Gen. Math. Notes, Vol. 3, No., September 25, pp.2-2 ISSN 229-784; Copyright c ICSRS Publication, 25 www.i-csrs.org Available free online at http://www.geman.in Basicity of System of Exponents with Complex Coefficients in Generalized Lebesgue Spaces Togrul Muradov Institute of Mathematics and Mechanics of NAS of Azerbaijan Department of Non-Harmonic Analysis 9 B.Vahabzadeh Str., AZ 4 Baku, Azerbaijan E-mail: togrulmuradov@gmail.com (Received: 7-7-5 / Accepted: 27-8-5) Abstract We consider a perturbed system of exponents e } iλnt, when the sequence n Z λ n } n Z has a definite asymptotics. We study basis properties (completeness, minimality, basicity) of this system in Lebesgue space of functions L p( ) (, π) with variable summability exponent p( ), subject to parameters contained in the asymptotics λ n } n Z. Keywords: System of exponents, Perturbation, Variable exponent, Basis properties. Introduction We consider the perturbed system of exponents e iλ nt } n Z, () where Z is the set of integrals and λ n } R is some sequence of real numbers for which, it holds the asymptotics ( λ n = n αsign n + O n β), n, (2) λ, β R are real parameters. Starting with Paley-Wiener s basic result [], a great number of papers have been devoted to studying basis properties (completeness, minimality, basicity)
Basicity of System of Exponents with Complex... 3 in L p = L p (; π) of the system of exponents of the form (). Some results in this direction may be found in R. Young s monograph [2]. The basicity criterion of the system () in L p, < p < +, in the case when λ n = n αsign n, was obtained in the papers [3;4]. The same problem in Lebesgue space L p( ) with variable summability exponent p ( ) at λ n = n αsign n, n Z, was studied in the papers [5-7]. The present paper is devoted to studying the basicity of the system () in the space L p( ) L p( ) (, π), when the sequence λ n } n Z has asymptotics (2). It should be noted that similar problems were studied in L p in the papers [8-2]. It also should be noted that partially, these results previously published in [6]. 2 Necessary Information Let p : [, π] [, + ) be some Lebesgue measurable function. We denote the class of all measurable on [, π] (with respect to Lebesgue measure) functions by L. Accept the denotation Assume Accept I p (f) def π f (t) p(t) dt. L f L : I p (f) < + }. p = inf vrai [,π] p (t), p+ = sup vrai [,π] p (t). For p + < +, with respect to ordinary linear operations of addition of functions and multiplication by a scalar, L turns into a linear space. By the norm ( ) } def f f p( ) inf λ > : I p, λ L is a Banach and we denote it by L p( ). Denote W L def p : p() = p(π); C >, p (t ) p (t 2 ) t, t 2 [, } π] : t t 2 2. C ln t t 2 Everywhere, q (t) denotes a function conjugated to p (t): holds the Holder generalized inequality + p(t) q(t). It
4 Togrul Muradov π f (t) g (t) dt c ( p ; p +) f p( ) g q( ), where c (p ; p + ) = +. p p + We need some notion and facts from the theory of close bases. The system x n } n N space X if n= a nx n = holds only for a n =, n N. While proving the main theorem we use the following fact. Let X be B-space with the norm. The systems x n } n N ; y n } n N X X is said to be ω-linear independent in Banach x n y n p < +. n= are said to be p-close if The basis x n } n N X in X with the biorthogonal system x n} n N X is called p- basis if it holds where c is some constant. The following statement is valid x n (x)} n N lp c x, x X, Statement 2. Let X be B-space with p-basis x n } n N X and y n } n N X be a system q-close to x n } n N. Then the following properties of the system y n } n N in X are equivalent. i) y n } n N is complete; ii) y n } n N is minimal; iii) y n } n N is ω- linear independent; iv) y n } n N forms a basis isomorphic to x n } n N. We also need the easily provable Lemma 2.2 Let X be a Banach space with the basis x n } n N X and F : X X be a Fredholm operator. Then the following properties of the system y n = F x n } n N in X are equivalent: ) y n } n N is complete; 2) y n } n N is minimal; 3) y n } n N is ω-linear independent; 4) y n } n N is a basis isomorphic to x n } n N. The validity of the following lemma is easily established. Lemma 2.3 Let X be B-space with the basis x n } n N and y n } n N X : card K = n : y n x n }} < +. Then the expression F x = n= x n (x) y n generates Fredholm operator F : X X, where x n} n N X.
