On Characterization of Bessel System

Transcription

1 Global Journal of Pure and Applied Mathematics. ISSN Volume 14, Number2 2018, pp Research India Publications On Characterization of Bessel System Al-jourany Khalid Hadi Hameed Ph.D. student, Department of Functions and Approximations Theory, