,𝛽 .

The basisness of systems that are quadratically near to a complete orthonormal system

was ﬁrst studied by Bary in [2]. A system {𝜓𝑛}which is quadratically near to a complete

orthonormal system {𝜙𝑛}is a Riesz basis provided that it is 𝜔-linearly independent, i.e., if

the strong convergence

∞

𝑛=1

𝜂𝑛𝜓𝑛= lim

𝑚→∞ ‖

𝑚

𝑛=1

𝜂𝑛𝜓𝑛‖= 0

for a sequence of scalars {𝜂𝑛}implies 𝜂𝑛= 0 for every 𝑛∈N. A proof of this stability result

by means of compact operators is given, e.g., in [21, Theorem V-2.20]. If the system {𝜓𝑛}

satisﬁes the more restrictive assumption of being strongly quadratically near to a complete

orthonormal system {𝜙𝑛}, then {𝜓𝑛}is also a Riesz basis, see, e.g., [21, Corollary V-2.22].

We establish a summation criterium for bounds on the mappings 𝛼and 𝛽of the Fučík system

6 F. BAUSTIAN AND V. BOBKOV

𝐹𝛼,𝛽 that yields the basisness of that system by means of the strong quadratic nearness to the

system of sine functions {𝜑𝑛}.

Theorem 1.4. Let 𝐹𝛼,𝛽 be a Fučík system. For any natural 𝑛≥2, we set

𝐶𝑛(𝑥, 𝑦) =

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

4(3 + 𝜋2)𝜋

9max(√𝑥, √𝑦)

𝑛−12

for even 𝑛,

4𝜋𝑛2(𝑛2+ 1)

(𝑛−1)4√𝑥

𝑛−12

for odd 𝑛with 𝑥≥𝑦,

5𝜋𝑛2(𝑛2+ 1)

(𝑛+ 1)4√𝑦

𝑛−12

for odd 𝑛with 𝑦 > 𝑥.

(1.5)

If the summation formula ∞

𝑛=2

𝐶𝑛(𝛼(𝑛), 𝛽(𝑛)) <𝜋

2(1.6)

is satisﬁed, then 𝐹𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).

The deﬁnition (1.5) of 𝐶𝑛is given by the bounds (3.2), (3.4), (3.5), (3.6) on the distances

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2that we will derive in Section 3below. The summation formula (1.6) guarantees

that the Fučík system 𝐹𝛼,𝛽 is quadratically near to the system of sine functions {𝜑𝑛}in the

sense that ∞

𝑛=1 ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2<𝜋

2.(1.7)

This inequality implies that the rescaled system 2/𝜋 𝐹𝛼,𝛽 is strongly quadratically near to

the complete orthonormal system {2/𝜋 𝜑𝑛}in 𝐿2(0, 𝜋), and thus it is a Riesz basis by [21,

Corollary V-2.22]. Hence, the initial Fučík system 𝐹𝛼,𝛽 is also a Riesz basis in 𝐿2(0, 𝜋).

We can make use of Theorem 1.4 to give asymptotic bounds on the mappings 𝛼and 𝛽.

Corollary 1.5. Let 𝐹𝛼,𝛽 be a Fučík system and let 𝜀 > 0be ﬁxed. Let the mappings 𝛼and 𝛽

satisfy

max 𝛼(𝑛),𝛽(𝑛)≤𝑛+√𝑐𝑛𝑛(1−𝜀)/2(1.8)

for every 𝑛≥2with non-negative constants

𝑐𝑛<9

8(3 + 𝜋2)·1

𝜁(1 + 𝜀)−1for even 𝑛,

𝑐𝑛<(𝑛−1)4

8𝑛2(𝑛2+ 1) ·1

𝜁(1 + 𝜀)−1for odd 𝑛with 𝛼(𝑛)≥𝛽(𝑛),(1.9)

𝑐𝑛<(𝑛+ 1)4

10𝑛2(𝑛2+ 1) ·1

𝜁(1 + 𝜀)−1for odd 𝑛with 𝛽(𝑛)> 𝛼(𝑛),(1.10)

where 𝜁is the Riemann zeta function deﬁned by 𝜁(𝑠) = ∞

𝑛=1 1

𝑛𝑠. Then 𝐹𝛼,𝛽 is a Riesz basis

in 𝐿2(0, 𝜋).

Remark 1.6. The upper bounds (1.9) and (1.10) can be replaced by the following weaker

ones which are independent from 𝑛:

𝑐𝑛<1

46 ·1

𝜁(1 + 𝜀)−1for odd 𝑛with 𝛼(𝑛)≥𝛽(𝑛),(1.9’)

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 7

𝑐𝑛<1

10 ·1

𝜁(1 + 𝜀)−1for odd 𝑛with 𝛽(𝑛)> 𝛼(𝑛).(1.10’)

On Figure 3we depict two regions of Fučík eigenvalues described by the inequality (1.8) with

the uniform constants 𝑐𝑛given by (1.9’).

(a) 𝜖= 0.1(b) 𝜖= 0.5

Figure 3. Shaded regions depict the result of Corollary 1.5 with 𝑐𝑛=1

46 ·

1

𝜁(1+𝜀)−1

Clearly, the general result given by Theorem 1.4 also covers such Fučík systems 𝐹𝛼,𝛽 in which

only some Fučík eigenfunctions diﬀer from the sine functions. However, in the particular case

when 𝑓𝑛

𝛼,𝛽 =𝜑𝑛for all odd 𝑛, we can obtain the following stronger result.

Theorem 1.7. Let 𝐹𝛼,𝛽 be a Fučík system with 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for every odd 𝑛. If the

mappings 𝛼and 𝛽satisfy

∞

𝑛=2 max(𝛼(𝑛),𝛽(𝑛))

𝑛−12

<∞,(1.11)

then 𝐹𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).

Theorem 1.7 will be proven in Section 4by showing that the corresponding Fučík system

𝐹𝛼,𝛽 is 𝜔-linearly independent and it is quadratically near to the system of sine functions

{𝜑𝑛}.

Corollary 1.8. Let 𝐹𝛼,𝛽 be a Fučík system with 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for every odd 𝑛. Let

𝜀, 𝑐 > 0be ﬁxed. If the mappings 𝛼and 𝛽satisfy

max 𝛼(𝑛),𝛽(𝑛)≤𝑛+√𝑐 𝑛(1−𝜖)/2(1.12)

for any even 𝑛, then 𝐹𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).

Let us now discuss the basisness of Fučík systems by means of the Paley-Wiener nearness to

the system of sine functions. We consider a Fučík system 𝐹𝛼,𝛽 for which the points (𝛼(𝑛), 𝛽(𝑛))

8 F. BAUSTIAN AND V. BOBKOV

for even 𝑛are on a line through the origin and, as in Theorem 1.7, the normalized Fučík

eigenfunctions 𝑓𝑛

𝛼,𝛽 are just 𝜑𝑛for odd 𝑛. We apply the method of separation of variables

from [15] to this speciﬁc Fučík system to obtain mappings 𝛼and 𝛽with better asymptotics

as 𝑛→ ∞ than in Theorem 1.7, see Figure 4.

Theorem 1.9. Let 𝐹𝛼,𝛽 be a Fučík system with 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for every odd 𝑛and

𝛼(𝑛) = 𝑛2𝛾

4, 𝛽(𝑛) = 𝑛2𝛾

(2√𝛾−2)2(1.13)

for every even 𝑛, where 𝛾∈[4,5.682] is an arbitrary ﬁxed constant. Then 𝐹𝛼,𝛽 is a Riesz basis

in 𝐿2(0, 𝜋).

The choice of the Fučík eigenfunctions for even 𝑛in Theorem 1.9 guarantees that these

functions form a dilated system in the sense that

𝑓𝑛

𝛼(𝑛),𝛽(𝑛)(𝑥) = 𝑓2

𝛼(2),𝛽(2) 𝑛𝑥

2for any even 𝑛.

This property is important in the proof of Theorem 1.9 which we give in Section 5.

The formulas (1.13) in Theorem 1.9 obviously guarantee that 𝛼(𝑛)> 𝛽(𝑛)for even 𝑛.

Moreover, the points (𝛼(𝑛), 𝛽(𝑛)) deﬁned by (1.13) are on the line

𝛽=4𝛼

(2√𝛾−2)2.(1.14)

We observe that (1.13) for the mapping 𝛼can be written as

𝛼(𝑛) = 𝑛+√𝛾

2−1𝑛for any even 𝑛,

which provides a better asymptotic than (1.12) of Corollary 1.8. Notice also that Theorem

1.9 remains valid if we exchange 𝛼and 𝛽due to the symmetry properties of the Fučík curves

Γ𝑛with even 𝑛.

(a) 𝜖= 0.1,

𝑐= 0.25;𝛾= 5.6

(b) 𝜖= 0.5,

𝑐= 0.4;𝛾= 5.6

Figure 4. Comparison of the results of Corollary 1.8 and Theorem 1.9

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 9

Let us emphasize that constants which appear in the function 𝐶𝑛in Theorem 1.4 and

the admissible range for the constant 𝛾in Theorem 1.9 are non-optimal since they follow

from a series of estimates convenient for simpliﬁcation of the derived expressions. Thus, we

anticipate that these constants might be substantially improved. On the other hand, the

asymptotic growth rate of 𝐶𝑛(𝛼(𝑛), 𝛽(𝑛)) is expected to be sharp for the quadratic nearness

considerations.

2. Norms – distances – scalar products

In this section, we derive explicit expressions for the 𝐿2-norms of the normalized Fučík

eigenfunctions 𝑓𝑛

𝛼,𝛽, the distances ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2which are important for the proof of Theorem

1.4, and some scalar products ⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑚⟩which will be used in the proof of Theorem 1.7. We

write these formulas for the case 𝛼≥𝑛2≥𝛽in dependence of 𝛼and 𝑛, and for the case

𝛽 > 𝑛2> 𝛼 in dependence of 𝛽and 𝑛. We will thoroughly treat only the ﬁrst case, and omit

details for the second case to shorten the exposition.

Recall the following notations from Section 1.1:

𝑙1=𝜋

√𝛼, 𝑙2=𝜋

√𝛽, 𝑙 =𝑙1+𝑙2.

Moreover, we have

𝛼=𝑛2𝛽

(2√𝛽−𝑛)2and 𝛽=𝑛2𝛼

(2√𝛼−𝑛)2

for all points (𝛼, 𝛽)∈Γ𝑛with even 𝑛, and

𝛼=(𝑛+ 1)2𝛽

(2√𝛽−(𝑛−1))2and 𝛽=(𝑛−1)2𝛼

(2√𝛼−(𝑛+ 1))2

for all points (𝛼, 𝛽)∈Γ𝑛with odd 𝑛.

2.1 The case 𝛼≥𝑛2≥𝛽.We begin with the derivation of the norms of the normalized

Fučík eigenfunctions 𝑓𝑛

𝛼,𝛽. We obtain

‖𝑓𝑛

𝛼,𝛽‖2=𝑛

2𝑙1

0

𝛽

𝛼sin2(√𝛼𝑥) d𝑥+𝑛

2𝑙2

0

sin2(𝛽𝑥) d𝑥=𝑛

4

𝛽

𝛼

𝜋

√𝛼+𝑛

4

𝜋

√𝛽

=𝜋𝑛3

4√𝛼(2√𝛼−𝑛)2+1

2𝜋−𝑛

2

𝜋

√𝛼=𝜋

2−𝜋𝑛(√𝛼−𝑛)

(2√𝛼−𝑛)2

for even 𝑛. In a similar way, we have

‖𝑓𝑛

𝛼,𝛽‖2=𝑛+ 1

4

𝛽

𝛼

𝜋

√𝛼+𝑛−1

4

𝜋

√𝛽=𝜋

2−𝜋(𝑛+ 1)(√𝛼−1)(√𝛼−𝑛)

√𝛼(2√𝛼−(𝑛+ 1))2

for odd 𝑛. Notice that all the sine functions have the same norm ‖𝜑𝑛‖=𝜋

2,𝑛∈N.

Let us now derive expressions for the distances ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2. We start with the case 𝑛= 2.

Using the formulas from Appendix A, we get

‖𝑓2

𝛼,𝛽 −𝜑2‖2=𝑙1

0√𝛽

√𝛼sin(√𝛼 𝑥)−sin(2𝑥)2

d𝑥+𝜋

𝑙1sin(𝛽(𝑥−𝑙1)) + sin(2𝑥)2d𝑥

10 F. BAUSTIAN AND V. BOBKOV

=𝜋

2+1

2

𝛽

𝛼𝑙1+1

2𝑙2+ 2 √𝛽

4−𝛼sin(2𝑙1)−2√𝛽

4−𝛽sin(2𝑙1)

=𝜋

2+1

2

𝛽

𝛼

𝜋

√𝛼+1

2

𝜋

√𝛽+ 2𝛽𝛼−𝛽

(4 −𝛼)(4 −𝛽)sin 2𝜋

√𝛼

=𝜋−𝜋2(√𝛼−2)

(2√𝛼−2)2−4𝛼2

(2√𝛼−2)(3√𝛼−2)(√𝛼+ 2)

sin 2𝜋

√𝛼

√𝛼−2.(2.1)

Now, for a general even 𝑛we have

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2=

𝑛

2−1

𝑖=0 (𝑖+1)𝑙

𝑖𝑙

(𝑓𝑛

𝛼,𝛽(𝑥)−sin(𝑛𝑥))2d𝑥=𝑛

2𝑙

0

(𝑓𝑛

𝛼,𝛽(𝑥)−sin(𝑛𝑥))2d𝑥

=𝜋

0

(𝑓2

4𝛼/𝑛2,4𝛽/𝑛2(𝑦)−sin(2𝑦))2d𝑦=‖𝑓2

4𝛼/𝑛2,4𝛽/𝑛2−𝜑2‖2,

where we used that 𝑓𝛼,𝛽(𝑥)−sin(𝑛𝑥)realizes the same values on each interval (𝑖𝑙, (𝑖+ 1)𝑙)for

every 𝑖∈N. Therefore, we deduce from (2.1) that

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2=𝜋−𝜋𝑛(√𝛼−𝑛)

(2√𝛼−𝑛)2−4𝛼2

(2√𝛼−𝑛)(3√𝛼−𝑛)(√𝛼+𝑛)

sin 𝑛𝜋

√𝛼

√𝛼−𝑛(2.2)

holds for any even 𝑛.

Now we consider the case of odd 𝑛≥3. This case requires more extensive calculations, and

we put the derivation of the following formula to Appendix A. We have

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2=𝜋−𝜋(𝑛+ 1)(√𝛼−1)

√𝛼(2√𝛼−(𝑛+ 1))2(√𝛼−𝑛)

−16(𝑛−1)√𝛼3

(2√𝛼−(𝑛+ 1))

(√𝛼−1)

(𝑛+√𝛼)(𝑛+ 1)((3𝑛−1)√𝛼−𝑛(𝑛+ 1))×

×

cos 𝜋

2

𝑛

√𝛼cos 𝜋

2

𝑛2+𝑛−2√𝛼

(𝑛−1)√𝛼

(√𝛼−𝑛) sin 𝜋√𝛼−𝑛

(𝑛−1)√𝛼.

