����������� ������ 56 (2004), 91–104

UDK 517.927

���������� ������ ��� research paper

ON UNIFORM CONVERGENCE ON CLOSED INTERVALS OF SPECTRAL EXPANSIONS AND THEIR DERIVATIVES, FOR FUNCTIONS FROM W_{p}^{(1)}

Nebojša L. Lažetić

Abstract. We consider the global uniform convergence of spectral expansions and their derivatives, ∑_{n=1}^{∞} f_{n} u_{n}^{(j)}(x), (j = 0, 1, 2), arising by an arbitrary one-dimensional self-adjoint Schrödinger operator, defined on a bounded interval G ⊂ ℝ. We establish the absolute and uniform convergence on \overline{G} of the series, supposing that f belongs to suitable defined subclasses of W_{p}^{(1+j)}(G) (1 < p ≤ 2). Also, some convergence rate estimates are obtained.

1. Introduction

Let L be an arbitrary self-adjoint differential operator of second order, defined by the differential expression

L(u)(x) = −u′′(x) + q(x)u(x),  x ∈ G,  (1)

and the self-adjoint boundary conditions

\begin{aligned}α_{10}u(a) + α_{11}u′(a) + β_{10}u(b) + β_{11}u′(b) & = 0, \\ α_{20}u(a) + α_{21}u′(a) + β_{20}u(b) + β_{21}u′(b) & = 0, \end{aligned}  (2)

where (α_{i0}, α_{i1}, β_{i0}, β_{i1}) ∈ ℝ^{4} (i = 1, 2) are linearly independent vectors, G = (a, b) is a bounded interval of the real axis ℝ, and q(x) ∈ L_{1}(G) is a real function. Denote by D(L) ⊂ L_{2}(G) the domain of the operator L : h(x) ∈ D(L) if functions h(x), h′(x) are absolutely continuous on \overline{G}, L(h)(x) ∈ L_{2}(G), and h(x) satisfies the conditions (2). If h(x) ∈ D(L), then L(h)(x) =^{def} L(h)(x) (see [1, §18]). Spectrum of L is discrete. We can suppose, with no loss of generality, that L is a positive operator.

Let {u_{n}(x)}_{n=1}^{∞} be complete (in L_{2}(G)) and orthonormal system of eigenfunctions of L, and let {λ_{n}}_{n=1}^{∞} be the corresponding system of positive eigenvalues, enumerated in non-decreasing order. (By definition, u_{n}(x) belongs to D(L) and satisfies the differential equation

AMS Subject Classification: 34 L 10, 47 E 05

Keywords and phrases: Spectral expansion, uniform convergence, Schrödinger operator

This paper is supported by Ministry of Sciences, Technology and Development of Republic of Serbia (project #1856–Structures of functional analysis and differential equations).

−u′′_{n}(x) + q(x)u_{n}(x) = λ_{n}u_{n}(x)  (3)

almost everywhere in G.) If f(x) ∈ L_{1}(x) and μ > 2, we can form the partial sum of order μ:

σ_{μ}(x, f) =^{def} ∑_{n=1}^{[μ]} f_{n}u_{n}(x), f_{n} =^{def} ∫_{a}^{b} f(x)u_{n}(x) dx.

([μ] denotes the entire part of μ.) The problem of behavior of functions σ_{μ}^{(j)}(x, f) on subsets K ⊆ \overline{G}, as μ → +∞, is the classical one. (Note, h^{(0)}(x) =^{def} h(x).) It is still actual and important, especially in the case of a non-self-adjoint operator L,by itself. The importance of the problem also follows from its relationship with the Fourier method for solving mixed boundary problems for one-dimensional hyperbolic (or parabolic) equations of second order, containing different classes of (non-) self-adjoint boundary conditions.

One of the most fruitful approaches to the problem is so-called “equiconvergence approach”: one studies the behavior on K of the difference σ_{μ}^{(j)}(x, f) − S_{μ}^{(j)}(x, f), as μ → +∞, where S_{μ}(x, f) is the corresponding partial sum of the trigonometrical Fourier series of function f(x). When j = 0 and K ⊂ G is a compact set, the problem has been completely solved by V. A. Il’in (see [2–5]) for a large class of non-self-adjoint boundary conditions. The L_{p}-equiconvergence on K ⊆ \overline{G} (1 ≤ p ≤ +∞, j = 0) was deeply studied by I. S. Lomov in papers [6–8]. A very extensive review of different results (case j = 0) can be found in [12]. As far as the convergence of derivatives of the spectral expansions concerned, it was less studied.

In this paper we consider the uniform convergence on the whole \overline{G}. A “naive” direct method is used:we point out some classes of functions f(x) such that the equalities

f^{(j)}(x) = ∑_{n=1}^{∞} f_{n}u_{n}^{(j)}(x), x ∈ \overline{G}, j = 0, 1, 2,

are valid, where series ∑_{n=1}^{∞} |f_{n}u_{n}^{(j)}(x)| converges uniformly on \overline{G}. (We (will) say that the series converges “absolutely and uniformly” on \overline{G}.) Also, we establish some uniform (on \overline{G}) asymptotic estimates for the differences f^{(j)}(x)−σ_{μ}^{(j)}(x, f),as μ → +∞.