Basicity of System of Exponents with Complex... 5 Indeed where F x = x n (x) (x n + y n x n ) = n= = x n (x) + x n (x) (y n x n ) = x + T x = (I + T ) x, n= n K T x = x n (x) (y n x n ). n K It is clear that T is a finite-dimensional operator and as a result of that F is Fredholm. For more detailed information on these results see the papers [2;-5]. 3 Basic Results Before proving the main result we prove the following theorem that we will use. Theorem 3. Let p W L, p > and A ± ; B ± L (, π). Then if the system () forms a basis in L p(t) L p(t) (, π), then it is isomorphic in L p(t) to classic system of exponents e int } n Z and the isomorphism is prescribed by the operator where Sf = A (t) ( ) f, e inx e int + B (t) (f, g) = π f (t) g (t)dt. 2π ( ) f, e inx e int, (3) Proof. Let the system A (t) e int ; B (t) e i(n+)t} (4) form a basis in L p( ) (, π). Denote by S an operator determined by formula (3). It is clear that S is a linear operator acting from L p( ) to L p( ). From basicity of the system of exponents e int } + and from condition (2) it follows that S is a bounded operator in L p( ). Furthermore, from (3) it immediately follows S [ e int] = A (t) e int, S [ e i(n+)t] = B (t) e i(n+)t, n =,.
6 Togrul Muradov Now prove that for g L p( ) the equation Sf = g has a solution. As the system (4) forms a basis in L p( ), then A (t) a n e int + B (t) b n e int = g (t), (5) where a n, b n are biorthogonal coefficients. Denote F (ξ) = a n ξ n, G (ξ) = bn+ ξ n, ξ =. As the series a ne int and b ne int converge in L p( ) (, π), we get ξ = ξ = F (ξ) ξ k dξ = G (ξ) ξ k dξ = a n bn+ ξ = ξ = ξ n+k dξ =, k ; (6) ξ n+k dξ =, k. From condition (6) it follows [6] that there exist the functions F (z) and G (z) from the Hardy class L p( ) such that F + (ξ) = F (ξ), G + (ξ) = G (ξ), where F + (ξ), G + (ξ) are the boundary values from within a unit circle of the functions F (z) and G (z), respectively. Thus, we get that the conjugate problem A (t) F + ( e it) + B (t) e it G + (e it ) = g (t), has a solution in L p( ) for g L p( ). As is known [6], for F (z) H p( ) it holds π lim F ( re it) F ( + e it) p( ) dt =. r Hence we get and so π 2π 2π π F ( + e it) e int an, n, dt =, n <, G + (e it ) e int bn+, n, dt =, n <, F + ( e it) = a n e int, (7)
Basicity of System of Exponents with Complex... 7 Denote It is clear that e it G + (e it ) = and Φ (z) H p( ). Introduce the following function b n+ e i(n+)t. Φ (z) = z G (z). π Φ 2π + (e it ) e int bn, n, dt =, n <, (8) f (t) = F + ( e it) + Φ + (e it ). From expressions (7), (8) it immediately follows that π f (t) e int an, n, dt = 2π b n, n <. Then from (5) it follows that Sf = g. As a result, from the Banach theorem we get that there exists a bounded inverse operator S. The theorem is proved. By using the above statements we prove the following basic theorem. Theorem 3.2 I. Let it hold asymptotics (2) and the inequalities 2p (π) < α < 2q (π), β > p (9) be fulfilled, where p = min p ; 2}. Then the following properties of system () in L p(t) are equivalent: i) system () is complete; ii) system () is minimal; iii) system () is ω-linear independent; iv) system () is a basis isomorphic to e int } n N ; v) λ i λ j for i j. II. Let β >. Then for α < the following properties are 2p(π) 2q(π) equivalent :. system () is complete; 2. system () is minimal; 3. λ i λ j for i j.
