PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 101(115) (2017), 169–182 https://doi.org/10.2298/PIM1715169K

ABSOLUTE AND UNIFORM CONVERGENCE OF SPECTRAL EXPANSION OF THE FUNCTION FROM THE CLASS W_{p}^{1}(G) , p > 1 , IN EIGENFUNCTIONS OF THIRD ORDER DIFFERENTIAL OPERATOR

V. M. Kurbanov and E. B. Akhundova

Abstract. We study an ordinary differential operator of third order and absolute and uniform convergence of spectral expansion of the function from the class W_{p}^{1}(G), G = (0, 1), p > 1, in eigenfunctions of the operator. Uniform convergence rate of this expansion is estimated.

1. Basic notion and formulation of results

Consider on the interval G = (0, 1) a formal differential operator

Lu = u^{(3)} + p_{1}(x)u^{(2)} + p_{2}(x)u^{(1)} + p_{3}(x)u

with summable complex valued coefficients p_{1}(x) ∈ L_{2}(G), p_{l}(x) ∈ L_{1}(G), l = 2, 3.

Denote by D(G) a class of functions absolutely continuous together with own derivatives to second order, inclusively on the closed interval Ḡ = [0, 1]. Following [1], under the eigenfunction of the operator L responding to the eigenvalue λ, we understand any not identically equal to zero function u(x) ∈ D(G) satisfying almost everywhere in G the equation Lu + λu = 0.

Let {u_{n}(x)}_{n=1}^{∞} be a complete orthonormalized in L_{2}(G) system consisting of eigenfunctions of the operator L, and {λ_{n}}_{n=1}^{∞} be an appropriate system of eigenvalues, moreover Reλ_{n} = 0 (it is supposed that the coefficients of the operator L admit the existence of such a system {u_{n}(x)}_{n=1}^{∞} ). Under rather smooth coefficients the existence of such systems follows from the monograph [2])

Denote by μ_{n} the number (∓iλ_{n})^{1/3} for ± Imλ_{n} ⩾ 0 . We will say that the function f(x) belongs to W_{p}^{1}(G), 1 ⩽ p ⩽ ∞ if f(x) is absolutely continuous on Ḡ and f′(x) ∈ L_{p}(G).

Introduce a partial sum of spectral expansion of the function f(x) ∈ W_{p}^{1}(G) in the system {u_{n}(x)}_{n=1}^{∞} by σ_{ν}(x, f) = ∑ _{μ_{n} ⩽ν} f_{n}u_{n}(x), ν > 0, where f_{n} = (f, u_{n}) = ∫_{0}^{1} f(x)\overline{u_{n}(x)}dx. Denote R_{ν} (x, f) = f(x) − σ_{ν}(x, f). We prove the following theorems.

2010 Mathematics Subject Classification: 34L10; 42A20.

Key words and phrases: absolute convergence; uniform convergence; eigenfunctions; spectral expansion.

Communicated by Gradimir Milovanović.

Theorem 1.1. Let p_{1}(x) ≡ 0, p_{l}(x) ∈ L_{1}(G), l = 2, 3; f(x) ∈ W_{p}^{1}(G), p > 1,

and the following condition be fulfilled

(1.1)  |f(x)\overline{u_{n}^{(2)}(x)}|_{0}^{1} | ⩽ C(f)μ_{n}^{α}∥u_{n}∥_{∞} , 0 ⩽ α < 2, μ_{n} ⩾ 1,

where C(f) > 0 is a constant dependent on the function f(x). Then the spectral expansion of the function f(x) in the system {u_{n}(x)}_{n=1}^{∞} converges absolutely and uniformly on Ḡ = [0, 1], and we have the estimation

(1.2) sup_{x∈Ḡ} |R_{ν}(x, f)|

⩽ const{ C(f)ν^{α−2} + ν^{−β} ∥f′∥_{p} + ν^{−1} (∥f∥_{∞} + ∥f′∥_{1}) ∑_{r=2}^{3} ν^{2−r} ∥p_{r}∥_{1}} ,

where β = min\{ \frac{1}{2}, \frac{1}{q}\}, \frac{1}{p} + \frac{1}{q} = 1, ν ⩾ 2, const is independent of f(x), and ∥ · ∥_{p} = ∥ · ∥_{L_{p} (G)} .

Corollary 1.1. If in Theorem 1.1, the function f(x) satisfies the condition f(0) = f(1) = 0, then obviously (1.1) is fulfilled and ve have the estimation

sup_{x∈Ḡ} |R_{ν}(x, f)| ⩽ const ν^{−β} ∥f′∥_{p}, ν ⩾ 2;

if C(f) = 0 or 0 ⩽ α < 2 − β, then sup_{x∈Ḡ} |R_{ν}(x, f)| = o(ν^{−β} ), ν → +∞.

Theorem 1.2. Let p_{1}(x) ∈ L_{2}(G), p_{l}(x) ∈ L_{1}(G), l = 2, 3; f(x) ∈ W_{2}^{1}(G)

and condition (1.1) be fulfilled. Then the spectral expansion of the function f(x) in system {u_{n}(x)}_{n=1}^{∞} converges absolutely and uniformly on Ḡ = [0, 1], and we have the estimation

sup_{x∈Ḡ} |R_{ν}(x, f)| ⩽ const\{ C(f)ν^{α−2} + ν^{− \frac{1}{2}} (∥f \overline{p_{1}}∥_{2} + ∥f′∥_{2})

+ ν^{−1} ∥f∥_{∞} ∑_{r=2}^{3} ν^{2−r} ∥p_{r}∥_{1}} , ν ⩾ 2.

Corollary 1.2. If in Theorem 1.2, C(f) = 0 or 0 ⩽ α < 3/2, then

sup_{x∈Ḡ} |R_{ν}(x, f)| = o(ν^{− \frac{1}{2}} ), ν → +∞.