2. Main results

Let AC(\overline{G}) be the class of (real-valued) absolutely continuous functions on the closed interval \overline{G} = [a, b], and let BV (\overline{G}) be the class of functions having the bounded variation on that interval. By W_{p}^{(k)}(G) we denote the set of functions h(x) such that h(x), h′(x), …, h^{(k−2)}(x) are continuously differentiable functions on \overline{G}, h^{(k−1)}(x) ∈ AC(\overline{G}), and h^{(k)}(x) ∈ L_{p}(G)(1 ≤ p < +∞, k ∈ ℕ). We say that function h(x) ∈ L_{p}(G) belongs to the class H_{p}^{α}(G)(0 < α ≤ 1) if there exists a constant C(h) > 0 such that

∥h(x + t) − h(t)∥_{L_{p}(G_{|t|})} ≤ C(h)|t|^{α}

for every t ∈ ((a − b)/2, (b − a)/2),where G_{|t|} =^{def} (a + |t|, b − |t|).

We can now state our results.

Theorem 1. Let q(x) ∈ L_{1}(G), f(x) ∈ W_{p}^{(1)}(G) (1 < p ≤ 2) and f(a) = 0 = f(b). Then:

(a) for x ∈ \overline{G} the equality

f(x) = ∑_{n=1}^{∞} f_{n}u_{n}(x)  (4)

is valid, and the series is absolutely and uniformly convergent on \overline{G}; (b) the estimate

max_{x∈\overline{G}} |f(x) − σ_{μ}(x, f)| = o(\frac{1}{μ^{1−1/p}} ),  μ → +∞.  (5)

holds.

Theorem 2. Let q(x) ∈ L_{1}(G) and f(x) ∈ D(L). Then for every x ∈ \overline{G} we have the equalities

f^{(j)}(x) = ∑_{n=1}^{∞} f_{n}u_{n}^{(j)}(x),  j = 0, 1,  (6)

the series (6) being absolutely and uniformly convergent on \overline{G}. Also, the following estimates are valid

max_{x∈\overline{G}} |f^{(j)}(x) − σ_{μ}^{(j)}(x, f)| = o(\frac{1}{μ^{3/2−j}} ),  j = 0, 1.  (7)

Theorem 3. Let q(x) ∈ W_{1}^{(1)}(G) and f(x) ∈ D(L) ∩ W_{1}^{(3)}(G). If L(f)(x) ∈ W_{p}^{(1)}(G) (1 < p ≤ 2) and L(f)(a) = 0 = L(f)(b), then:

(a) the equalities

f^{(j)}(x) = ∑_{n=1}^{∞} f_{n}u_{n}^{(j)}(x),  j = 0, 1, 2,  (8)

hold on \overline{G}, and the series (8) converge absolutely and uniformly on \overline{G}; (b) we have the estimates

max_{x∈\overline{G}} |f^{(j)}(x) − σ_{μ}^{(j)}(x, f)| = o(\frac{1}{μ^{3−j−1/p}} ),  j = 0, 1, 2.  (9)

Remark 1. In some special cases estimate (5) can be improved. For example, if f′(x) ∈ BV (\overline{G}), then

max_{x∈\overline{G}} |f(x) − σ_{μ}(x, f)| = O(\frac{1}{μ}).

Also, if f′(x) ∈ L_{∞}(G) ∩ H_{1}^{α}(G) (0 < α ≤ 1), then

max_{x∈\overline{G}} |f(x) − σ_{μ}(x, f)| = O(\frac{1}{μ^{α}} ) + o(\frac{1}{μ^{1/2}} ).

These estimates were obtained in [13].

Remark 2. Propositions of Theorem 2 were actually proved in papers [14] and [13]. In order to keep the completeness of exposition of all our results obtained in the considered classes of smoothness for q(x) and f(x), we cite the propositions here. Note, the uniform convergence on \overline{G} of the series ∑_{n=1}^{∞} f_{n}u_{n}(x), for every function f(x) ∈ D(L), is a very well known fact (see [1, p.90]).

Related to Theorem 2, we can add the following. Let q(x) ∈ L_{1}(G), f(x) ∈ W_{1}^{(2)}(G), L(f)(x) ∈ L_{p}(G) (1 < p ≤ 2), and f^{(j)}(a) = 0 = f^{(j)}(b), j = 0, 1. Then the first proposition of the theorem holds, and the estimates

max_{x∈\overline{G}} |f^{(j)}(x) − σ_{μ}^{(j)}(x, f)| = o(\frac{1}{μ^{2−j−1/p}} ),  j = 0, 1,  (10)

are valid, instead of estimates (7).

Remark 3. The estimates (9) can be also improved in some cases. If L(f)′(x) ∈ BV (G), supposing the other conditions of Theorem 3 are satisfied, then

max_{x∈\overline{G}} |f^{(j)}(x) − σ_{μ}^{(j)}(x, f)| = O(\frac{1}{μ^{3−j}} ), j = 0, 1, 2.

Further, if L(f)′(x) ∈ L_{∞}(G) ∩ H_{1}^{α}(G), the following estimates hold:

max_{x∈\overline{G}} |f^{(j)}(x) − σ_{μ}^{(j)}(x, f)| = O(\frac{1}{μ^{2−j+α}} ) + o(\frac{1}{μ^{5/2−j}} ), j = 0, 1, 2,

(see paper [13]).

Remark 4. Suppose that the coefficients of boundary conditions (2) satisfy the relation

α_{11}β_{21} − α_{21}β_{11} ≠ 0,  (11)

as in the case of separated boundary conditions, for example. Then assumptions f(a) = 0 = f(b) (in Theorem 1) and L(f)(a) = 0 = L(f)(b) (in Theorem 3) can be dropped.

If one of the boundary conditions (2) has the form α_{j1}u′(a) + β_{j1}u′(b) = 0, then f(a) = 0 = f(b) (in Theorem 1) can be replaced by β_{j1}f(a) + α_{j1}f(b) = 0, and in Theorem 3 we can suppose that β_{j1}L(f)(a) + α_{j1}L(f)(b) = 0 instead of L(f)(a) = 0 = L(f)(b).