(2.3)

for any odd 𝑛≥3. Notice that the assumption 𝛼≥𝑛2guarantees that each multiplier on the

right-hand side of (2.3) is nonnegative.

Finally, we derive some scalar products ⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑚⟩needed for the proof of Theorem 1.7. For

the special case 𝑛=𝑚≥2, we can express the scalar product in terms of the formulas above

as

⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑛⟩=1

2‖𝑓𝑛

𝛼,𝛽‖2+‖𝜑𝑛‖2− ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2,(2.4)

and then easily obtain

⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑛⟩=2𝛼2

(2√𝛼−𝑛)(3√𝛼−𝑛)(√𝛼+𝑛)

sin 𝑛𝜋

√𝛼

√𝛼−𝑛(2.5)

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 11

for even 𝑛, and

⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑛⟩=8(𝑛−1)√𝛼3

(2√𝛼−(𝑛+ 1))

(√𝛼−1)

(𝑛+√𝛼)(𝑛+ 1)((3𝑛−1)√𝛼−𝑛(𝑛+ 1))×

×

cos 𝜋

2

𝑛

√𝛼cos 𝜋

2

𝑛2+𝑛−2√𝛼

(𝑛−1)√𝛼

(√𝛼−𝑛) sin 𝜋√𝛼−𝑛

(𝑛−1)√𝛼

for odd 𝑛≥3.

If 𝑛̸=𝑚, then the scalar product ⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑚⟩vanishes for some combinations of 𝑛and 𝑚.

In particular, we have ⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑚⟩= 0 for odd 𝑛and even 𝑚by a simple symmetry argument.

When both 𝑛and 𝑚are even with 𝑛>𝑚, the scalar product also vanishes. Indeed, we get

⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑚⟩=(𝛽−𝛼)√𝛽

(𝛼−𝑚2)(𝛽−𝑚2)sin 𝑛

2−1𝑚

2𝑙+ sin 𝑛

2−1𝑚

2𝑙+𝑚𝑙1sin 𝑚𝑛

4𝑙

sin 𝑚

2𝑙= 0

(2.6)

due to 𝑛

2𝑙=𝜋. Details on the derivation of this formula are given in Appendix A.

2.2 The case 𝛽 > 𝑛2> 𝛼.For the norm of the normalized Fučík eigenfunctions 𝑓𝑛

𝛼,𝛽, we

have

‖𝑓𝑛

𝛼,𝛽‖2=𝑛

2𝑙1

0

sin2(√𝛼𝑥) d𝑥+𝑛

2𝑙2

0

𝛼

𝛽sin2(𝛽𝑥) d𝑥

=𝑛

4

𝜋

√𝛼+𝑛

4

𝛼

𝛽

𝜋

√𝛽=𝜋

2−𝜋𝑛(√𝛽−𝑛)

(2√𝛽−𝑛)2

for even 𝑛, and

‖𝑓𝑛

𝛼,𝛽‖2=𝑛+ 1

4

𝜋

√𝛼+𝑛−1

4

𝛼

𝛽

𝜋

√𝛽=𝜋

2−𝜋(𝑛−1)(√𝛽+ 1)(√𝛽−𝑛)

√𝛽(2√𝛽−(𝑛−1))2

for odd 𝑛.

Now we derive expressions for the distances ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2. As in Section 2.1, we start with

the case 𝑛= 2:

‖𝑓2

𝛼,𝛽 −𝜑2‖2=𝑙1

0

(sin(√𝛼 𝑥)−sin(2𝑥))2d𝑥+𝜋

𝑙1√𝛼

√𝛽sin(𝛽(𝑥−𝑙1)) + sin(2𝑥)2

d𝑥

=𝜋

2+1

2

𝜋

√𝛼+1

2

𝛼

𝛽

𝜋

√𝛽+ 2√𝛼𝛼−𝛽

(4 −𝛼)(4 −𝛽)sin 2𝜋

√𝛼

=𝜋−𝜋2(√𝛽−2)

(2√𝛽−2)2−4𝛽2

(2√𝛽−2)(3√𝛽−2)(√𝛽+ 2)

sin 2𝜋

√𝛽

√𝛽−2.

Therefore, we obtain

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2=𝜋−𝜋𝑛(√𝛽−𝑛)

(2√𝛽−𝑛)2−4𝛽2

(2√𝛽−𝑛)(3√𝛽−𝑛)(√𝛽+𝑛)

sin 𝑛𝜋

√𝛽

√𝛽−𝑛(2.7)

12 F. BAUSTIAN AND V. BOBKOV

for even 𝑛. Furthermore, we have

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2=𝜋−𝜋(𝑛−1)(√𝛽+ 1)

√𝛽(2√𝛽−(𝑛−1))2(𝛽−𝑛)

−16(𝑛+ 1)√𝛽3

(2√𝛽−(𝑛−1))

(√𝛽+ 1)

(𝑛+√𝛽)(𝑛−1)((3𝑛+ 1)√𝛽−𝑛(𝑛−1))×

×

cos 𝜋

2

𝑛

√𝛽cos 𝜋

2

(2√𝛽+𝑛2−𝑛)

(𝑛+1)√𝛽

(√𝛽−𝑛) sin 𝜋𝑛(√𝛽+1)

(𝑛+1)√𝛽

(2.8)

for odd 𝑛≥3. As in (2.3), each multiplier on the right-hand side of (2.8) is nonnegative in

view of the assumption 𝛽 > 𝑛2.

Finally, for the derivation of the scalar product ⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑛⟩, we use (2.4) and get

⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑛⟩=2𝛽2

(2√𝛽−𝑛)(3√𝛽−𝑛)(√𝛽+𝑛)

sin 𝑛𝜋

√𝛽

√𝛽−𝑛(2.9)

for even 𝑛, and

⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑛⟩=8(𝑛+ 1)√𝛽3

(2√𝛽−(𝑛−1))

(√𝛽+ 1)

(𝑛+√𝛽)(𝑛−1)((3𝑛+ 1)√𝛽−𝑛(𝑛−1))×

×

cos 𝜋

2

𝑛

√𝛽cos 𝜋

2

(2√𝛽+𝑛2−𝑛)

(𝑛+1)√𝛽

(√𝛽−𝑛) sin 𝜋𝑛(√𝛽+1)

(𝑛+1)√𝛽

for odd 𝑛≥3. The scalar product ⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑚⟩vanishes for even 𝑚provided that either 𝑛is

odd, or 𝑛is even with 𝑛>𝑚, as in the case 𝛼≥𝑛2≥𝛽in Section 2.1.

3. Asymptotics of distances

In this section, we establish upper bounds on the distances ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2which allows to

describe their asymptotic behaviour for 𝑛→ ∞, as well as for √𝛼→𝑛or √𝛽→𝑛.

3.1 The case 𝛼≥𝑛2≥𝛽.We begin with estimating the distance ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2given by

(2.2) for even 𝑛. Using the lower bound

sin 𝑥≥𝑥−𝑥3

6=𝑥

6(√6−𝑥)(√6 + 𝑥), 𝑥 ≥0,(3.1)

and the assumption 𝛼≥𝑛2, we see that

sin 𝑛𝜋

√𝛼= sin 𝜋−𝜋𝑛

√𝛼≥𝜋

6√𝛼3(√𝛼−𝑛)((√6−𝜋)√𝛼+𝑛𝜋)((√6 + 𝜋)√𝛼−𝑛𝜋).

