c Pleiades Publishing, Ltd., 2007. ISSN 0001-4346, Mathematical Notes, 2007, Vol. 81, No. 5, pp. 632–637. c Yu. S. Kolomoitsev, 2007, published in Matematicheskie Zametki, 2007, Vol. 81, No. 5, pp. 707–712. Original Russian Text
Completeness of the Trigonometric System for the Classes ϕ(L) Yu. S. Kolomoitsev* Donetsk National University, Ukraine Received January 24, 2006; in final form, October 5, 2006
Abstract—We obtain a necessary and sufficient condition for the completeness of the trigonometric system with gaps for the classes ϕ(L). DOI: 10.1134/S0001434607050082 Key words: trigonometric system with gaps, 2π-periodic function, Jensen’s inequality, trigonometric polynomial, Parseval’s equality, Riesz–Fischer theorem.
Introduction. Suppose that Φ is the set of even nonnegative, finite, and nondecreasing functions ϕ on [0, ∞) such that lim ϕ(t) = ϕ(∞) = ∞.
t→∞
In what follows, we shall assume that the function ϕ satisfies the following conditions: ϕ ∈ Φ,
ϕ(0) = 0,
ϕ(t) > 0 for t > 0 and
ϕ ∈ C[0, ∞).
By ϕ(L) we denote the set of all measurable 2π-periodic functions f for which ϕ(|f (t)|) dt < ∞, T
where T = (−π, π] is the unit circle. If the functions f and g belong to the class ϕ(L), then the quantity ϕ(|f (t) − g(t)|) dt ρϕ (f, g) = T
will be conditionally called ϕ-distance. Also set ρϕ (f ) := ρϕ (f, 0). A sequence of functions {fn }∞ n=1 from the class ϕ(L) is said to be convergent with respect to ϕ-distance to a function f ∈ ϕ(L) if (f − fn ) ∈ ϕ(L) for n ≥ n0 for some n0 ∈ N and limn→∞ ρϕ (f − fn ) = 0. The classes ϕ(L) were studied in numerous works by Ulanov. For example, as shown in [1], the class ϕ(L) is separable in the sense of convergence with respect to ϕ-distance if and only if ϕ satisfies the condition ϕ(t + 1) = O{ϕ(t)} as t → ∞. In the present paper, we study the question of the completeness of a trigonometric system with gaps, i.e., of a system of the form {ek }k∈A (ek = eikx ), where A ⊂ Z and A = Z for the classes ϕ(L). First, note that the trigonometric system with gaps is not complete in the space L. Therefore, it follows from Jensen’s integral inequality (see Lemma 1) that this system is also not complete in the space ϕ(L) if ϕ is a convex (downward) function. The same can be said about this system if we assume that the function ϕ satisfies the Δ2 -condition (i.e., ϕ(2t) = O{ϕ(t)} as t → ∞) and limt→∞ ϕ(t)/t > 0. This follows from the papers of Ivanov [2, Theorem 2] and Filippov [3, Theorem 1.2]. Besides, note that the problem of completeness of the system {ek }k∈A for A ⊂ Z and A = Z in the spaces Lp , 0 < p < 1, and ϕ(L) was studied in [4]–[9]. The existence of an infinite set B ⊂ Z such *
E-mail:
[email protected].