Theorem 1.3. Let p_{1}(x) ∈ L_{2}(G), p_{l}(x) ∈ L_{1}(G), l = 2, 3; f(x) ∈ W_{p}^{1}(G),

1 < p < 2, condition (1.1) be fulfilled, and the system {u_{n}(x)}_{n=1}^{∞} be uniformly bounded. Then the spectral expansion of the function f(x) in system {u_{n}(x)}_{n=1}^{∞} converges absolutely and uniformly on Ḡ, and we have the estimation

sup_{x∈Ḡ} |R_{ν}(x, f)| ⩽ const\{ C(f)ν^{α−2} + ν^{− \frac{1}{2}} ∥f\overline{p_{1}}∥_{2}

+ ν^{− \frac{1}{q}} ∥f′∥_{p} + ν^{−1} ∥f∥_{∞} ∑_{r=2}^{3} ν^{2−r} ∥p_{r}∥_{1}\} , μ ⩾ 2,

where p^{−1} + q^{−1} = 1.

Corollary 1.3. If in Theorem 1.3, C(f) = 0 or 0 ⩽ α < 2 − q^{−1} , then

sup_{x∈Ḡ} |R_{ν}(x, f)| = o(ν^{− \frac{1}{q}} ), ν → +∞.

Note that such results for the Schrodinger operator L_{1} = −\frac{d^{2}}{dx^{2}} + q(x) were obtained in [3–7]. In the case f(x) ∈ W_{1}^{1}(G) the absolute and uniform convergence of spectral expansion in eigenfunctions of a third order operator was studied in [8].

2. Some auxiliary statements

To prove, the formulated results it is necessary to estimate the Fourier coefficients of the function f(x) in the system {u_{n}(x)}_{n=1}^{∞} . To this end we established the representation for the eigenfunction u_{n}(x). Introduce the notation

K(z) = ∑_{j=1}^{3} ω_{j} exp(iω_{j} (sgnImλ_{n})z),

M(u_{n}(ξ)) = \frac{1}{3}μ_{n}^{−2} ∑_{r=1}^{3} p_{r}(ξ)u_{n}^{(3−r)} (ξ),

X_{j}^{±} (x) = \frac{1}{3}μ_{n}^{−2} ∑_{m=0}^{2} (±iμ_{n})^{m}ω_{j}^{m+1} u_{n}^{(2−m)} (x),

where ω_{1} = −1, ω_{2} = exp(−iπ/3), ω_{3} = exp(iπ/3).

Lemma 2.1. For the eigenfunction u_{n}(x), the following representations are valid (λ_{n} ≠ 0, l = \overline{0, 2})

(2.1)  u_{n}^{(l)}(x + t) = ∑_{j=1}^{3} (−iω_{j} μ_{n})^{l}X_{j}^{−} (x) exp(−iω_{j} μ_{n}t)

− ∫_{x}^{x+t} M(u_{n}(ξ))K_{t}^{(l)}(ξ − x − t)dξ, if Imλ_{k} > 0;

(2.2)  u_{n}^{(l)}(x + t) = ∑_{j=1}^{3} (iω_{j} μ_{n})^{l}X_{j}^{+} (x) exp(iω_{j} μ_{n}t)

− ∫_{x}^{x+t} M(u_{n}(ξ))K_{t}^{(l)}(ξ − x − t)dξ, if Imλ_{n} < 0.

Proof. For definiteness we consider the case Imλ_{n} < 0. Multiply each side of the equation Lu_{n}(ξ) + λ_{n}u_{n}(ξ) = 0 by the function K_{t}^{(l)}(ξ − x − t) and integrate the obtained equality with respect to ξ from x to x + t, where x, x + t ∈ G

∑_{j=1}^{3} ω_{j} (iω_{j} μ_{n})^{l} ∫_{x}^{x+t} u_{n}^{(3)}(ξ) exp(−iω_{j} μ_{n}(ξ − x − t))dξ (2.3)

+ ∫_{x}^{x+t} {p_{1}(ξ)u_{n}^{(2)}(ξ) + p_{2}(ξ)u_{n}^{(1)}(ξ) + p_{3}(ξ)u_{n}(ξ)}
× K_{t}^{(l)}(ξ − x − t)dξ + λ_{n} ∫_{x}^{x+t} u_{n}(ξ)K_{t}^{(l)}(ξ − x − t)dξ = 0.

Integrating by parts and using ∑ _{j=1}^{3} ω_{j}^{s} = 3δ_{3s} (δ_{ks} is the Kronecker symbol), we transform the first expression in equality (2.3) in the following way

∑_{j=1}^{3} ω_{j} (iω_{j} μ_{n})^{l} ∫_{x}^{x+t} u_{n}^{(3)}(ξ) exp(−iω_{j} μ_{n}(ξ − x − t))dξ

= ∑_{j=1}^{3} ω_{j} (iω_{j} μ_{n})^{l} ∑_{m=0}^{2} (−1)^{m}(−iω_{j} μ_{n})^{m}

× [u_{n}^{(2−m)} (x + t) − u_{n}^{(2−m)} (x) exp(iω_{j} μ_{n}t)]

− ∑_{j=1}^{3} ω_{j} (iω_{j} μ_{n})^{l}(−ω_{j} μ_{n})^{3} ∫_{x}^{x+t} u_{n}(ξ) exp(−iω_{j} μ_{n}(ξ − x − t))dξ

= 3μ_{n}^{2}u_{n}^{(l)}(x + t)

− ∑_{j=1}^{3} ω_{j} (iω_{j} μ_{n})^{l} ∑_{m=0}^{2} (iω_{j} μ_{n})^{m}u_{n}^{(2−m)} (x) exp(iω_{j} μ_{n}t) + (−iμ_{n})^{3}

× ∑_{j=1}^{2} ω_{j} (iω_{j} μ_{n})^{l} ∫_{x}^{x+t} u_{n}(ξ) exp(−iω_{j} μ_{n}(ξ − x − t)dξ.

Taking this into account in (2.3) and taking into attention the equality (−iμ_{n})^{3} = −λ_{n}, we get

3μ_{n}^{2}u_{n}^{(l)}(x + t) − ∑_{j=1}^{3} (iω_{j} μ_{n})^{l}[ ∑_{m=0}^{2} ω_{j}^{m+1} (iμ_{n})^{m}u_{n}^{(2−m)} (x)] exp(iω_{j} μ_{n}t)

+ ∫_{x}^{x+t} { ∑_{τ=1}^{3} p_{r}(ξ)u_{n}^{(3−r)} (ξ)} K_{t}^{(l)}(ξ − x − t)dξ = 0.

Hence we find u_{n}^{(l)}(x + t) and get formula (2.2). Lemma 2.1 is proved. □

For x = 0 we write formulas (2.1) and (2.2) in a more convenient form

(2.4)  μ_{n}^{−l} u_{n}^{(l)}(t) = ∑_{j=1}^{2} X_{j}^{−} (0)(−iω_{j})^{l} exp(−iω_{j} μ_{n}t)

− (−iω_{3})B_{3}^{−} exp(iω_{3}μ_{n}(1 − t))

− ∑_{j=1}^{2} (−i)^{l}ω_{j}^{l+1} ∫_{0}^{t} M(u_{n}(ξ)) exp(iω_{j} μ_{n}(ξ − t))dξ

+ (−i)^{l}ω_{3}^{l+1} ∫_{t}^{1} M(u_{n}(ξ)) exp(iω_{3}μ_{n}(ξ − t))dξ,

Imλ_{n} > 0, l = \overline{0, 2}, where

B_{3}^{−} = X_{3}^{−} (0) exp(−iω_{3}μ_{n}) − ω_{3} ∫_{0}^{1} M(u_{n}(ξ)) exp(−iω_{3}μ_{n}(ξ − 1))dξ;

(2.5)  μ_{n}^{−l} u_{n}^{(l)}(t) = ∑_{j=1,j≠2}^{2} (iω_{j})^{l}X_{j}^{+} (0) exp(iω_{j} μ_{n}t)

+ (iω_{2})^{l}B_{2}^{+} exp(−iω_{2}μ_{n}(1 − t))

− ∑_{j=1,j≠2}^{3} (i)^{l}ω_{j}^{l+1} ∫_{0}^{t} M(u_{n}(ξ)) exp(−iω_{j} μ_{n}(ξ − t))dξ

+ (i)^{l}ω_{3}^{l+1} ∫_{t}^{1} M(u_{n}(ξ)) exp(−iω_{2}μ_{n}(ξ − t))dξ,

where Imλ_{n} < 0, l = \overline{0, 2}

B_{2}^{+} = X_{2}^{+} (0) exp(iω_{2}μ_{n}) − ω_{2} ∫_{0}^{1} M(u_{n}(ξ)) exp(−iω_{2}μ_{n}(ξ − t))dξ.

For the coefficients in formulas (2.4) and (2.5) we note that if the system {u_{n}(x)}_{n=1}^{∞} is a Bessel system in L_{2}(G), then for, them the following estimations are fulfilled (see [9, 10])

(2.6)  \begin{matrix}|X_{1}^{±} (0)| & ⩽ ∥u_{n}∥_{2}, & |X_{2}^{±} (0)| & ⩽ C∥u_{n}∥_{∞} \\ |B_{2}^{+} | & ⩽ C∥u_{n}∥_{∞} , & |B_{3}^{−} | & ⩽ C∥u_{n}∥_{∞} . \end{matrix}

Now, using formulas (2.4) and (2.5), estimate the Fourier coefficients of the function f(x) ∈ W_{p}^{1}(G), p > 1, in system {u_{n}(x)}_{n=1}^{∞} .

Lemma 2.2. Let the function f(x) ∈ W_{p}^{1}(G), p > 1, and the system {u_{n}(x)}_{n=1}^{∞}

satisfy condition (1.1). Then for the Fourier coefficients f_{n} the estimations (μ_{n} ⩾ 1) are valid

(2.7) |f_{n}| ⩽ const\{ C(f)μ_{n}^{α−3} ∥u_{n}∥_{∞} + μ_{n}^{−1} |(f \overline{p_{1}}, μ_{n}^{−2} u_{n}^{(2)})|

+ μ_{n}^{−1} |(f′, μ_{n}^{−2} u_{n}^{(2)})| + μ_{n}^{−2} ( ∑_{r=2}^{3} μ_{n}^{2−r} ∥p_{r}∥_{1}) ∥f∥_{∞} ∥u∥_{∞} } ;

(2.8) |f_{n}| ⩽ const{ [C(f)μ_{n}^{α−3} + μ_{n}^{−1} |(f′, exp(iω_{3}μ_{n}t))|

+ μ_{n}^{−1} |(f′, exp(−iω_{2}μ_{n}(1 − t)))| + (∥f∥_{∞} + ∥f′∥_{1})μ_{n}^{−2} ∑_{r=2}^{3} μ_{n}^{2−r} ∥p_{r}∥_{1}]

× ∥u_{n}∥_{∞} + μ_{n}^{−1} |(f′, exp(−iμ_{n}t))|} , if p_{1}(x) ≡ 0, Imλ_{n} < 0;

(2.9) |f_{n}| ⩽ const{ [C(f)μ_{n}^{α−3} + μ_{n}^{−1} |(f′, exp(−iω_{2}μ_{n}t))|

+ μ_{n}^{−1} |(f′, exp(iω_{3}μ_{n}(1 − t)))| + (∥f∥_{∞} + ∥f′∥_{1})μ_{n}^{−2} ∑_{r=2}^{3} μ_{n}^{2−r} ∥p_{r}∥_{1}]

× ∥u_{n}∥_{∞} + μ_{n}^{−1} |(f′, exp(iμ_{n}t))|} ,

if p_{1}(x) ≡ 0, Imλ_{n} > 0.