Therefore, we get

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤𝜋−𝜋𝑛(√𝛼−𝑛)

(2√𝛼−𝑛)2−2𝜋√𝛼((√6−𝜋)√𝛼+𝑛𝜋)((√6 + 𝜋)√𝛼−𝑛𝜋)

3(2√𝛼−𝑛)(3√𝛼−𝑛)(√𝛼+𝑛)

=𝜋

3

4(3 + 𝜋2)𝛼+√𝛼𝑛(15 −2𝜋2)−6𝑛2

(2√𝛼−𝑛)2(3√𝛼−𝑛)(√𝛼+𝑛)(√𝛼−𝑛)2

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 13

≤𝜋

3

4(3 + 𝜋2)𝛼−(2𝜋2−9)𝑛2

𝑛2(3√𝛼−𝑛)(√𝛼+𝑛)(√𝛼−𝑛)2

=4(3 + 𝜋2)𝜋

93√𝛼−3√2𝜋2−9

2√3+𝜋2𝑛√𝛼+√2𝜋2−9

2√3+𝜋2𝑛

𝑛2(3√𝛼−𝑛)(√𝛼+𝑛)(√𝛼−𝑛)2

≤4(3 + 𝜋2)𝜋

9

(√𝛼−𝑛)2

𝑛2=4(3 + 𝜋2)𝜋

9√𝛼

𝑛−12

(3.2)

for even 𝑛.

Let us now estimate ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2given by (2.3) for odd 𝑛. Recalling that all multipliers in

(2.3) are nonnegative and applying the rough upper bound

sin 𝜋√𝛼−𝑛

(𝑛−1)√𝛼≤𝜋√𝛼−𝑛

(𝑛−1)√𝛼,

we get

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤𝜋−𝜋(𝑛+ 1)(√𝛼−1)

√𝛼(2√𝛼−(𝑛+ 1))2(√𝛼−𝑛)

−16(𝑛−1)2𝛼2

𝜋(2√𝛼−(𝑛+ 1))

(√𝛼−1)

(𝑛+√𝛼)(𝑛+ 1)((3𝑛−1)√𝛼−𝑛(𝑛+ 1))×

×

cos 𝜋

2

𝑛

√𝛼cos 𝜋

2

𝑛2+𝑛−2√𝛼

(𝑛−1)√𝛼

(√𝛼−𝑛)2.

(3.3)

Then, using (3.1), we have

cos(𝑥) = sin 𝜋

2−𝑥≥1

6𝜋

2−𝑥√6−𝜋

2+𝑥√6 + 𝜋

2−𝑥, 𝑥 ≤𝜋

2,

and hence, by 𝛼 > 𝑛2, we obtain

cos 𝜋

2

𝑛

√𝛼≥𝜋(√𝛼−𝑛)

48√𝛼3((2√6−𝜋)√𝛼+𝜋𝑛)((2√6 + 𝜋)√𝛼−𝜋𝑛)

and

cos 𝜋

2

𝑛2+𝑛−2√𝛼

(𝑛−1)√𝛼≥𝜋(𝑛+ 1)(√𝛼−𝑛)

48(𝑛−1)3√𝛼3((2√6(𝑛−1) −𝜋(𝑛+ 1))√𝛼+𝜋𝑛(𝑛+ 1))

×((2√6(𝑛−1) + 𝜋(𝑛+ 1))√𝛼−𝜋𝑛(𝑛+ 1)).

Substituting these estimates into (3.3), we get

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤𝜋−𝜋(𝑛+ 1)(√𝛼−1)

√𝛼(2√𝛼−(𝑛+ 1))2(√𝛼−𝑛)

−𝜋(√𝛼−1)((2√6−𝜋)√𝛼+𝜋𝑛)((2√6 + 𝜋)√𝛼−𝜋𝑛)

144(𝑛−1)𝛼(2√𝛼−(𝑛+ 1))(𝑛+√𝛼)((3𝑛−1)√𝛼−𝑛(𝑛+ 1))×

×((2√6(𝑛−1) −𝜋(𝑛+ 1))√𝛼+𝜋𝑛(𝑛+ 1))

×((2√6(𝑛−1) + 𝜋(𝑛+ 1))√𝛼−𝜋𝑛(𝑛+ 1))

≤4𝜋𝑛2(𝑛2+ 1)

(𝑛−1)4√𝛼

𝑛−12

≤23𝜋√𝛼

𝑛−12

(3.4)

14 F. BAUSTIAN AND V. BOBKOV

for odd 𝑛. Several intermediate estimates used to derive these upper bounds are given in

Appendix A.

3.2 The case 𝛽 > 𝑛2> 𝛼.Let us estimate ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2given by (2.7) for even 𝑛. Noting

that (2.7) is the same formula as (2.2) up to a replacement of 𝛼by 𝛽, we get

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤4(3 + 𝜋2)𝜋

9√𝛽

𝑛−12

(3.5)

for even 𝑛, as in (3.2).

Now we provide an upper bound on ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2given by (2.8) for odd 𝑛. Using (3.1) as in

Section 3.1 and recalling that 𝛽 > 𝑛2, we estimate the trigonometric terms in (2.8) as follows:

cos 𝜋

2

𝑛

√𝛽≥𝜋(√𝛽−𝑛)

48𝛽3((2√6−𝜋)𝛽+𝜋𝑛)((2√6 + 𝜋)𝛽−𝜋𝑛),

cos 𝜋

2

(2√𝛽+𝑛2−𝑛)

(𝑛+ 1)√𝛽≥𝜋(𝑛−1)(√𝛽−𝑛)

48(𝑛+ 1)3𝛽3((2√6(𝑛+ 1) −𝜋(𝑛−1))𝛽+𝜋𝑛(𝑛−1))

×((2√6(𝑛+ 1) + 𝜋(𝑛−1))𝛽−𝜋𝑛(𝑛−1)),

sin 𝜋𝑛(√𝛽+ 1)

(𝑛+ 1)√𝛽= sin 𝜋−𝜋𝑛(√𝛽+ 1)

(𝑛+ 1)√𝛽= sin 𝜋(√𝛽−𝑛)

(𝑛+ 1)√𝛽≤𝜋(√𝛽−𝑛)

(𝑛+ 1)√𝛽.

Substituting these estimates into (2.8), we deduce that

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤𝜋−𝜋(𝑛−1)(√𝛽+ 1)

√𝛽(2√𝛽−(𝑛−1))2(𝛽−𝑛)

−𝜋(√𝛽+ 1)((2√6−𝜋)√𝛽+𝜋𝑛)((2√6 + 𝜋)√𝛽−𝜋𝑛)

144(𝑛+ 1)𝛽(2√𝛽−(𝑛−1))(𝑛+√𝛽)((3𝑛+ 1)√𝛽−𝑛(𝑛−1))

×((2√6(𝑛+ 1) −𝜋(𝑛−1))𝛽+𝜋𝑛(𝑛−1))

×((2√6(𝑛+ 1) + 𝜋(𝑛−1))𝛽−𝜋𝑛(𝑛−1))

≤5𝜋𝑛2(𝑛2+ 1)

(𝑛+ 1)4√𝛽

𝑛−12

≤5𝜋√𝛽

𝑛−12

(3.6)

for odd 𝑛. Several intermediate estimates in the derivation of these upper bounds can be

found in Appendix A.