Remark 5. In paper [9] I. S. Lomov has proved that the assertion of proposition (a) of Theorem 1 holds in the case of biorthogonal series ∑_{n=1}^{∞}(f, v_{k})u_{k}(x), for f(x) ∈ W_{2}^{(1)}(G), generated by a non-self-adjoint Schrödinger operator L, with a complex-valued potential q(x) ∈ L_{1}(G). The operator L is defined on G = (0, 1) by multi-point boundary conditions.

Proofs of Theorems 1 and 3 are based on suitable estimates for the Fourier coefficients of f(x). These estimates are obtained by the known asymptotics for the eigenfunctions and eigenvalues of the operator L. On the other hand, the propositions concerned with Theorem 2 can be proved by using only some uniform estimates for the eigenfunctions and their derivatives, and an estimate for a number of the eigenvalues (see the next section).

3. Auxiliary results

In our proofs we will use the following results.

Proposition 1 [15]. If q(x) ∈ L_{1}(G), then there exists a constant C_{0} > 0, independent of n ∈ ℕ, such that

max_{x∈\overline{G}} |u_{n}(x)| ≤ C_{0},  n ∈ ℕ.  (12)

Proposition 2 [15], [16]. If q(x) ∈ L_{1}(G), then there exists a constant A > 0 such that

∑_{t≤\sqrt{λ_{n}}≤t+1} 1 ≤ A,  (13)

for each t ≥ 0, where A does not depend on t.

Proposition 3 [17]. (a) Let q(x) ∈ L_{1}(G). There is a constant C_{1} > 0, not depending on n ∈ ℕ, such that the following estimates are valid:

max_{x∈\overline{G}} |u′_{n}(x)| ≤ C_{1} · (\sqrt{λ_{n}} + 1),  n ∈ ℕ.  (14)

(b) Let q(x) ∈ C(\overline{G}). Then the eigenfunctions u_{n}(x) have continuous second derivative, satisfy the equation (3) everywhere on G, and there exists a constant C_{2} > 0, independent on n ∈ ℕ, such that

max_{x∈\overline{G}} |u′′_{n}(x)| ≤ C_{2} · (λ_{n} + 1),  n ∈ ℕ.  (15)

We will also need the following known asymptotics for the eigenfunctions and eigenvalues of the (self-adjoint) operator L. Let q(x) ∈ L_{1}(G), where G = (−1, 1). There is a number n_{0} ∈ ℕ such that for every n > n_{0} the equalities

u_{n}(x) = A_{n}(cos \sqrt{λ_{n}}x)(1 + \frac{r_{1n}(x)}{\sqrt{λ_{n}}}) + B_{n}(sin \sqrt{λ_{n}}x)(1 + \frac{r_{2n}(x)}{\sqrt{λ_{n}}}),  (16)
u′_{n}(x) = −A_{n}\sqrt{λ_{n}}(sin \sqrt{λ_{n}}x)(1 + \frac{r_{3n}(x)}{\sqrt{λ_{n}}}) + B_{n}\sqrt{λ_{n}}(cos \sqrt{λ_{n}}x)(1 + \frac{r_{4n}(x)}{\sqrt{λ_{n}}}) (17)

are valid on [−1, 1]. Here |A_{n}|^{2} + |B_{n}|^{2} ∈ [c^{−1}, c] for some constant c > 0, and there is a constant D > 0 such that for every n > n_{0} we have

max_{x∈\overline{G}}\{|r_{in}(x)||i = 1, 2, 3, 4\} ≤ D.

Also, for every n > n_{0} the following holds:

\sqrt{λ_{n}} = nπ + γ + \frac{ρ_{n}}{n^{β−1}} ,  (18)

where β = 2 if the boundary conditions (2) satisfy θ_{0}^{2} − 4θ_{−1}θ_{1} ≠ 0, and β = 3/2 if θ_{0}^{2} − 4θ_{−1}θ_{1} = 0 (see [1, pp. 66–67, 74], [10], [11]). Here γ > 0 is a constant, and {ρ_{n}}_{n=n_{0}}^{∞} is a bounded sequence: |ρ_{n}| ≤ ρ.

4. Proof of Theorem 1

In order to avoid technicalities, we will prove the theorem supposing that G = (−1, 1). Transition from the interval (−1, 1) to an arbitrary interval (a, b) can be realized by the following change of variable:

x = \frac{b − a}{2}t + \frac{a + b}{2}, −1 ≤ t ≤ 1.

Consequently, in formulae (16)–(17) one should put (2x − a − b)/(b − a) instead of x, and π in equality (18) should be replaced by π/(b − a).

An estimate for the Fourier coefficients. The first step in the proof is to establish an appropriate estimate for the Fourier coefficients f_{n}(n > n_{0}).

Let λ_{n} be an arbitrary eigenvalue such that n > n_{0}. Using the equation (3), the integration by parts, and assumption f(−1) = 0 = f(1), we obtain the equality

f_{n} = \frac{1}{λ_{n}} · (∫_{−1}^{1} f′(x)u′_{n}(x) dx + ∫_{−1}^{1} f(x)q(x)u_{n}(x) dx).  (19)

Consider the integral ∫_{−1}^{1} f′(x)u′_{n}(x) dx. By virtue of equalities (17)–(18) it can be represented in the form

∫_{−1}^{1} f′(x)u′_{n}(x) dx = −A_{n}\sqrt{λ_{n}}·(I_{1}(n, β)+I_{2}(n, β))+B_{n}\sqrt{λ_{n}}·(I_{3}(n, β)+I_{4}(n, β))