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that the system {ek }k∈Z\B is complete in Lp for 0 < p < 1 was, apparently, first proved in [4]. Sufficient conditions for the completeness of the trigonometric system with gaps were obtained in [5]–[9]. An approach for obtaining necessary and sufficient conditions for completeness was proposed in [7], while, in [8], the following necessary condition for completeness was obtained: the system {ek }k∈Z\B is not complete in Lp for any 0 < p < 1 if B is an infinite (in both directions) arithmetic progression. In [10], the results from [8] were extended to some classes ϕ(L). 1. Before passing to the main result of the paper, let us state some auxiliary statements. We shall often use Jensen’s integral inequality. Lemma 1 ([11, p. 45 (Russian transl.)]). Suppose that ϕ is a convex (upward) function and f , q are integrable positive functions, with q(x) = 0 on T. Then (x))q(x) dx f (x)q(x) dx T ϕ(f ≤ ϕ T . T q(x) dx T q(x) dx Lemma 2 ([2, Theorem 5]). Suppose that ϕ(L) ⊂ L. Then the completeness of the system {gk }k≥1 in ϕ(L) implies that, for each N ≥ 1, the system {gk }k≥N is also complete in ϕ(L). Besides, note that if the function ϕ satisfies the Δ2 -condition, then it follows from the papers of Ivanov [2, Theorem 2] and Filippov [3, Theorem 1.2] that the condition ϕ(L) ⊂ L is equivalent to limt→∞ ϕ(t)/t = 0. We shall use the following notation: 1 f (x)e−inx dx, n∈Z f (n) = 2π T for the Fourier coefficients of an integrable function f ; by spec f = {n ∈ Z : fˆ(n) = 0} we denote the spectrum of the function f . The main result of the paper is the following theorem. Theorem 1. Suppose that the function ϕ is convex (upward), limt→∞ ϕ(t)/t = 0, and A ⊂ Z. Then the system {ek }k∈A is complete in ϕ(L) if and only if, for any trigonometric polynomial T , there exists a function f ∈ L2 (T) such that f(n) = T(n) for all n ∈ Z \ A and f (x + h) − f (x − h) dx = 0. (1) ϕ lim h h→0 T Proof. Sufficiency. By the properties of the functions ϕ, it suffices to consider the case T = em , where m ∈ Z \ {0}. Suppose that the function f ∈ L2 satisfies condition (1), fˆ(m) = 1, and spec f ⊂ A ∪ {m}. We introduce the auxiliary function f (x + δ) − f (x − δ) . 2i sin mδ ´ We define a kernel of Vallee-Poussin type by setting 2n k ek , v Kn = n gδ (x) =
k=−2n
where v ∈
C ∞ (R)
and v(x) = 1 for |x| ≤ 1 and v(x) = 0 for |x| ≥ 2. Obviously, the polynomial 1 Kn (x − t)gδ (t) dt = Kn ∗ gδ (x), n ≥ |m| Tn,δ (x) = 2π T
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satisfies the relations Tn,δ (m) = 1 and spec Tn,δ ⊂ A ∪ {m}. Let us show that, for any δ > 0,
lim
n→∞ T
ϕ(|Tn,δ (x)|) dx ≤
T
ϕ(|gδ (x)|) dx.