Proof. By definition of the eigenfunction u_{n}(x), the Fourier coefficients f_{n} for μ_{n} ⩾ 1 are calculated by the formula

(2.10)  f_{n} = (f, u_{n}) = −\frac{1}{\overline{λ_{n}}} (f, Lu_{n})

= −\frac{1}{\overline{λ_{n}}} (f, u^{(3)}) − \frac{1}{\overline{λ_{n}}} (f, p_{1}u_{n}^{(2)}) − \frac{1}{\overline{λ_{n}}} (f, p_{2}u_{n}^{(1)}) − \frac{1}{\overline{λ_{n}}} (f, p_{3}u_{n}).

Applying the estimation (see [10])

(2.11)  ∥u_{n}^{(s)} ∥_{∞} ⩽ const(1 + μ_{m})^{s+ \frac{1}{p}} ∥u_{n}∥_{p}, p ⩾ 1, s = \overline{0, 2}

we find

(2.12)  \frac{1}{|λ_{n}|} |(f, p_{2}u_{n}^{(1)})| + \frac{1}{|λ_{n}|} |(f, p_{3}u_{n})|

⩽ const∥f∥_{∞} μ_{n}^{−2} ( ∑_{r=2}^{3} μ_{n}^{2−r} ∥p_{r}∥_{1}) ∥u_{n}∥_{∞} .

Making integration in parts in the first summand on the right-hand side of equality (2.10) and taking into account condition (1.1), we get

(2.13)  \frac{1}{|λ_{n}|} |(f, u_{n}^{(3)})| ⩽ C(f)μ_{n}^{α−3} ∥u_{n}∥_{∞} + μ_{n}^{−3} |(f′, u_{n}^{(2)})|.

From (2.10), (2.12) and (2.13) it follows estimation (2.7).

Estimate the expression μ_{n}^{−3} |(f′, u_{n}^{(2)})| in the case p_{1}(x) ≡ 0. For that we use formulas (2.4) and (2.5) depending on the sign of Imλ_{n}. For definiteness we consider the case Imλ_{n} < 0 and apply formula (2.5) for l = 2. Then by estimations (2.6), (2.11) and

|M(u_{n}(ξ))| ⩽ \frac{1}{3}μ_{n}^{−2} ∑_{r=2}^{3} |p_{r}(ξ)||u_{n}^{(3−r)} (ξ)|

⩽ constμ_{n}^{−1} ( ∑_{r=2}^{3} |p_{r}(ξ)|μ_{n}^{2−r} ) ∥u_{n}∥_{∞}

we get that

μ_{n}^{−3} |(f′, u_{n}^{(2)})| = μ_{n}^{−1} |(f′, μ_{n}^{−2} u_{n}^{(2)})|

⩽ μ_{n}^{−1} ∑_{j=1,j≠2}^{3} |X_{j}^{+} (0)||(f′, exp(iω_{j} μ_{n}t))|

+ μ_{n}^{−1} |B_{2}^{+} ||(f′, exp(−iω_{2}μ_{n}(1 − t)))|

+ μ_{n}^{−1} ∑_{j=1,j≠2}^{3} |( f′, ∫_{0}^{t} M(u_{n}(ξ)) exp(−iω_{j} μ_{n}(ξ − t)dξ)) |

+ μ_{n}^{−1} |( f′, ∫_{t}^{1} M(u_{n}(ξ)) exp(−iω_{2}μ_{n}(ξ − t)dξ)) |

⩽ constμ_{n}^{−1} {|(f′, exp(−iμ_{n}t))| + [|(f′, exp(iω_{3}μ_{n}t))|

+ |(f′, exp(−iω_{2}μ_{n}(1 − t)))|]∥u_{n}∥_{∞} }

+ constμ_{n}^{−2} ∥f′∥_{1}( ∑_{r=2}^{3} μ_{n}^{2−r} ∥p_{r}∥_{1}) ∥u_{n}∥_{∞} .

Thus, in the case p_{1}(x) ≡ 0, Imλ_{n} < 0 the following estimation is fulfilled

(2.14) μ_{n}^{−3} |(f′, u_{n}^{(3)})| ⩽ constμ_{n}^{−1} { |(f′, exp(−iμ_{n}t))| + [|(f′, exp(iω_{3}μ_{n}t))|

+ |(f′, exp(iω_{2}μ_{n}(t − 1)))| + ∥f′∥_{1}μ_{n}^{−1} ∑_{r=2}^{3} μ_{n}^{2−r} ∥p_{r}∥_{1}]∥u_{n}∥_{∞} } .

Consequently, estimation (2.8) follows from (2.10), (2.12)–(2.14). Estimation (2.9) for p_{1}(x) ≡ 0, Imλ_{n} > 0 is proved in the same way. The Lemma 2.2 is proved. □

Lemma 2.3 (see [10]). Let p_{1}(x) ∈ L_{2}(G), p_{l}(x) ∈ L_{1}(G), l = 2, 3 and {u_{n}(x)} be an orthonormalized in L_{2}(G) system of eigenfunctions of the operator L. Then the following estimations are fulfilled:

(2.15)  ∑_{τ⩽μ_{n} ⩽tau+1} 1 ⩽ const, for all τ ⩾ 0,

(2.16)  ∑_{0⩽μ_{n} ⩽τ} ∥u_{n}∥_{∞}^{2} ⩽ const(1 + τ), for all τ > 0.

Lemma 2.4 (see [9]). If conditions of Lemma 2.3 are fulfilled, then the system {μ_{n}^{−2} u_{n}^{(2)}(x)}, μ_{n} ⩾ 1, is a Bessel system, i.e., for any g(x) ∈ L_{2}(G) the following inequality is fulfilled

∑_{μ_{n} ⩾1} |(f, μ_{n}^{−2} u_{n}^{(2)}(x))|^{2} ⩽ const∥g∥_{2}^{2}.

Lemma 2.5. Under conditions of Lemma 2.3, the system {exp(−iμ_{n}t)} for Imλ_{n} < 0 and the system {exp(iμ_{n}t)} for Imλ_{n} > 0 satisfies the Riesz inequality for 1 < p ⩽ 2.