4. Proof of Theorem 1.7

To prove Theorem 1.7, we ﬁrst establish the 𝜔-linear independence of Fučík systems 𝐹𝛼,𝛽

with 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for every odd 𝑛. As the second step, we use the bounds (3.2) and

(3.5) to deduce that the assumption (1.11) guarantees the quadratic nearness of 𝐹𝛼,𝛽 to the

system of sine functions {𝜑𝑛}.

Let {𝜂𝑛}be a sequence of scalars such that

lim

𝑚→∞ ‖

𝑚

𝑛=1

𝜂𝑛𝑓𝑛

𝛼,𝛽‖= 0.

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 15

In order to show that 𝐹𝛼,𝛽 is 𝜔-linear independent, we have to prove 𝜂𝑛= 0 for each 𝑛. Fix

any 𝑘∈Nand consider the functions

𝑔𝑚,𝑘(𝑥) = sin(𝑘𝑥)

𝑚

𝑛=1

𝜂𝑛𝑓𝑛

𝛼,𝛽(𝑥)for 𝑚∈N.

We see that

⃒⃒⃒𝜋

0

𝑔𝑚,𝑘(𝑥) d𝑥⃒⃒⃒≤𝜋

0|sin(𝑘𝑥)|⃒⃒⃒

𝑚

𝑛=1

𝜂𝑛𝑓𝑛

𝛼,𝛽(𝑥)⃒⃒⃒d𝑥≤𝜋

2‖

𝑚

𝑛=1

𝜂𝑛𝑓𝑛

𝛼,𝛽‖ → 0as 𝑚→ ∞,

which yields

∞

𝑛=1

𝜂𝑛⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑘⟩= lim

𝑚→∞ 𝜋

0

𝑔𝑚,𝑘(𝑥) d𝑥= 0.(4.1)

Taking 𝑘= 2 and recalling that ⟨𝑓𝑛

𝛼,𝛽, 𝜑2⟩= 0 for any 𝑛̸= 2 (see Section 2), we get

𝜂2⟨𝑓2

𝛼,𝛽, 𝜑2⟩= 0. Since ⟨𝑓2

𝛼,𝛽, 𝜑2⟩ ̸= 0 by (2.5) and (2.9), we obtain 𝜂2= 0. By an induc-

tive argument, we derive in the same way 𝜂2𝑚= 0 for every 𝑚≥1. Let us remark that until

now we did not use the special form of the Fučík system 𝐹𝛼,𝛽 , namely, 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for

all odd 𝑛≥3. Since this assumption reads as 𝑓𝑛

𝛼,𝛽 =𝜑𝑛for any odd 𝑛, we further obtain

from (4.1) that 𝜂2𝑚−1= 0 for every 𝑚≥1by the orthogonality of {𝜑𝑛}. This concludes the

𝜔-linear independence of the Fučík system 𝐹𝛼,𝛽.

Using the uppers bounds (3.2) and (3.5) on the distances ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2, we get

∞

𝑛=1 ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤4(3 + 𝜋2)𝜋

9

∞

𝑛=2 max(𝛼(𝑛),𝛽(𝑛))

𝑛−12

.

Since the right-hand side is bounded in view of the assumption (1.11), the Fučík system 𝐹𝛼,𝛽

is quadratically near to the system of sine functions {𝜑𝑛}. Therefore, the rescaled system

2/𝜋 𝐹𝛼,𝛽 is quadratically near to the complete orthonormal system {2/𝜋 𝜑𝑛}, and hence

it is a Riesz basis in 𝐿2(0, 𝜋)by [21, Theorem V-2.20]. Clearly, 𝐹𝛼,𝛽 is also a Riesz basis.

5. Proof of Theorem 1.9

In this section, we provide the proof of our third main result, Theorem 1.9. For this purpose,

we use the method of separation of variables from [15] to show that a speciﬁc Fučík system 𝐹𝛼,𝛽

is Paley-Wiener near to the system of sine functions {𝜑𝑛}, see Deﬁnition 1.2. The classical

result of Paley and Wiener [27] then yields the basisness of 𝐹𝛼,𝛽 in 𝐿2(0, 𝜋).

Recall that we choose 𝐹𝛼,𝛽 to satisfy 𝑓𝑛

𝛼(𝑛),𝛽(𝑛)=𝜑𝑛for any odd 𝑛, and

𝛼(𝑛) = 𝑛2𝛾

4, 𝛽(𝑛) = 𝑛2𝛾

(2√𝛾−2)2(1.13)

for every even 𝑛, where 𝛾is a ﬁxed constant in the interval [4,5.682]. Denoting 𝑙1(𝑛) = 𝜋

√𝛼(𝑛)

and 𝑙2(𝑛) = 𝜋

√𝛽(𝑛), we see that (1.13) implies

𝑙1(2)

𝑙1(𝑛)=𝑛

2and 𝑙2(2)

𝑙2(𝑛)=𝑛

2

16 F. BAUSTIAN AND V. BOBKOV

for every even 𝑛. It can be easily deduced from the piecewise deﬁnition (1.2) that 𝑓𝑛

𝛼(𝑛),𝛽(𝑛)

for even 𝑛form a sequence of dilated functions in the sense that

𝑓𝑛

𝛼(𝑛),𝛽(𝑛)(𝑥) = 𝑓2

𝛼(2),𝛽(2) 𝑛𝑥

2≡𝑓2

𝛾,𝛾/(√𝛾−1)2𝑛𝑥

2.(5.1)

We claim that the assumptions of [15, Theorem D] (with 𝑓𝑛=𝜑𝑛and 𝑔𝑛=𝑓𝑛

𝛼(𝑛),𝛽(𝑛)) are

satisﬁed for any 𝛾∈[4,5.682], namely, there exist a matrix of constants {𝐶𝑛,𝑘}and a sequence

of bounded linear operators {𝑇𝑘}such that each 𝑓𝑛

𝛼(𝑛),𝛽(𝑛)has the representation

𝑓𝑛

𝛼(𝑛),𝛽(𝑛)(𝑥) = sin(𝑛𝑥) + ∞

𝑘=1

𝐶𝑛,𝑘𝑇𝑘sin(𝑛𝑥),(5.2)

where we have |𝐶𝑛,𝑘 | ≤ 𝑐𝑘and ‖𝑇𝑘‖*≤𝑡𝑘with constants 𝑐𝑘and 𝑡𝑘satisfying ∞

𝑘=1 𝑐𝑘𝑡𝑘<1.

Here, ‖ · ‖*is the operator norm. If our claim is true, then the Fučík system 𝐹𝛼,𝛽 is a Riesz

basis in 𝐿2(0, 𝜋). We remark that although the system {𝜑𝑛}is not orthonormal, [15, Theorem

D] is applicable by simple rescaling arguments since all 𝜑𝑛are of the same norm.

Let 𝑛be even. Then, in view of (5.1), we have

𝑓𝑛

𝛼(𝑛),𝛽(𝑛)(𝑥) = 𝑓2

𝛼(2),𝛽(2) 𝑛𝑥

2=∞

𝑘=1

𝐴𝑘sin 𝑘𝑛𝑥

2,(5.3)

where 𝐴𝑘are the coeﬃcients of the odd Fourier expansion of 𝑓2

𝛼(2),𝛽(2). The comparison of

(5.2) and (5.3) suggests to deﬁne bounded linear operators 𝑇𝑘,𝑘∈N, that satisfy the following

property:

𝑇𝑘sin(𝑛𝑥) = sin 𝑘𝑛𝑥

2for every even 𝑛. (5.4)

To this end, for any 𝑔∈𝐿2(0, 𝜋)we deﬁne its antiperiodic extension 𝑔*as

𝑔*(𝑥)=(−1)𝜅𝑔(𝑥−𝜋𝜅)for 𝜋𝜅 ≤𝑥≤𝜋(𝜅+ 1), 𝜅 ∈N0.