− A_{n} ∫_{−1}^{1} f′(x)r_{3n}(x) sin \sqrt{λ_{n}} x dx + B_{n} ∫_{−1}^{1} f′(x)r_{4n}(x) cos \sqrt{λ_{n}} x dx, (20)

where I_{j}(n, β) are defined as follows:

I_{1}(n, β) = ∫_{−1}^{1} f′(x)[sin nπx · cos(γ + \frac{ρ_{n}}{n^{β−1}} )x] dx,
I_{2}(n, β) = ∫_{−1}^{1} f′(x)[cos nπx · sin(γ + \frac{ρ_{n}}{n^{β−1}} )x] dx,
I_{3}(n, β) = ∫_{−1}^{1} f′(x)[cos nπx · cos(γ + \frac{ρ_{n}}{n^{β−1}} )x] dx,
I_{4}(n, β) = ∫_{−1}^{1} f′(x)[sin nπx · sin(γ + \frac{ρ_{n}}{n^{β−1}} )x] dx.

Let us introduce ρ_{n,β} =^{def} ρ_{n}n^{1−β}. In the further transformations of integrals I_{j}(n, β) we will use the following fact: for any point x ∈ [−1, 1] there exist numbers θ_{1}(x), θ_{2}(x) ∈ (0, 1) such that

cos ρ_{n,β}x = 1 − \frac{ρ_{n,β}^{2}x^{2}}{2} cos(ρ_{n,β}θ_{1}(x)x),
sin ρ_{n,β}x = ρ_{n,β}x − \frac{ρ_{n,β}^{2}x^{2}}{2} sin(ρ_{n,β}θ_{2}(x)x).

Now, for the integral I_{1}(n, β) we obtain

I_{1}(n, β) = ∫_{−1}^{1}[f′(x) cos γx] sin nπx · cos(ρ_{n,β}x) dx

− ∫_{−1}^{1}[f′(x) sin γx] sin nπx · sin(ρ_{n,β}x) dx
= ∫_{−1}^{1}[f′(x) cos γx] sin nπx dx − ρ_{n,β} · ∫_{−1}^{1}[f′(x)x sin γx] sin nπx dx

− \frac{ρ_{n,β}^{2}}{2} · ∫_{−1}^{1}[f′(x)x^{2} cos γx · cos(ρ_{n,β}θ_{1}(x)x)] sin nπx dx

+ \frac{ρ_{n,β}^{2}}{2} · ∫_{−1}^{1}[f′(x)x^{2} sin γx · sin(ρ_{n,β}θ_{2}(x)x)] sin nπx dx.

If we define functions and Fourier coefficients

g_{jc}(x) =^{def} f′(x)x^{j} cos γx, g_{js}(x) =^{def} f′(x)x^{j} sin γx, j = 0, 1, 2;

a_{n}(h) =^{def} ∫_{−1}^{1} h(x) cos nπx dx, b_{n}(h) =^{def} ∫_{−1}^{1} h(x) sin nπx dx, n ∈ ℕ,

then from the preceding equalities it follows that

I_{1}(n, β) = b_{n}(g_{0c}) − \frac{ρ_{n}}{n^{β−1}} · b_{n}(g_{1s}) + O(\frac{1}{n^{2(β−1)}} ).

Analogously, one can obtain the asymptotic relations

I_{2}(n, β) = a_{n}(g_{0s}) + \frac{ρ_{n}}{n^{β−1}} · a_{n}(g_{1c}) + O(\frac{1}{n^{2(β−1)}} ),
I_{3}(n, β) = a_{n}(g_{0c}) − \frac{ρ_{n}}{n^{β−1}} · a_{n}(g_{1s}) + O(\frac{1}{n^{2(β−1)}} ),
I_{4}(n, β) = b_{n}(g_{0s}) + \frac{ρ_{n}}{n^{β−1}} · b_{n}(g_{1c}) + O(\frac{1}{n^{2(β−1)}} ),

where |O(n^{2(1−β)})| ≤ ρ^{2}∥f′∥_{L_{1}(G)}n^{2(1−β)} in all the cases.

Let us return to the equality (20). Applying the preceding representations of the integrals I_{j}(n, β),we can get the following estimate:

|∫_{−1}^{1} f′(x)u′_{n}(x) dx| ≤ \sqrt{cλ_{n}}[|a_{n}(g_{0c} + g_{0s})| + |b_{n}(g_{0c} + g_{0s})|

+ \frac{ρ}{n^{β−1}} (|a_{n}(g_{1c} + g_{1s})| + |b_{n}(g_{1c} + g_{1s})|) + \frac{4ρ^{2}∥f′∥_{L_{1}(G)}}{n^{2(β−1)}} ] + 2\sqrt{c} D∥f′∥_{L_{1}(G)}. (21)

That is why from (19) and (21) it follows, by virtue of estimate (12) and equality (18), that the final (desired) estimate

|f_{n}| ≤ \frac{D_{1}}{n} · (|a_{n}(g_{0c} + g_{0s})| + |b_{n}(g_{0c} + g_{0s})|)

+ \frac{D_{2}}{n^{β}} · (|a_{n}(g_{1c} + g_{1s})) + |b_{n}(g_{1c} + g_{1s})|) + \frac{D_{3}}{n^{2}} , n > n_{0}, (22)

holds, where the constants D_{j} have the following values:

D_{1} =^{def} \frac{\sqrt{c}}{π}, D_{2} =^{def} \frac{ρ\sqrt{c}}{π},

D_{3} =^{def} (\frac{4ρ^{2}\sqrt{c}}{πn_{0}^{2β−3}} + \frac{2\sqrt{c}D}{π^{2}} ) · ∥f′∥_{L_{1}(G)} + \frac{C_{0}}{π^{2}} · ∥f∥_{C(\overline{G})}∥q∥_{L_{1}(G)}. (23)

Proof of equality (4). We first prove the absolute and uniform convergence on \overline{G} = [−1, 1] of the series (4), and the equality (4).