Proof. As these system satisfy the Bessel inequality in L_{2}(G) (see [11]) subject to condition (2.15), furthermore, for any g(x) ∈ L_{1}(G) it is valid

| ∫_{0}^{1} g(x)\overline{φ_{n}(x)}dx| ⩽ const∥g∥_{1}

where {φ_{n}(x)} is any from the above mentioned systems, then by the Riesz–Torin theorem (see [12]), for these systems the Riesz inequality is valid, i.e.,

∑_{n} | ∫_{0}^{1} g(x)\overline{φ_{n}(x)}dx|^{q} ⩽ const∥g∥_{p}^{q}

for any g(x) ∈ L_{p}(G), 1 < p ⩽ 2. Lemma 2.5 is proved. □

Lemma 2.6. Let the conditions of Lemma 2.3 be fulfilled. Then

(2.17)  ∑_{μ_{n} ⩾μ} \frac{∥u_{n}∥_{∞}^{2}}{μ_{n}^{θ+1}} ⩽ constμ^{−θ} , θ > 0, for all μ ⩾ 2.

Proof. By estimations (2.15), (2.16) and Abel transformation for any l ∈ N

∑_{μ⩽μ_{n} ⩽[μ]+l} \frac{∥u_{n}∥_{∞}^{2}}{μ_{n}^{1+θ}} ⩽ ∑_{k=[μ]}^{[μ]+l} \frac{1}{k^{1+θ}} ( ∑_{k⩽μ_{n} <k+1} ∥u_{n}∥_{∞}^{2} )

⩽ ∑_{k=[μ]}^{[μ]+l−1} ( ∑_{1⩽μ_{n} ⩽k+1} ∥u_{n}∥_{∞}^{2} ) ( \frac{1}{k^{1+θ}} − \frac{1}{(k + 1)^{1+θ}} )

+ ( ∑_{1⩽μ_{n} ⩽[μ]+l} ∥u_{n}∥_{∞}^{2} ) ([μ] + l)^{−(1+θ)} + ( ∑_{1⩽μ_{n} ⩽[μ]−1} ∥u_{n}∥_{∞}^{2} ) [μ]^{−(1+θ)}

⩽ const ∑_{k=[μ]}^{[μ]+l−1} (k + 1)\frac{(k + 1)^{θ}(1 + θ)}{(k(k + 1))^{1+θ}}

+ const([μ] + l)^{−θ} + const[μ]^{−θ} ⩽ const( ∑_{k=[μ]}^{∞} \frac{1 + θ}{k^{1+θ}} + [μ]^{−θ} ) .

Hence, by the arbitrariness of the natural number l we get estimation (2.17). □

Lemma 2.7. Let the conditions of Lemma 2.3 be fulfilled. Then

(2.18)  ∑_{μ_{n} ⩾μ} \frac{∥u_{n}∥_{∞}^{p}}{μ_{n}^{p}} ⩽ constμ^{1−p} , 1 < p ⩽ 2, for all μ ⩾ 2.

Proof. For p = 2 estimation (2.18) follows from (2.17) for θ = 1. Consider the case p ≠ 2 and apply the Holder inequality for p′ = \frac{2}{p}, q′ = \frac{2}{2−p} :

∑_{μ_{n} ⩾μ} \frac{∥u_{n}∥_{∞}^{p}}{μ_{n}^{p}} = ∑_{μ_{n} ⩾μ} \frac{∥u_{n}∥_{∞}^{p}}{μ_{n}^{p− \frac{1}{2}}} \frac{1}{μn^{\frac{1}{2}}} ⩽ ( ∑_{μ_{n} ⩾μ} \frac{∥u_{n}∥_{∞}^{2}}{μ_{n}^{2− \frac{1}{p}}} ) ^{p/2}( ∑_{μ_{n} ⩾μ} \frac{1}{μn^{\frac{1}{2−p}}} ) ^{\frac{2−p}{2}}

⩽ ( ∑_{μ_{n} ⩾μ} \frac{∥u_{n}∥_{∞}^{2}}{μ_{n}^{2− \frac{1}{p}}} ) ^{p/2}( ∑_{k=[μ]}^{∞} \frac{1}{k^{\frac{1}{2−p}}} ( ∑_{k⩽μ_{n} ⩽k+1} 1) ) ^{\frac{2−p}{2}} .

Having applied here Lemma 2.6 for θ = 1 − \frac{1}{p} and estimation (2.15), we get

∑_{μ_{n} ⩾μ} \frac{∥u_{n}∥_{∞}^{2}}{μ_{n}^{p}} ⩽ const(μ^{( \frac{1}{p} −1)} )^{p/2}[μ]^{\frac{1−p}{2}} ⩽ constμ^{1−p} .

Lemma 2.7 is proved. □

Lemma 2.8 (see [9]). Let {α_{m}}_{m=0}^{∞} be a numerical sequence with the elements α_{m} ⩾ 0, β be a complex number for which Reβ > 0. Then for the inequality

( ∑_{m=0}^{∞} α_{m} | ∫_{G} f(x) exp(−mβx)dx|^{q}) ^{\frac{1}{q}} ⩽ M_{p}∥f∥_{p}

to be fulfilled for any function f(x) ∈ L_{p}(G), 1 < p ⩽ 2, q = \frac{p}{p−1} , it is necessary and sufficient the existence of a constant K such that for all N = 1, 2, … the following estimation is valid ∑ _{m=0}^{N} α_{m} ⩽ KN.

Lemma 2.9. If under the conditions of Lemma 2.3, we have that for each system

\{∥u_{n}∥∞^{\frac{2}{q}} e^{iω_{3} μ_{n} t} \}_{n=1}^{∞} and \{∥u_{n}∥∞^{\frac{2}{q}} e^{−iω_{2} μ_{n} (1−t)} \}_{n=1}^{∞} for Imλ_{n} < 0, for each system

\{∥u_{n}∥∞^{\frac{2}{q}} e^{−iω_{2} μ_{n} t} \}_{n=1}^{∞} and \{∥u_{n}∥∞^{\frac{2}{q}} e^{iω_{3} μ_{n} (1−t)} \}_{n=1}^{∞} for Imλ_{n} < 0 the Riesz inequal-

ity is fulfilled for 1 < p ⩽ 2, where p^{−1} + q^{+1} = 1.