In particular, we see that if 𝑔(𝑥) = sin(𝑥)for 𝑥∈(0, 𝜋), then 𝑔*(𝑥) = sin(𝑥)for any 𝑥∈R+.

Now we choose 𝑇𝑘:𝐿2(0, 𝜋)→𝐿2(0, 𝜋)as

𝑇𝑘𝑔(𝑥) = 𝑔*𝑘𝑥

2.(5.5)

Clearly, 𝑇2is just the identity operator and each 𝑇𝑘satisﬁes (5.4). Moreover, each 𝑇𝑘is linear

with the norm ‖𝑇𝑘‖*= 1 for even 𝑘and ‖𝑇𝑘‖*=1+1/𝑘 for odd 𝑘, see Appendix B.

Thus, in accordance with (5.2) and (5.3), for even 𝑛we set 𝐶𝑛,1=𝐴1,𝐶𝑛,2=𝐴2−1, and

𝐶𝑛,𝑘 =𝐴𝑘for 𝑘≥3, while for odd 𝑛we simply choose 𝐶𝑛,𝑘 = 0 for 𝑘∈N. For this choice of

constants, we can set 𝑐1=|𝐴1|,𝑐2=|𝐴2−1|, and 𝑐𝑘=|𝐴𝑘|for 𝑘≥3.

Let us now estimate the constants 𝑐𝑘. Notice that

𝐴𝑘=2

𝜋𝜋

0

𝑓2

𝛼(2),𝛽(2)(𝑥) sin(𝑘𝑥) d𝑥=2

𝜋

𝛾2

√𝛾−1

(2 −√𝛾) sin 𝑘𝜋

√𝛾

(𝑘2−𝛾)(𝑘2(√𝛾−1)2−𝛾).

Thus, for 𝑐1we easily obtain

𝑐1=|𝐴1| ≤ 2

𝜋

𝛾2(√𝛾−2)

(√𝛾−1)2(√𝛾+ 1)(2√𝛾−1).

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 17

For 𝑐2we recall that 𝛾≥4and use the upper bound

sin 2𝜋

√𝛾= sin 𝜋−2𝜋

√𝛾= sin 𝜋(√𝛾−2)

√𝛾≤𝜋(√𝛾−2)

√𝛾(5.6)

to deduce

𝐴2−1 = 2

𝜋

𝛾2

√𝛾−1

sin 2𝜋

√𝛾

(𝛾−4)(3√𝛾−2) −1≤2√𝛾3

(√𝛾−1)(√𝛾+ 2)(3√𝛾−2) −1≤0.

Hence, using (3.1) instead of (5.6), we derive that

𝑐2= 1 −𝐴2≤1−√𝛾((√6−𝜋)√𝛾+ 2𝜋)((√6 + 𝜋)√𝛾−2𝜋)

3(√𝛾−1)(√𝛾+ 2)(3√𝛾−2)

=((3 + 𝜋2)𝛾+ (9 −2𝜋2)√𝛾−6)(√𝛾−2)

3(√𝛾−1)(√𝛾+ 2)(3√𝛾−2) .

Finally, for 𝑐𝑘with 𝑘≥3we get

𝑐𝑘=|𝐴𝑘|=2

𝜋

𝛾2(√𝛾−2)

√𝛾−1⃒⃒sin 𝑘𝜋

√𝛾⃒⃒

(𝑘2−𝛾)(𝑘2−𝛾+𝑘2√𝛾(√𝛾−2)) ≤2

𝜋

𝛾2(√𝛾−2)

√𝛾−1

1

(𝑘2−𝛾)2.

Using the estimates above, we deduce that

∞

𝑘=1

𝑐𝑘‖𝑇𝑘‖*≤

4

𝑘=1

𝑐𝑘‖𝑇𝑘‖*+6

5·2

𝜋

𝛾2(√𝛾−2)

√𝛾−1

∞

𝑘=5

1

(𝑘2−9)2

≤√2·2

𝜋

𝛾2(√𝛾−2)

(√𝛾−1)2(√𝛾+ 1)(2√𝛾−1)

+((3 + 𝜋2)𝛾+ (9 −2𝜋2)√𝛾−6)(√𝛾−2)

3(√𝛾−1)(√𝛾+ 2)(3√𝛾−2)

+4

3·2

𝜋

𝛾2(√𝛾−2)

√𝛾−1

1

(32−𝛾)2+2

𝜋

𝛾2(√𝛾−2)

√𝛾−1

1

(42−𝛾)2

+6

5·2

𝜋

𝛾2(√𝛾−2)

√𝛾−1𝜋2

108 −536741

6350400=: 𝐸(𝛾).

By straightforward calculations, it is not hard to show that each summand in 𝐸(𝛾)is strictly

increasing with respect to 𝛾≥4and 𝐸(4) = 0. At the same time, we have 𝐸(5.682) =

0.9992..., which shows that 𝐸(𝛾)<1for any 𝛾∈[4,5.682]. This completes the proof of

Theorem 1.9.

6. Final remarks

1. The quadratic nearness assumption (1.7) used in the proof of Theorem 1.4 can be

weakened to the inequality

∞

𝑛=1 ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2−|⟨𝑓𝑛

𝛼,𝛽 −𝜑𝑛, 𝑓 𝑛

𝛼,𝛽⟩|2

‖𝑓𝑛

𝛼,𝛽‖2<𝜋

2(6.1)

18 F. BAUSTIAN AND V. BOBKOV

and still guarantee that the Fučík system 𝐹𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋), see [21, Theorem

V-2.21]. Noting that each summand in (6.1) can be written as

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2−‖𝑓𝑛

𝛼,𝛽‖2− ‖𝜑𝑛‖2+‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖22

4‖𝑓𝑛

𝛼,𝛽‖2,(6.2)

one can apply the formulas from Section 2to derive the explicit expression for (6.2) and

estimate it from above in the same way as in Section 3. However, this does not improve the

asymptotic behaviour of the function 𝐶𝑛(𝛼(𝑛), 𝛽(𝑛)) as 𝑛→ ∞ in Theorem 1.4, but only

slightly improves the constants.

2. We anticipate that the 𝜔-linear independence is satisﬁed for general Fučík systems.

Provided this claim is true, the assumptions of Theorem 1.4 can be signiﬁcantly weakened in

the sense that the sum in (1.6) only has to converge.

3. In the proof of Theorem 1.9 we used the fact that normalized Fučík eigenfunctions 𝑓𝑛

𝛼,𝛽,

with (𝛼(𝑛), 𝛽(𝑛)) for even 𝑛on a straight line through the origin, form a dilated sequence in

the sense of (5.1). It is not hard to show that the normalized Fučík eigenfunctions for odd 𝑛

with (𝛼(𝑛), 𝛽(𝑛)) on the same line satisfy

𝑓𝑛

𝛼(𝑛),𝛽(𝑛)(𝑥) = 𝑓2

𝛼(2),𝛽(2) 𝑛−1

2+1

𝛼(2)𝑥.

Nevertheless, this property is less suitable for application of [15, Theorem D] since the opera-

tors 𝑇𝑘in (5.2) have to be independent of 𝑛.