Note that the trigonometrical system \{1/\sqrt{2}, cos nπx, sin nπx | n ∈ ℕ\} is uniformly bounded and orthonormal in L_{2}(G). Therefore, the corresponding Riesz inequality holds: for every function h(x) ∈ L_{p}(G)(1 < p ≤ 2) we have

(∑_{n=0}^{∞}(|a_{n}(h)|^{r} + |b_{n}(h)|^{r}))^{1/r} ≤ ∥h∥_{L_{p}(G)},

where 1/p + 1/r = 1 and a_{0}(h) =^{def} 2^{−1} ∫_{−1}^{1} h(x) dx, b_{0}(h) =^{def} 0 (see [18]).

Now, the mentioned convergence of series (4) is a consequence of the following formal chain of equalities and inequalities:

∑_{n=1}^{∞} |f_{n}u_{n}(x)| = ∑_{n=1}^{n_{0}} |f_{n}u_{n}(x)| + ∑_{n=n_{0}+1}^{∞} |f_{n}u_{n}(x)| ≤ n_{0}C_{0}^{2} · ∥f∥_{L_{1}(G)}

+ C_{0}D_{1} · ∑_{n=n_{0}+1}^{∞}[\frac{1}{n}(|a_{n}(g_{0c} + g_{0s})| + |b_{n}(g_{0c} + g_{0s})|)]

+ C_{0}D_{2} · ∑_{n=n_{0}+1}^{∞}[\frac{1}{n^{β}} (|a_{n}(g_{1c} + g_{1s})| + |b_{n}(g_{1c} + g_{1s})|)] + ∑_{n=n_{0}+1}^{∞} \frac{C_{0}D_{3}}{n^{2}}

≤ E_{1} + E_{2}(∑_{n=1}^{∞} \frac{1}{n^{p}} )^{1/p}(∑_{k=0}^{∞}(|a_{k}(·)|^{r} + |b_{k}(·)|^{r}))^{1/r}

+ E_{3}(∑_{n=1}^{∞} \frac{1}{n^{βp}} )^{1/p}(∑_{k=0}^{∞}(|a_{k}(··)|^{r} + |b_{k}(··)|^{r}))^{1/r} + E_{4} · ∑_{n=1}^{∞} \frac{1}{n^{2}}

≤ E_{1} + E_{2}(∑_{n=1}^{∞} \frac{1}{n^{p}} )^{1/p} · ∥g_{0c} + g_{0s}∥_{L_{p}(G)}

+ E_{3}(∑_{n=1}^{∞} \frac{1}{n^{βp}} )^{1/p} · ∥g_{1c} + g_{1s}∥_{L_{p}(G)} + E_{4} · ∑_{n=1}^{∞} \frac{1}{n^{2}} ,

the constants E_{1}–E_{4} having an obvious meaning. Estimates (12),(22) and inequalities of Hölder and Riesz are used.

The equality (4) follows from the completeness of the system {u_{n}(x)}_{n=1}^{∞} in L_{2}(G) and from the continuity of f(x) on \overline{G}.

Proof of estimate (5). Let μ > n_{0} and x ∈ [−1, 1]. Using equality (4), estimates (12) and (22), and the Hölder inequality, we obtain the relations

|f(x) − σ_{μ}(x, f)| = | ∑_{n=[μ]+1}^{∞} f_{n}u_{n}(x)|

≤ E_{2}( ∑_{n=[μ]+1}^{∞} \frac{1}{n^{p}} )^{1/p}( ∑_{n=[μ]+1}^{∞}(|a_{n}(g_{0c} + g_{1c})|^{r} + |b_{n}(g_{0c} + g_{1s})|^{r}))^{1/r}

+ E_{3}( ∑_{n=[μ]+1}^{∞} \frac{1}{n^{βp}} )^{1/p}( ∑_{n=[μ]+1}^{∞}(|a_{n}(g_{1c} + g_{1s})|^{r} + |b_{n}(g_{1c} + g_{1s})|^{r}))^{1/r}

+ E_{4} ∑_{n=[μ]+1}^{∞} \frac{1}{n^{2}} ≤ E_{2}(∫_{[μ]}^{+∞} \frac{dt}{t^{p}})^{1/p} · α_{1,r}(μ) + E_{3}(∫_{[μ]}^{+∞} \frac{dt}{t^{βp}} )^{1/p} · α_{2,r}(μ)

+ E_{4} ∫_{[μ]}^{+∞} \frac{dt}{t^{2}} ≤ \frac{1}{μ^{1−1/p}} · (E_{5} · α_{1,r}(μ) + \frac{E_{6}}{μ^{β−1}} · α_{2,r}(μ) + \frac{2}{μ^{1/p}} ),

where functions α_{1,r}(μ) and α_{2,r}(μ) are defined in the following way:

α_{1,r}(μ) = ( ∑_{n=[μ]+1}^{∞}(|a_{n}(g_{0c} + g_{0s})|^{r} + |b_{n}(g_{0c} + g_{0s})|^{r}))^{1/r},

α_{2,r}(μ) = ( ∑_{n=[μ]+1}^{∞}(|a_{n}(g_{1c} + g_{1s})|^{r} + |b_{n}(g_{1c} + g_{1s})|^{r}))^{1/r}.

The Riesz inequality gives lim_{μ→+∞} α_{1,r}(μ) = 0 = lim_{μ→+∞} α_{2,r}(μ). Therefore, estimate (5) is a consequence of relations mentioned above.