Proof. Let us consider the first one of these systems and prove for it the Riesz inequality (the remaining systems are considered similarly). Since μ_{n} ∈ [0, +∞) and ω_{3} = i\frac{\sqrt{3}}{2} + \frac{1}{2}, then iω_{3}μ_{n}t = (i\frac{1}{2} − \frac{\sqrt{3}}{2})μ_{n}t and |e^{iω_{3} μ_{n} t} | = e^{− \frac{\sqrt{3}}{2} μ_{n} t} . Taking this into account, we get that for any f(x) ∈ L_{p}(G)

∑_{Im λ_{n} <0} ∥u_{n}∥_{∞}^{2} | ∫_{0}^{1} \overline{f(t)}e^{iω_{3} μ_{n} t} dt|^{q} ⩽ ∑_{Im λ_{n} <0} ∥u_{n}∥_{∞}^{2} | ∫_{0}^{1} \overline{f(t)}e^{(i \frac{1}{2} − \frac{\sqrt{3}}{2} )μ_{n} t} dt|^{q}

⩽ ∑_{n=1}^{∞} ∥u_{n}∥_{∞}^{2} ( ∫_{0}^{1} |f(t)|e^{− \frac{1}{2} μ_{n} t} dt) ^{q}

⩽ ∑_{k=0}^{∞} ∑_{k⩽μ_{n} <k+1} ∥u_{n}∥_{∞}^{2} ( ∫_{0}^{1} |f(t)|e^{− \frac{1}{2} μ_{n} t} dt) ^{q}

⩽ ∑_{k=0}^{∞} ( ∑_{k⩽μ_{n} <k+1} ∥u_{n}∥_{∞}^{2} ) ( ∫_{0}^{1} |f(t)|e^{− \frac{1}{2} kt} dt) ^{q}

= ∑_{k=0}^{∞} α_{k}( ∫_{0}^{1} |f(t)|e^{− \frac{1}{2} kt} dt) ^{q},

where α_{k} = ∑ _{k⩽μ_{n} <k+1} ∥u_{n}∥_{∞}^{2} .

By inequality (2.16), for any natural number N it is fulfilled:

∑_{k=0}^{N} α_{k} = ∑_{k=0}^{N} ( ∑_{k⩽μ_{n} <k+1} ∥u_{n}∥_{∞}^{2} ) = ∑_{0⩽μ_{n} <N+1} ∥u_{n}∥_{∞}^{2} ⩽ const·N.

Consequently, the condition of Lemma 2.8 is fulfilled. Therefore, the following inequality is valid

\{ ∑_{k=1}^{∞} α_{k}( ∫_{0}^{1} |f(t)|e^{− \frac{1}{2} kt} dt) ^{q}\}^{1/q} ⩽ M(p)∥f∥_{p}.

Lemma 2.9 is proved. □

3. Proof of the basic results

Proof of Theorem 1.1. If suffices to consider the case 1 < p ⩽ 2. Prove the uniform convergence of the series ∑ _{n=1}^{∞} |f_{n}||u_{n}(x)| on Ḡ. To this end we partition this series in two sums: ∑ _{0⩽μ_{n} ⩽2} |f_{n}||u_{n}(x)| and ∑ _{μ_{n} >2} |f_{n}||u_{n}(x)|. The first sum by inequality (2.16) doesn’t exceed the quantity const∥f∥_{1}. For investigating the second series we apply Lemma 2.2, i.e., estimations (2.8) and (2.9) depending on the sign of Imλ_{n}. For that we represent the given series in the form

∑_{μ_{n} >2} |f_{n}||u_{n}(x)| = ∑_{n∈J_{1}} |f_{n}||u_{n}(x)| + ∑_{n∈J_{2}} |f_{n}||u_{n}(x)| = I_{1} + I_{2},

where J_{1} = {n : μ_{n} > 2, Imλ_{n} < 0}, J_{2} = {n : μ_{n} > 2, Imλ_{n} > 0}. By estimation (2.8)

I_{1} = ∑_{n∈J_{1}} |f_{n}||u_{n}(x)| ⩽ const ∑_{n∈J_{1}} C(f)μ_{n}^{α−3} ∥u_{n}∥_{∞}^{2}

+ const ∑_{n∈J_{1}} μ_{n}^{−1} |(f′, e^{iω_{3} μ_{n} t} )|∥u_{n}∥_{∞}^{2}

+ const ∑_{n∈J_{1}} μ_{n}^{−1} |(f′, e^{−iω_{2} μ_{n} (1−t)} )|∥u_{n}∥_{∞}^{2}

+ const(∥f∥_{∞} + ∥f′∥_{1}) ∑_{n∈J_{1}} μ_{n}^{−2} ( ∑_{r=2}^{3} μ_{n}^{2−r} ∥p_{r}∥_{1}) ∥u_{n}∥_{∞}^{2}

+ const ∑_{n∈J_{1}} μ_{n}^{−1} |(f, e^{−iμ_{n} t} )|∥u_{n}∥_{∞}

= const(I_{1}^{1} + I_{1}^{2} + I_{1}^{3} + I_{1}^{4} + I_{1}^{5}).

Estimate the series I_{1}^{j}, j = \overline{1, 5}. By Lemma 2.6 and condition 0 ⩽ α < 2 we find

(3.1)  I_{1}^{1} = C(f) ∑_{n∈J_{1}} μ_{n}^{α−3} ∥u_{n}∥_{∞}^{2} ⩽ C(f) ∑_{μ_{n⩾2}} \frac{∥u_{n}∥_{∞}^{2}}{μ_{n}^{1+(2−α)}}
⩽ constC(f)2^{α−2} < ∞.