Using Lemma 1 with the function q(x) ≡ 1/(2π), as well as the semiadditivity of the convex (upward) function ϕ (see, for example, [2, p. 19]), we obtain ϕ(|Tn,δ (x)|) dx − ϕ(|gδ (x)|) dx ≤ ϕ(|Tn,δ (x) − gδ (x)|) dx T T T 1 |Tn,δ (x) − gδ (x)| dx , ≤ 2πϕ 2π T but since it is well known that, for each integrable function h, the relation Kn ∗ h − h 1 → 0 as n → ∞ holds, then, for a sufficiently small δ, we find f (x + δ) − f (x − δ) dx lim ϕ(|Tn,δ (x)|) dx ≤ ϕ(|gδ (x)|) dx = ϕ n→∞ T 2i sin mδ T T f (x + δ) − f (x − δ) dx. ϕ ≤ δ T Thus, the sufficiency in Theorem is proved. Necessity. Now, suppose that T is a fixed trigonometric polynomial of degree at most M0 . We can also assume that T(0) = 0. Let us construct a series of the form T+
ak ek
k∈A,|k|>M0,
for which there exists a subsequence of partial sums which converges to zero with respect to ϕ-distance. Choose some ε1 > 0. Since, by Lemma 2, the system {ek }k∈A,|k|>M0 is complete in ϕ(L), there exists a trigonometric polynomial ak ek such that ρϕ (T + pN1 ) < ε1 . p N1 = k∈A,M0 Nj
The theorem is proved. 2. In this section, we obtain corollaries of Theorem 1. But, first, let us prove an auxiliary statement. Lemma 3. Suppose that ϕ is a convex (upward) function and limt→∞ ϕ(t)/t = 0. If the function f is of finite variation on T and f (x) = 0 a.e., then f (x + h) − f (x − h) dx = 0. ϕ lim h h→0 T
Proof. Denote ghi (x) := (f (x + hi ) − f (x − hi ))/hi , where hi → 0 as i → ∞. Choose ε > 0 and set Ei = {x ∈ T : |ghi (x)| > ε}. In Lemma 1, taking the characteristic function of the set Ei for the function q, we obtain ϕ(|ghi (x)|) dx = ϕ(|ghi (x)|) dx + ϕ(|ghi (x)|) dx T T\Ei Ei f (x + hi ) − f (x − hi ) 1 dx ≤ 2πϕ(ε) + |Ei |ϕ |Ei | T hi V (f ) , ≤ 2πϕ(ε) + |Ei |ϕ |Ei | where V (f ) is the total variation of the function f on T, and |E| is the Lebesgue measure of the set E. At the last step, we apply a well-known result due to Hardy and Littlewood (see, for example, [13, Theorem 4.6.14]). Since the choice of ε > 0 is arbitrary and |Ei | → 0 as i → ∞, we obtain the proof of the lemma. MATHEMATICAL NOTES
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The following corollary was obtained by Aleksandrov [7, Theorem 8.2] in the special case of the spaces Lp for 0 < p < 1. Corollary 1. Suppose that ϕ is a convex (upward) function, limt→∞ ϕ(t)/t = 0, and A ⊂ Z. Then the existence, for each n ∈ Z, of an a ∈ N such that ([−a − n, −a + n] ∪ [a − n, a + n]) ∩ Z ⊂ A implies that the system {ek }k∈A is complete in ϕ(L). Proof. For each m ∈ N, choose an infinite sequence {nm,j }∞ j=0 ⊂ N possessing the following properties: m = nm,0 , nm,j+1 /nm,j ≥ 3 for all j ∈ Z+ , ±m + ε1 nm,1 + · · · + εk nm,k ∈ A, for each k ∈ N, where εj ∈ {−1, 0, 1}. Consider the function
fm (x) = lim
|m| < |ε1 nm,1 + · · · + εk nm,k |
k x
(1 + cos nm,j t) dt.
k→∞ 0 j=0
This function is singular (see, for example, [11, p. 334 (Russian transl.)]). But, in that case, and the function ± (x) = m(fm (x + π/(2m)) ± ifm (x)) gm ± (±m) = 1 and spec g ± ⊂ A ∪ {±m}. Thus, using Lemma 3 and Theorem 1, is also singular; besides, gˆm m we can prove Corollary 1.