Proof of Theorem 1 is completed.

5. On Remark 2

In the second part of Remark 2 we have stated a proposition essentially related to Theorem 2. Proof of the proposition “contains” a proof of the theorem, and it is not based on asymptotic relations (16)–(18).

Let f(x) ∈ W_{1}^{(2)}(G) and L(f)(x) ∈ L_{p}(G) (1 < p ≤ 2). Using differential equation (3) and the integration by parts twice, we can write

f_{n} = \frac{1}{λ_{n}} · ∫_{a}^{b} f(x)L(u_{n})(x) dx

= \frac{1}{λ_{n}} (−f(x)u′_{n}(x)|_{a}^{b} + f′(x)u_{n}(x)|_{a}^{b} + ∫_{a}^{b} L(f)(x)u_{n}(x) dx).

By virtue of f^{(j)}(a) = 0 = f^{(j)}(b) (j = 0, 1), the double replacements vanish, and we get the basic relation

f_{n} = \frac{1}{λ_{n}} · ∫_{a}^{b} L(f)(x)u_{n}(x) dx ≡ \frac{1}{λ_{n}} · L(f)_{n},  (24)

playing in the following a role analogous to the one of estimate (22).

Let x ∈ \overline{G}. Using estimates (12)–(14), equality (24), and the Riesz inequality, we obtain

∑_{n=1}^{∞} |f_{n}u_{n}^{(j)}(x)| = ∑_{0<\sqrt{λ_{n}}≤1}(·) + ∑_{\sqrt{λ_{n}}>1}(·) ≤ AC_{0}C_{j}∥f∥_{L_{1}(G)}

+ C_{j}( ∑_{\sqrt{λ_{n}}>1} |L(f)_{n}|^{r})^{1/r}[∑_{k=1}^{∞}( ∑_{k≤\sqrt{λ_{n}}<k+1} \frac{1}{λ_{n}^{p(1−j/2)}} )]^{1/p}

≤ E_{7} + A^{1/p}C_{0}^{(2/p−1)}C_{j}∥L(f)∥_{L_{p}(G)} · (∑_{k=1}^{∞} \frac{1}{k^{p(2−j)}} )^{1/p},

where j = 0, 1 and 1/p + 1/r = 1. Therefore, it follows that series (6) converge absolutely and uniformly on \overline{G}.

Proofs of equalities (6) are the standard ones. These equalities allow us to show that the estimates

|f^{(j)}(x) − σ_{μ}^{(j)}(x, f)| ≤ A^{1/p}C_{j}( ∑_{n=[μ]+1}^{∞} |L(f)_{n}|^{r})^{1/r}( ∑_{n=[μ]+1}^{∞} \frac{1}{n^{p(2−j)}} )^{1/p}

hold on \overline{G}, wherefrom the estimates (10) easily follow.

Let us suppose now that f(x) ∈ D(L). Then L(f)(x) ∈ L_{2}(G), and equalities (24) are valid. The double replacements still vanish, as a consequence of the selfadjointness of the operator L. So, we see that the propositions of theorem 2 follow from the previous arguments (case p = 2).

6. Proof of Theorem 3

This proof is completely analogous to the one of Theorem 1; we suppose that G = (−1, 1), and use relations (16)–(18).

Estimate for the Fourier coefficients. The starting point is equality (24), which is obviously valid in the case considered. So, using equation (3), the integration by parts, and assumption L(f)(−1) = 0 = L(f)(1), we obtain the equalities

\begin{aligned}f_{n} & = \frac{1}{λ_{n}^{2}} · ∫_{−1}^{1} L(f)(x)L(u_{n})(x) dx \\ & = \frac{1}{λ_{n}^{2}} (∫_{−1}^{1} L(f)′(x)u′_{n}(x) dx + ∫_{−1}^{1} L(f)(x)q(x)u_{n}(x) dx). \end{aligned}  (25)

We need a suitable estimate for the integral ∫_{−1}^{1} L(f)′(x)u′_{n}(x)dx, n > n_{0}. Repeating, step by step, the arguments used in the first part of the proof of Theorem 1, we get the estimate

|∫_{−1}^{1} L(f)′(x)u′_{n}(x) dx| ≤ \sqrt{cλ_{n}}[|a_{n}(h_{0c} + h_{0s})| + |b_{n}(h_{0c} + h_{0s})|

+ \frac{ρ}{n^{β−1}} (|a_{n}(h_{1c} +h_{1s})|+|b_{n}(h_{1c} +h_{1s})|)+ \frac{4ρ^{2}∥L(f)′∥_{L_{1}(G)}}{n^{2(β−1)}} ]+2\sqrt{c}D∥L(f)′∥_{L_{1}(G)}, (26)

with functions h_{jc}(x) and h_{js}(x) defined as follows:

h_{jc}(x) = L(f)′(x)x^{j} cos γx, h_{js}(x) = L(f)′(x)x^{j} sin γx, j = 0, 1.

From (25)–(26) we obtain, having in mind estimate (12) and equality (18), the necessary estimate for f_{n}:

|f_{n}| ≤ \frac{D_{4}}{n^{3}} · (|a_{n}(h_{0c} + h_{0s})| + |b_{n}(h_{0c} + h_{0s})|)

+ \frac{D_{5}}{n^{β+2}} · (|a_{n}(h_{1c} + h_{1s})| + |b_{n}(h_{1c} + h_{1s})|) + \frac{D_{6}}{n^{4}} , n > n_{0}, (27)

where the constants D_{j} have the values:

D_{4} =^{def} \frac{2\sqrt{c}}{π^{3}}, D_{5} =^{def} \frac{ρ\sqrt{c}}{π^{3}},

D_{6} =^{def} (\frac{4ρ^{2}\sqrt{c}}{π^{3}n_{0}^{2β−3}} + \frac{2\sqrt{c}D}{π^{4}} ) · ∥L(f)′∥_{L_{1}(G)} + \frac{2C_{0}}{π^{4}} · ∥L(f)q∥_{L_{1}(G)}. (28)

Proof of equalities (8). First we prove that series (8) converge absolutely and uniformly on [−1, 1].