For estimating the series I_{1}^{2} we apply at first the Holder inequality for the sum, and then Lemmas 2.6 and 2.9:

I_{1}^{2} = ∑_{n∈J_{1}} ∥u_{n}∥_{∞}^{2/p}μ_{n}^{−1} (∥u_{n}∥_{∞}^{2/q}|(f′, e^{iω_{3} μ_{n} t} )|)

⩽ \{ ∑_{n∈J_{1}} \frac{∥u_{n}∥^{2}}{μ_{n}^{p}} \}^{1/p}\{ ∑_{n∈J_{1}} ∥u_{n}∥_{∞}^{2} |(f′, e^{iω_{3} μ_{n} t} )|^{q}\}^{1/q}

⩽ \{ ∑_{μ_{n} ⩾2} \frac{∥u_{n}∥^{2}}{μ_{n}^{p}} \}^{1/p}\{ ∑_{n∈J_{1}} ∥u_{n}∥_{∞}^{2} |(f′, e^{iω_{3} μ_{n} t} )|^{q}\}^{1/q}

⩽ const2^{−1/q} M(p)∥f′∥_{p} < ∞.

The series I_{1}^{3} is estimated in the same way as the series I_{1}^{2}. For estimating the series I_{1}^{4} we apply Lemma 2.6.

(3.2)  I_{1}^{4} = (∥f∥_{∞} + ∥f′∥_{1}) ∑_{n∈J_{1}} μ_{n}^{−2} ( ∑_{r=2}^{3} μ_{n}^{2−r} ∥p_{r}∥_{1}) ∥u∥_{∞}^{2}

⩽ const(∥f∥_{∞} + ∥f′∥_{1})( ∑_{r=2}^{3} ∥p_{r}∥_{1}2^{1−r} ) < ∞.

Now estimate the series I_{1}^{5}. For that we apply the Holder inequality, and then Lemmas 2.5 and 2.7:

I_{1}^{5} = ∑_{n∈J_{1}} \frac{∥u_{n}∥_{∞}}{μ_{n}} |(f′, e^{−iμ_{n} t} )|

⩽ ( ∑_{n∈J_{1}} \frac{∥u_{n}∥_{∞}^{p}}{μ_{n}^{p}} ) ^{1/p}( ∑_{n∈J_{1}} |(f′, e^{−iμ_{n} t} )|^{q}) ^{\frac{1}{q}} ⩽ const∥f′∥_{p}2^{− \frac{1}{q}} < ∞.

Thus, the series I_{1} uniformly converges on Ḡ. Applying estimation (2.9) for the coefficients f_{n} in the same way we prove the uniform convergence of the series I_{2} on Ḡ. Consequently, the series ∑ _{n=1}^{∞} |f_{n}| |u_{n}(x)| uniformly converges on Ḡ. By the completeness of the system {u_{n}(x)}_{n=1}^{∞} in L_{2}(G) and continuity of the function f(x) on Ḡ the series ∑ _{n=1}^{∞} f_{n}u_{n}(x) uniformly converges to f(x), i.e. it holds the equality

(3.3)  f(x) = ∑_{n=1}^{∞} f_{n}u_{n}(x), x ∈ Ḡ

Now establish estimate (1.2). By equality (3.3)

|R_{ν}(x, f)| = |f(x) − σ_{ν}(x, f)| = | ∑_{μ_{n} >ν} f_{n}u_{n}(x)|

⩽ ∑_{μ_{n} >ν} |f_{n}||u_{n}(x)| = ∑_{n∈A_{1} (ν)} + ∑_{n∈A_{2} (ν)} = T_{1}(ν) + T_{2}(ν),

where A_{1}(ν) = {n : μ_{n} ⩾ ν, Imλ_{n} < 0}, A_{2}(ν) = {n : μ_{n} ⩾ ν, Imλ_{n} < 0}. The series T_{1}(ν), T_{2}(ν) are estimated by the scheme demonstrated by estimating the series I_{1}. As a result we find

T_{j} (ν) ⩽ const\{ C(f)ν^{α−2} + ν^{− \frac{1}{q}} ∥f′∥_{p} + ν^{−1} (∥f∥_{∞} + ∥f′∥_{1})( ∑_{r=2}^{3} ∥p_{r}∥_{1}ν^{2−r} ) \} ,

j = 1, 2, x ∈ Ḡ.

Consequently, estimation (1.2) is valid for 1 < p ⩽ 2. For p > 2 the validity of estimation (1.2) follows from the embedding L_{p}(G) ⊂ L_{2}(G).

Theorem 1.1 is proved. □

Corollary 1.1 follows from Theorem 1.1 with regard to the inequality ∥f∥_{∞} ⩽ ∥f′∥_{1} that is fulfilled for the functionf(x) ∈ W_{p}^{1}(G), f(0) = f(1) = 0.

Proof of Theorem 1.2. In the present case it is necessary to prove the uniform convergence of the series ∑ _{μ_{n} ⩾2} |f_{n}||u_{n}(x)| on Ḡ. By estimation (2.7) we find

∑_{μ_{n} ⩾2} |f_{n}||u_{n}(x)| ⩽ const{ C(f) ∑_{μ_{n} ⩾2} μ^{α−3} ∥u_{n}∥_{∞}^{2}

+ ∑_{μ_{n} ⩾2} μ_{n}^{−1} |( f \overline{p_{1}}, \frac{u_{n}^{(2)}}{μ_{n}^{2}} ) |∥u_{n}∥_{∞}^{2} + ∑_{μ_{n} ⩾2} μ_{n}^{−1} ∥u_{n}∥_{∞} |( f′, \frac{u_{n}^{(2)}}{μ_{n}^{2}} ) |

+ ∥f∥_{∞} ∑_{μ_{n} ⩾2} μ_{n}^{−2} ∥u_{n}∥_{∞}^{2} ( ∑_{r=2}^{3} μ_{n}^{2−r} ∥p_{r}∥_{1}) }

⩽ const{V_{1} + V_{2} + V_{3} + V_{4}}.

The series V_{1} and V_{4} are estimated in the same way as the series I_{1}^{1} and I_{1}^{4}. For V_{1} estimation (3.1) is fulfilled, and for V_{4} estimation (3.2) is fulfilled by changing the multiplier (∥f∥_{∞} + ∥f′∥_{1}) by the multiplier ∥f∥_{∞} .