Corollary 2. Suppose that ϕ is a convex (upward) function, limt→∞ ϕ(t)/t = 0, B = {nk }k∈Z , nk+1 − 2nk + nk−1 ≥ 0 and n−k = −nk for k ∈ N. The system {ek }k∈Z\B is complete in ϕ(L) if and only if limk→∞ (nk+1 − nk ) = ∞. Proof. Sufficiency immediately follows from Corollary 1. Let us prove the necessity. Assume the converse: namely, there exists a number q ∈ N such that nk+1 − nk ≤ q for all k ∈ N. Then, by the convexity of B, we find that, for all k ≥ k0 , the relation nk+1 − nk = q holds; therefore, nk0 +l = lq + nk0 , l ∈ N. Hence the set B contains an infinite arithmetic progression. Using a result of Ivanov and Yudin [8, Theorem 4] (see, also [10, Theorem 4]), we find that the system {ek }k∈Z\B is not complete in ϕ(L). A contradiction. The corollary is proved. The following assertion was obtained by Ivanov, who used another method (see Theorem 8 and Lemma 14 in [9]). Corollary 3. Suppose that ϕ is a convex (upward) function and limt→∞ ϕ(t)/t = 0. There exists a sequence C = {nk }k∈Z such that lim|k|→∞(nk+1 − nk ) = ∞ and the system {ek }k∈C is complete in ϕ(L). Proof. To construct the set C, we use the following procedure. Consider the sequence of trigonometric polynomials (0)
t1 ,
(±1)
t1
,
(±1)
t2
,
(0)
t2 ,
(±2)
t1
,
(±2)
t2
,
(±2)
t3
,
(0)
t3 ,
(±1)
t3
,
(±3)
t1
,
...,
where (m) tN (x)
imx
:= e
N
(1 + cos mN,j x).
j=1 (m)
In these polynomials, we choose the sequences {mN,j } (mN,j+1 ≥ 3mN,j ) so that tˆN (m) = 1 and the (m)
set C = ∪m ∪N (spec tN \ {m}) satisfies our assumptions (i.e., the difference of the adjacent elements of the set C tends to infinity). The proof of the corollary follows from the fact that, for each fixed m ∈ Z, (m) ρ(tN ) → 0 as N → ∞. The last relation can be obtained from the proof of Lemma 3 and Theorem 7.6 from [11, Chap. 5]. MATHEMATICAL NOTES
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REFERENCES
1. P. L. Ul anov, “The representation of functions by series and the classes ϕ(L),” Uspekhi Mat. Nauk 27 (2), 3–52 (1972) [Russian Math. Surveys 27 (2), 1–54 (1972)]. 2. V. I. Ivanov, “Representation of functions by series in metric symmetric spaces without linear functionals,” in Proc. Steklov Inst. Math. (Nauka, Moscow, 1989), Vol. 189, pp. 34–77 [in Russian]. 3. V. I. Filippov, “Linear continuous functionals and representation of functions by series in the spaces Eϕ ,” Anal. Math. 27 (4), 239–260 (2001). 4. A. A. Talalyan, “Representation of functions of classes Lp [0, 1], 0 < p < 1, by orthogonal series,” Acta Math. Academ. Sci. Hungar. 21 (1–2), 1–9 (1970). 5. K. de Leeuw, “The failure of spectral analysis in Lp for 0 < p < 1,” Bull. Amer. Math. Soc. 82 (1), 111–114 (1976). 6. J. H. Shapiro, “Subspaces of Lp (G) spanned by characters: 0 < p < 1,” Israel J. Math. 29 (2–3), 248–264 (1978). 7. A. B. Aleksandrov, “Essays on nonlocally convex Hardy classes,” in Complex Analysis and Spectral Theory, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1981), Vol. 864, pp. 1–89. 8. V. I. Ivanov and V. A. Yudin, “On the trigonometric system in Lp , 0 < p < 1,” Mat. Zametki 28 (6), 859–868 (1980). 9. V. I. Ivanov, “Representation of measurable functions by multiple trigonometric series,” in Proc. Steklov Inst. Math. (Nauka, Moscow, 1983), Vol. 164, pp. 100–123 [in Russian]. 10. D. Ya. Spivakovskaya, “On the trigonometric system in the metric spaces LΨ ,” Vestnik Dnepropetrovsk. Nats. Univ., No. 6, 101–115 (2001). 11. A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1959, 1960; Mir, Moscow, 1965), Vols. 1, 2. 12. M. A. Krasnoselskii and Ya. B. Rutitskii, Convex Functions and Orlicz Spaces (Fizmatgiz, Moscow, 1958) [in Russian]. 13. R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions (Kluwer Acad. Publ., Dordrecht, 2004).
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