Having in mind estimates (12), (14), (15), (27), the equality (18), and inequalities of Hölder and Riesz, we obtain the following formal chain:

∑_{n=1}^{∞} |f_{n}u_{n}^{(j)}(x)| = ∑_{n=1}^{n_{0}}(·) + ∑_{n=n_{0}+1}^{∞}(·) ≤ n_{0}C_{0}C_{j}λ_{n_{0}}^{j/2} · ∥f∥_{L_{1}(G)}

+ (2π)^{j}C_{j}D_{4} · ∑_{n=n_{0}+1}^{∞} \frac{1}{n^{3−j}} · (|a_{n}(h_{0c} + h_{0s})| + |b_{n}(h_{0c} + h_{0s})|)

+ (2π)^{j}C_{j}D_{5} · ∑_{n=n_{0}+1}^{∞} \frac{1}{n^{β+2−j}} · (|a_{n}(h_{1c} + h_{1s})| + |b_{n}(h_{1c} + h_{1s})|)

+ (2π)^{j}C_{j}D_{6} · ∑_{n=n_{0}+1}^{∞} \frac{1}{n^{4−j}}

≤ E_{8} + E_{9}(∑_{n=1}^{∞} \frac{1}{n^{(3−j)p}} )^{1/p} · ∥h_{0c} + h_{0s}∥_{L_{p}(G)}

+ E_{10}(∑_{n=1}^{∞} \frac{1}{n^{(β+2−j)p}} )^{1/p} · ∥h_{1c} + h_{1s}∥_{L_{p}(G)} + E_{11} · ∑_{n=1}^{∞} \frac{1}{n^{4−j}} .

The numerical series on the right-hand side of the last inequality being convergent for j = 0, 1, 2, the convergence of series (8) holds as it has been claimed.

In the proof of equality (8) (case j = 0) one have to use, in a standard way, the continuity of f(x) and the completeness of the system {u_{n}(x)}_{n=1}^{∞}. The classical theorem on differentiability of the sum of a functional series helps in proving equality (8) in the cases j = 1, 2.

Proof of estimates (9). Let μ > n_{0} and x ∈ [−1, 1]. Starting from equalities (8), we can obtain, as in the preceding part of the proof, the inequalities

|f^{(j)}(x) − σ_{μ}^{(j)}(x, f)| ≤ ∑_{n=[μ]+1}^{∞} |f_{n}u_{n}^{(j)}(x)|

≤ E_{9}( ∑_{n=[μ]+1}^{∞} \frac{1}{n^{(3−j)p}} )^{1/p} · α_{3,r}(μ)

+ E_{10}( ∑_{n=[μ]+1}^{∞} \frac{1}{n^{(β+2−j)p}} )^{1/p} · α_{4,r}(μ) + E_{11} · ∑_{n=[μ]+1}^{∞} \frac{1}{n^{4−j}}

≤ \frac{1}{μ^{3−j−1/p}} (E_{12} · α_{3,r}(μ) + \frac{E_{13}}{μ^{β−1}} · α_{4,r}(μ) + \frac{E_{14}}{μ^{1/p}} ), (29)

where constants E_{j} have obvious meanings, and

α_{3,r}(μ) =^{def} ( ∑_{n=[μ]+1}^{∞}(|a_{n}(h_{0c} + h_{0s})|^{r} + |b_{n}(h_{0c} + h_{0s})|^{r}))^{1/r},

α_{4,r}(μ) =^{def} ( ∑_{n=[μ]+1}^{∞}(|a_{n}(h_{1c} + h_{1s})|^{r} + |b_{n}(h_{1c} + h_{1s})|^{r}))^{1/r}.

By virtue of lim_{μ→+∞} α_{3,r}(μ) = 0 = lim_{μ→+∞} α_{4,r}(μ), estimates (9) follow from relations (29).

Theorem 3 is proved.

7. On Remark 4

If the coefficients in boundary conditions (2) satisfy (11), then equations (2) can be solved with respect to the “variables” u′_{n}(a) and u′_{n}(b):

u′_{n}(a) = R_{1a}(α_{ij}, β_{ij})u_{n}(a) + R_{1b}(α_{ij}, β_{ij})u_{n}(b),
u′_{n}(b) = R_{2a}(α_{ij}, β_{ij})u_{n}(a) + R_{2b}(α_{ij}, β_{ij})u_{n}(b),

where constants R do not depend on n. Let us suppose that f(a) ≠ 0 or/and f(b) ≠ 0. Then we have, because of the above equalities and estimate (12), the inequality

|f(a)u′_{n}(b) − f(b)u′_{n}(b)| ≤ 2C_{0}R_{0}(|f(a)| + |f(b)|),  (30)

where constant R_{0} is independent of n.

Return now to the equality (19). In the general case considered, that equality must have the following form:

f_{n} = \frac{1}{λ_{n}} · ∫_{a}^{b} f(x)L(u_{n})(x) dx = \frac{1}{λ_{n}} [−f(a)u′_{n}(a) + f(b)u′_{n}(b)]

+ \frac{1}{λ_{n}} (∫_{a}^{b} f′(x)u′_{n}(x) dx + ∫_{a}^{b} f(x)q(x)u_{n}(x)dx). (31)

By virtue of estimate (30), the first member on the right-hand side of the second equality (31) has a “satisfactory” order with respect to λ_{n}. That is why the basic estimate (22) is still valid in this case, the constant D_{3} being changed in the corresponding way.