4. Theorem 1.9 might suggest that the basisness of a Fučík system 𝐹𝛼,𝛽 also holds when

each point (𝛼(𝑛), 𝛽(𝑛)) ∈Γ𝑛,𝑛≥2, belongs to the angular sector in between the line (1.14)

and its reﬂection with respect to the main diagonal 𝛼=𝛽.

5. Recall that the constants in the function 𝐶𝑛in Theorem 1.4, as well as the upper

bound for 𝛾in Theorem 1.9, are not optimal due to the employed methods and the estimation

procedure.

Appendix A.

We start by providing several useful formulas. For the derivation of expressions in Section

2, we need to calculate several integrals of the general form

𝑐sin(√𝛿(𝑥−𝑥0)) ±sin(𝑛𝑥)2d𝑥

with constants 𝑐∈1,√𝛼/√𝛽, √𝛽/√𝛼,𝛿∈ {𝛼, 𝛽}, and certain 𝑥0∈[0, 𝜋]. The antideriva-

tive of this integral can be expressed through the following formulas:

sin2(𝑛𝑥) d𝑥=1

2𝑥−1

𝑛sin(𝑛𝑥) cos(𝑛𝑥)+𝐶,

sin2(√𝛿(𝑥−𝑥0)) d𝑥=1

2𝑥−𝑥0−1

√𝛿sin(√𝛿(𝑥−𝑥0)) cos(√𝛿(𝑥−𝑥0))+𝐶,

sin(√𝛿(𝑥−𝑥0)) sin(𝑛𝑥) d𝑥=√𝛿

𝑛2−𝛿cos(√𝛿(𝑥−𝑥0)) sin(𝑛𝑥)

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 19

−𝑛

𝑛2−𝛿sin(√𝛿(𝑥−𝑥0)) cos(𝑛𝑥) + 𝐶.

We observe that sin(√𝛿(𝑥−𝑥0)) vanishes for 𝑥=𝑥0and 𝑥=𝑥0+𝜋/√𝛿, and cos(√𝛿(𝑥−𝑥0))

evaluated in these points becomes either 1or −1, which signiﬁcantly simpliﬁes corresponding

deﬁnite integrals occurring in the sections above.

Let us now provide a more detailed derivation of the expression ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2for odd 𝑛≥3

and 𝛼≥𝑛2≥𝛽, see (2.3). Recall that each normalized Fučík eigenfunction 𝑓𝑛

𝛼,𝛽 with odd 𝑛

has 𝑛+1

2positive bumps and 𝑛−1

2negative bumps. Therefore, using the formulas from above,

we deduce that

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2=

𝑛+1

2−1

𝑖=0 𝑖𝑙+𝑙1

𝑖𝑙 √𝛽

√𝛼sin(√𝛼(𝑥−𝑖𝑙)) −sin(𝑛𝑥)2

d𝑥

+

𝑛−1

2−1

𝑖=0 (𝑖+1)𝑙

𝑖𝑙+𝑙1sin(𝛽(𝑥−𝑖𝑙 −𝑙1)) + sin(𝑛𝑥)2d𝑥

=𝜋

2+𝑛+ 1

4

𝛽

𝛼𝑙1+𝑛−1

4𝑙2

+ 4𝛽𝛼−𝛽

(𝑛2−𝛼)(𝑛2−𝛽)

𝑛−1

2−1

𝑖=0

[sin(𝑛(𝑖𝑙 +𝑙1)) + sin(𝑛(𝑖+ 1)𝑙)]

=𝜋

2+𝑛+ 1

4

𝛽

𝛼𝑙1+𝑛−1

4𝑙2

+ 8𝛽𝛼−𝛽

(𝑛2−𝛼)(𝑛2−𝛽)

sin (𝑛−1)𝑛𝑙

8cos 𝑛𝑙2

2

sin 𝑛𝑙

2sin (𝑛−1)𝑛𝑙

8+𝑛𝑙1

2

=𝜋−𝜋(𝑛+ 1)(√𝛼−1)

√𝛼(2√𝛼−(𝑛+ 1))2(√𝛼−𝑛)

+16(𝑛−1)√𝛼3

(2√𝛼−(𝑛+ 1))

(√𝛼−1)

(𝑛+√𝛼)(𝑛+ 1)(𝑛2−3𝑛√𝛼+𝑛+√𝛼)×

×

cos 𝜋

2

𝑛

√𝛼cos 𝜋

2

𝑛(2√𝛼−𝑛−1)

(𝑛−1)√𝛼

(√𝛼−𝑛) sin 𝜋𝑛(√𝛼−1)

(𝑛−1)√𝛼.(A.1)

Here, we used the summation formula

𝑘−1

𝑖=0

sin(𝑐𝑖 +𝑑) = sin 𝑘𝑐

2sin (𝑘−1) 𝑐

2+𝑑

sin 𝑐

2.(A.2)

Notice that the arguments of the last cosine and sine in (A.1) satisfy

𝜋

2≤𝜋

2

𝑛(2√𝛼−𝑛−1)

(𝑛−1)√𝛼≤3𝜋

2and 𝜋≤𝜋𝑛(√𝛼−1)

(𝑛−1)√𝛼≤3𝜋

2.

20 F. BAUSTIAN AND V. BOBKOV

That is, the cosine and sine of these arguments are negative. To make it easier to control the

total sign in (A.1), we apply the formulas

cos(𝑥) = −cos(𝜋−𝑥)and sin(𝑥) = −sin(𝑥−𝜋).

This gives the expression (2.3).

The derivation of the expression (2.6) for the scalar product ⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑚⟩for even 𝑛and 𝑚

with 𝑛>𝑚follows similar steps as above:

⟨𝑓𝑛

𝛼,𝛽, 𝜑𝑚⟩=

𝑛

2−1

𝑘=0

√𝛽

√𝛼𝑘𝑙+𝑙1

𝑘𝑙

sin(√𝛼(𝑥−𝑘𝑙)) sin(𝑚𝑥) d𝑥

−

𝑛

2−1

𝑘=0 (𝑘+1)𝑙

𝑘𝑙+𝑙1

sin(𝛽(𝑥−𝑘𝑙 −𝑙1)) sin(𝑚𝑥) d𝑥

=𝛽1

𝛼−𝑚2−1

𝛽−𝑚2𝑛

2−1

𝑘=0

sin(𝑚𝑘𝑙) + sin(𝑚(𝑘𝑙 +𝑙1)),

and we conclude by applying the summation formula (A.2).

Let us now provide details on the derivation of the upper bound for ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2with odd

𝑛≥3and 𝛼≥𝑛2≥𝛽, given by (3.4). Reducing the terms on the right-hand side of the ﬁrst

inequality in (3.4) to a common denominator, we arrive at

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤𝜋(√𝛼−𝑛)2

144(𝑛−1)𝛼(2√𝛼−𝑛−1)2(𝑛+√𝛼)((3𝑛−1)√𝛼−𝑛(𝑛+ 1))

×144√𝛼(𝑛2−1)(2√𝛼+ 4√𝛼3+𝑛+𝑛2−2√𝛼𝑛(𝑛+ 2) + 𝛼(5𝑛−7))

+ 48𝜋2𝛼(√𝛼−1)(2√𝛼−𝑛−1)(𝑛2+ 1)

−𝜋4(√𝛼−1)(√𝛼−𝑛)2(2√𝛼−𝑛−1)(𝑛+ 1)2.