By estimating the series V_{2} and V_{3}, we apply the Bessel inequality for the system {u_{n}^{(2)}(x)/μ_{n}^{2}}, μ_{n} ⩾ 2, whose validity is established in the paper [9], and also Lemma 2.7 for p = 2.

As a result, we find

V_{2} = ∑_{μ_{n} ⩾2} μ_{n}^{−1} ∥u_{n}∥_{∞} ( f \overline{P_{1}}, \frac{u_{n}^{(2)}}{μ_{n}^{2}} ) ⩽ ( ∑_{μ_{n} ⩾2} \frac{∥u_{n}∥_{∞}^{2}}{μ_{n}^{2}} ) ^{1/2}

× ( ∑_{μ_{n} ⩾2} |( f \overline{p_{1}}, \frac{u_{n}^{(2)}}{μ_{n}^{2}} ) |^{2}) ^{1/2} ⩽ const2^{− \frac{1}{2}} ∥f \overline{p_{1}}∥_{2};

V_{3} ⩽ ( ∑_{μ_{n} ⩾2} \frac{∥u_{n}∥_{∞}^{2}}{μ_{n}^{2}} ) ^{1/2} ( ∑_{μ_{n} ⩾2} |( f′, \frac{u_{n}^{(2)}}{μ_{n}^{2}} ) |^{2}) ^{1/2} ⩽ const2^{− \frac{1}{2}} ∥f′∥_{2}.

Consequently, the series ∑ _{n=1}^{∞} f_{n}u_{n}(x) converges absolutely and uniformly on Ḡ, and the following equality f(x) = ∑ _{n=1}^{∞} f_{n}u_{n}(x), x ∈ G, is valid. It is easy to see that for the remainder R_{ν}(x, f) of this series the following estimation will be valid (in the remainder the summation is conducted according to numbers n, for which μ_{n} > ν)

sup|R_{ν}(x, f)| ⩽ const\{C(f)ν^{α−2} + ν^{− \frac{1}{2}} (∥f \overline{p_{1}}∥_{2} + ∥f′∥_{2}) + ν^{−1} ∑_{r=2}^{3} ν^{2−r} ∥p_{r}∥_{1}\} ,

ν ⩾ 2.

Theorem 1.2 is proved. □

For justification of Corollary 1.2, it suffices to take into account that the sequence of remainders of the converging series tends to zero, more exactly,

∑_{μ_{n} ⩾ν} |( f \overline{p_{1}}, \frac{u_{n}^{(2)}}{μ_{n}^{2}} ) |^{2} = o(1), ν → +∞;

∑_{μ_{n} ⩾ν} |( f′, \frac{u_{n}^{(2)}}{μ_{n}^{2}} ) |^{2} = o(1), ν → +∞.

Proof of Theorem 1.3. By orthonormality of the system {u_{n}(x)}_{n=1}^{∞} in L_{2}(G), condition (2.15) is fulfilled. On the other hand,

1 = |(u_{n}, u_{n})| ⩽ ∥u_{n}∥_{p}∥u_{n}∥_{q} ⩽ ∥u_{n}∥_{∞} ∥u_{n}∥_{q}.

Hence we find ∥u_{n}∥_{q}^{−q} ⩽ ∥u_{n}∥_{∞}^{q} . Therefore, by inequality (2.15) and uniform boundedness of the system {u_{n}(x)}_{n=1}^{∞} we find

∑_{0⩽μ_{n} ⩽τ} ∥u_{n}∥_{∞}^{q} ∥u_{n}∥_{q}^{−q} ⩽ ∑_{0⩽μ_{n} ⩽τ} ∥u_{n}∥_{∞}^{2q} ⩽ C ∑_{0⩽μ_{n} ⩽τ} 1 ⩽ constτ, for all τ > 0.

Thus, for the system {u_{n}(x)}_{n=1}^{∞} , all the conditions of the sufficiency part of [9, Theorem 3] are fulfilled. Therefore, for the system \{\frac{u_{n} (x)}{μ_{n}^{2}} \}, μ_{n} ⩾ 1 the Riesz inequality is valid.

For proving Theorem 1.3, it suffices to estimate the series V_{3} (all remaining series V_{1}, V_{2}, V_{4} were estimated in Theorem 1.2 without requirement of uniform boundedness of the system {u_{n}(x)}_{n=1}^{∞} ). At first, apply the Hölder inequality, and then the Riesz inequality and Lemma 2.7. As a result, for V_{3} and its remainder we get

V_{3} = ∑_{μ_{n} ⩾2} μ_{n}^{−1} ∥u_{n}∥_{∞} |( f′, \frac{u_{n}^{(2)}}{μ_{n}^{2}} ) |

⩽ ( ∑_{μ_{n} ⩾2} \frac{∥u_{n}∥_{∞}^{p}}{μ_{n}^{p}} ) ^{1/p}( ∑_{μ_{n} ⩾2} |( f′, \frac{u_{n}^{(2)}}{μ_{n}^{2}} ) |^{q}) ^{1/q} ⩽ const2^{− \frac{1}{q}} ∥f′∥_{2};

∑_{μ_{n} ⩾ν} \frac{∥u_{n}∥_{∞}}{μ_{n}} |( f, \frac{u_{n}^{(2)}}{μ_{n}^{2}} ) |

⩽ ( ∑_{μ_{n} ⩾ν} \frac{∥u_{n}∥_{∞}^{p}}{μ_{n}^{p}} ) ^{1/p}( ∑_{μ_{n} ⩾ν} |( f′, \frac{u_{n}^{(2)}}{μ_{n}} ) |^{q}) ^{1/q} ⩽ constν^{− \frac{1}{q}} ∥f′∥_{p}.

Theorem 1.3 is proved. □

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Institute of Mathematics and Mechanics of NAS of Azerbaijan (Received 09 05 2015) Baku, Azerbaijan

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