8 Togrul Muradov For α = system () doesn t form a basis in L 2p(π) p(t). III. Let β > and λ i λ j for i j. Then system () is complete and minimal in L p(t) only for α < 2p(π) 2q(π) complete, but minimal; while for α L p(t). 2q(π) Proof. Consider case I. Assume µ n = n αsignn, n Z. It is clear that it holds λ n µ n cn β, n Z\ },, and for α < it is not 2p(π) is complete but not minimal in where c (and later on also may be different at various places) is a constant independent of n. We have e iλ nt e iµnt = e i(λ n µ n)t π λn µ n + π λ n µ n 2 2! ( = π λ n µ n + π λ ) n µ n +... mn β, 2! where m is some constant. Consequently e iλ nt e iµnt p c p( ) n n +... = n βp < +, () if β >, let p = p(π) andq = q(π). From condition (9) it follows that the p system e iµnt } n Z forms a basis in L p L p (, π) (see e.i. [4]). Then by the results of the paper [], the system e iµnt } n Z is isomorphic in L p (, π) to the classic system of exponents e int } n Z, and as a result it is clear that the Hausdorff-Young type statement is valid for it, i.e. if < p 2 and f L p, the sequence of biorthogonal coefficients f n } n Z of the function f in the system e iµnt } n Z belongs to l q and it holds fn } lq n Z c f p, f L p. Thus, if p p, there hold the continuous imbeddings f p c f p c 2 f p( ), f L p( ). Consequently fn } n Z lq c f p( ), f L p( ). Having paid attention to relation (), we get that the systems e iλnt} n Z and e iµnt } n Z are p-close in L p( ), and e iµnt } n Z forms a q-basis in L p( ). The basicity of the system e iµnt } n Z in L p( ) follows from the result of the papers
Basicity of System of Exponents with Complex... 9 [2;3]. Then the validity of part I immediately follows from Statement 2.. Case II differs from case I by the value α =. In this case, by the results 2p(π) of the papers [5;2;3], the system e iµnt } n Z is complete and minimal in L p( ), but doesn t form a basis in it. Then it will ascertain similar to the previous case by using Theorem 3.. Case III is derived from case II, using periodicity of the system of exponents. The theorem is proved. Acknowledgements: The author express his deep gratitude to Professor B.T. Bilalov, corresponding member of the National Academy of Sciences of Azerbaijan, for his inspiring guidance and valuable suggestions during the work. References [] R. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., Providence, RI, 9(934), 84. [2] R.M. Young, An Introduction to Non-Harmonic Fourier Series, AP NYLTSSF, (98), 246. [3] A.M. Sedletskii, Biorthogonal expansions in series of exponents on the intervals of real axis, Usp. Mat. Nauk, 37(5) (227) (982), 5-95. [4] E.I. Moiseev, On basicity of the system of sines and cosines, DAN SSSR, 275(4) (984), 794-798. [5] B.T. Bilalov and Z.G. Guseynov, Bases from exponents in Lebesque spaces of functions with variable summability exponent, Trans. of NAS of Azerbaijan, XXVIII() (28), 43-48. [6] B.T. Bilalov and Z.G. Guseynov, K -Bessel and K -Hilbert systems and K -bases, Dokl. Math., 8(3) (29), 826-828. [7] B.T. Bilalov and Z.G. Guseynov, Basicity of a system of exponents with a piece-wise linear phase in variable spaces, Mediterr. J. Math., 9(3) (22), 487-498. [8] B.T. Bilalov, Basicity of some systems of exponents, cosines and sines, Diff. Uravneniya, 26() (99), -6. [9] B.T. Bilalov, On basicity of systems of exponents, cosines and sines in L p, Dokl. RAN, 365() (999), 7-8.
2 Togrul Muradov [] B.T. Bilalov, On basicity of some systems of exponents, cosines and sines in L p, Dokl. RAN, 379(2) (2), 7-9. [] B.T. Bilalov, Bases of exponents, cosines and sines which are Eigen functions of differential operators, Diff. Uravneniya, 39(5) (23), -5. [2] B.T. Bilalov, Basis properties of some systems of exponents, cosines and sines, Sibirskiy Matem. Jurnal, 45(2) (24), 264-273. [3] B.T. Bilalov, On isomorphism of two bases in L p, Fundam. Prikl. Mat., (4) (995), 9-94. [4] Ch. Heil, A Basis Theory Primer, Springer, (2), 536. [5] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhaeuser, Boston, Basel, Berlin, (23), XX+44 S. [6] T.R. Muradov, On bases from perturbed system of exponents in Lebesgue spaces with variable summability exponent, Journal of Inequalities and Applications, (24), 495. (accepted paper) http://www.journalofinequalitiesandapplications.com/content/24//495.