Using now the following simple estimates:

(2√𝛼−𝑛−1)2≥(𝑛−1)2

𝑛2𝛼, (𝑛+√𝛼)((3𝑛−1)√𝛼−𝑛(𝑛+ 1)) ≥4𝑛2(𝑛−1),

𝑛2−1≤𝑛2+ 1, 𝑛 +𝑛2−2√𝛼𝑛(𝑛+ 2) + 𝛼(5𝑛−7) ≤𝛼(5𝑛−7) ≤5√𝛼3,

(√𝛼−1)(2√𝛼−𝑛−1) ≤2𝛼, −𝜋4(√𝛼−1)(√𝛼−𝑛)2(2√𝛼−𝑛−1)(𝑛+ 1)2≤0,

we get

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤𝜋(√𝛼−𝑛)2(𝑛2+ 1)(288 + (1296 + 96𝜋2)𝛼)

576(𝑛−1)4𝛼.

Finally, recalling that √𝛼≥𝑛≥3and roughly estimating

288 + (1296 + 96𝜋2)𝛼

576 ≤4𝛼,

we obtain

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤4𝜋(𝑛2+ 1)(√𝛼−𝑛)2

(𝑛−1)4≤4𝜋𝑛2(𝑛2+ 1)

(𝑛−1)4√𝛼

𝑛−12

≤23𝜋√𝛼

𝑛−12

.

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 21

For the case 𝛽 > 𝑛2> 𝛼, we use a similar procedure as above to estimate the upper bound

for ‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2with odd 𝑛≥3given by (3.6). We convert the right-hand side of the ﬁrst

inequality in (3.6) to a common denominator, to get

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤𝜋(√𝛽−𝑛)2

144(𝑛+ 1)𝛽(2√𝛽−𝑛+ 1)2(𝑛+√𝛽)((3𝑛+ 1)√𝛽−𝑛(𝑛−1))

×144𝛽(𝑛2−1)(2𝛽+ 4𝛽3+𝑛−𝑛2−2𝛽𝑛(𝑛−2) + 𝛽(5𝑛+ 7))

+ 48𝜋2𝛽(𝛽+ 1)(2𝛽−𝑛+ 1)(𝑛2+ 1)

−𝜋4(𝛽+ 1)(𝛽−𝑛)2(2𝛽−𝑛+ 1)(𝑛−1)2.

With the simple estimates

(2𝛽−𝑛+ 1)2≥(𝑛+ 1)2

𝑛2𝛽, (𝑛+𝛽)((3𝑛+ 1)𝛽−𝑛(𝑛−1)) ≥4𝑛2(𝑛+ 1),

𝑛2−1≤𝑛2+ 1, 𝑛 −𝑛2−2𝛽𝑛(𝑛−2) + 𝛽(5𝑛+ 7) ≤𝛽(5𝑛+ 7) ≤8𝛽3,

(𝛽+ 1)(2𝛽−𝑛+ 1) ≤2𝛽, −𝜋4(𝛽+ 1)(𝛽−𝑛)2(2𝛽−𝑛+ 1)(𝑛−1)2≤0,

we obtain

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤𝜋(√𝛽−𝑛)2(𝑛2+ 1)(288 + (1728 + 96𝜋2)𝛽)

576(𝑛+ 1)4𝛽.

Finally, estimating

288 + (1728 + 96𝜋2)𝛽

576 ≤5𝛽,

we obtain

‖𝑓𝑛

𝛼,𝛽 −𝜑𝑛‖2≤5𝜋(𝑛2+ 1)(√𝛽−𝑛)2

(𝑛+ 1)4≤5𝜋𝑛2(𝑛2+ 1)

(𝑛+ 1)4√𝛽

𝑛−12

≤5𝜋√𝛽

𝑛−12

.

Appendix B.

In this section, we calculate the norms of the operators 𝑇𝑘deﬁned by (5.5). First, let us

show that each 𝑇𝑘is linear. Indeed, taking any 𝑔, ℎ ∈𝐿2(0, 𝜋), we have

𝑇𝑘(𝑔+ℎ)(𝑥)=(𝑔+ℎ)*𝑘𝑥

2= (−1)𝜅(𝑔+ℎ)𝑘𝑥

2−𝜋𝜅

for 𝜋𝜅 ≤𝑘𝑥

2≤𝜋(𝜅+ 1),𝜅∈N0. At the same time, for such 𝑥we get

(−1)𝜅(𝑔+ℎ)𝑘𝑥

2−𝜋𝜅= (−1)𝜅𝑔𝑘𝑥

2−𝜋𝜅+ (−1)𝜅ℎ𝑘𝑥

2−𝜋𝜅

=𝑔*𝑘𝑥

2+ℎ*𝑘𝑥

2=𝑇𝑘𝑔(𝑥) + 𝑇𝑘ℎ(𝑥),

which concludes the linearity.

Assume now that 𝑘is even, i.e., 𝑘= 2𝑚,𝑚≥1. For any 𝑔∈𝐿2(0, 𝜋)we obtain

𝜋

0

(𝑇2𝑚𝑔(𝑥))2d𝑥=𝜋

0

(𝑔*(𝑚𝑥))2d𝑥=1

𝑚𝜋𝑚

0

(𝑔*(𝑥))2d𝑥

22 F. BAUSTIAN AND V. BOBKOV

=1

𝑚

𝑚−1

𝜅=0 𝜋(𝜅+1)

𝜋𝜅

𝑔2(𝑥−𝜋𝜅) d𝑥=1

𝑚

𝑚−1

𝜅=0 𝜋

0

𝑔2(𝑥) d𝑥=𝜋

0

𝑔2(𝑥) d𝑥.

Therefore, we get

‖𝑇2𝑚‖*= sup

𝑔∈𝐿2(0,𝜋)∖{0}

‖𝑇2𝑚𝑔‖

‖𝑔‖= 1.

Assume that 𝑘is odd, i.e., 𝑘= 2𝑚+ 1,𝑚≥0. We have

𝜋

0

(𝑇2𝑚+1𝑔(𝑥))2d𝑥=𝜋

0𝑔*(2𝑚+ 1)𝑥

22

d𝑥=2

2𝑚+ 1 𝜋(𝑚+1

2)

0

(𝑔*(𝑥))2d𝑥

=2

2𝑚+ 1

𝑚−1

𝜅=0 𝜋(𝜅+1)

𝜋𝜅

𝑔2(𝑥−𝜋𝜅) d𝑥+2

2𝑚+ 1 𝜋(𝑚+1

2)

𝜋𝑚

𝑔2(𝑥−𝜋𝑚) d𝑥

=2𝑚

2𝑚+ 1 𝜋

0

𝑔2(𝑥) d𝑥+2

2𝑚+ 1 𝜋

2

0

𝑔2(𝑥) d𝑥≤2𝑚+ 2

2𝑚+ 1 𝜋

0

𝑔2(𝑥) d𝑥.

Notice that this estimate is sharp since equality holds for any 𝑔with the support on (0, 𝜋/2).

Thus, we deduce that

‖𝑇2𝑚+1‖*= sup

𝑔∈𝐿2(0,𝜋)∖{0}

‖𝑇2𝑚+1𝑔‖

‖𝑔‖=2𝑚+ 2

2𝑚+ 1.

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(F. Baustian)

Department of Mathematics, University of Rostock,

Ulmenstraße 69, 18057 Rostock, Germany

Email address:falko.baustian@uni-rostock.de

(V. Bobkov)

Department of Mathematics and NTIS, Faculty of Applied Sciences,

University of West Bohemia, Univerzitní 8, 301 00 Plzeň, Czech Republic,

Institute of Mathematics, Ufa Federal Research Centre, RAS,

Chernyshevsky str. 112, 450008 Ufa, Russia

Email address:bobkov@kma.zcu.cz