Analogous arguments work in the cases L(f)(a) ≠ 0 or/and L(f)(b) ≠ 0. If one of boundary conditions (2) has the form α_{j1}u′(a) + β_{j1}u′(b) = 0 and

f(x) satisfies β_{j1}f(a) + α_{j1}f(b) = 0, then the double replacements in (31) vanish, and we get estimate (22) again. The corresponding remark holds in the case of function L(f)(x) from Theorem 3.

Note that theorems 1 and 3 were reported at the Seminar for Functional Analysis and Operator Theory (Faculty of Mathematics, Belgrade) on March 21st, 2002.

Acknowledgment. We thank the referee for valuable suggestions.

Added in proof. Recently, V. M. Kurbanov and R. A. Safarov [19] have obtained, by a different approach, results stated in Theorem 1, with O(μ^{1/p−1}) standing in the estimate (5) (instead of o(μ^{1/p−1}).

REFERENCES

[1] M. A. Naímark, Lineínye differentsial’nye operatory, Nauka, Moskva, 1969.

[2] V. A. Il’in, Neobkhodimye i dostatochnye usloviya bazisnosti i ravnoskhodimosti s trigonometricheskim ryadom spektral’nykh razlozhenií. I, Diff. uravneniya 16, 5 (1980), 771–794.

[3] V. A. Il’in, Neobkhodimye i dostatochnye usloviya bazisnosti i ravnoskhodimosti s trigonometricheskim ryadom spektral’nykh razlozhenií. II, Diff. uravneniya 16, 6 (1980), 980–1009.

[4] V. A. Il’in, Neobkhodimye i dostatochnye usloviya bazisnosti v L_{p} i ravnoskhodimosti s trigonometricheskim ryadom spektral’nykh razlozhenií i razlozhenií po sistemam eksponent, Dokl.AN SSSR 273, 4 (1983), 789–793.

[5] V. A. Il’in, Ravnoskhodimost’ s trigonometricheskim ryadom razlozhenií po kornevym funktsiyam odnomernogo operatora Shredingera s kompleksnym potentsialom iz klasa L_{1}, Diff. uravneniya 27, 4 (1991), 577–597.

[6] I. S. Lomov, Conditions for convergence of biorthogonal expansions of functions on a closed interval, Diff. equations 37, 4 (2001), 562–565.

[7] I. S. Lomov, The local convergence of biorthogonal series related to differential operators with nonsmooth coefficients. I, Diff. equations 37, 3 (2001), 328–342.

[8] I. S. Lomov, The local convergence of biorthogonal series related to differential operators with nonsmooth coefficients. II, Diff. equations 37, 5 (2001), 648–660.

[9] I. S. Lomov, Ravnomernaya skhodimost’ biortogonal’nogo ryada dlya operatora Shredingera s mnogozhnachnymi kraevymi usloviyami, Diff. uravneniya 38, 7 (2002), 890–896.

[10] A. M. Minkin, Teorema ravnoskhodomosti dlya razlozheniyi po oboshchennoyi spektral’noyi funktsii simmetricheskogo differentsial’nogo operatora i v integral Fur’e, Funktsional’niyi analiz. Spektral’naya teoriya, t. 14, 1980, 109–112.

[11] A. M. Minkin, Almost orthogonality of Birkhoff’s solutions, Results in Math., 24, 3/4 (1999), 280–287.

[12] A. M. Minkin, Equiconvergence theorems for differential operators, Journal of Math. Sci. 96, 6 (1999), 3631–3715.

[13] N. L. Lažetić, On uniform convergence of spectral expansions arising by self-adjoint extensions of an one-dimensional Schrödinger operator, Publ. de l’Inst. Math. (Beograd) 69 (83) (2001), 59–77.

[14] N. L. Lazhetich, O sushchestvovanii klassicheskogo resheniya smeshannoí zadachi dlya odnomernogo giperbolicheskogo uravneniya vtorogo poryadka(II), Publ. de l’Inst. Math. (Beograd) 64(78) (1998), 69–84.

[15] V. A. Il’in, I. Ío, Ravnomernaya otsenka sobstvennykh funktsií i otsenka sverkhu chisla sobstvennykh znachenií odnomernogo operatora Shturma-Liuvillya s potentsialom iz klassa L_{p}, Diff. uravneniya 15, 7 (1979), 1165–1174.

[16] L. V. Krickov, Ravnomernaya otsenka poryadka prisoedinennykh funktsií i raspredelenie sobstvennykh znachenií odnomernogo operatora Shredingera, Diff. uravneniya 25, 7 (1989), 1121– 1129.

[17] N. Lazhetich, Ravnomernye otsenki dlya proizvodnykh sobstvennykh funktsií samosopryazhennogo operatora Shturma-Liuvillya, Diff. uravneniya 17, 11 (1981), 1978–1983.

[18] A. Zygmund, Trigonometrical Series, Volume I and II, Cambridge, 1968.

[19] V. M. Kurbanov, R. A. Safarov, On uniform convergence of orthogonal expansions in eigenfunctions of Sturm-Liouville operator, Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 24, 1 (2004), Math. Mech., 161–168.

(received 21.05.2003, in revised form 02.11.2004)

Faculty of Mathematics, Studentski trg 16, P.O. Box 550, 11000 Beograd, Serbia

E-mail: lazetic@matf.bg.ac.yu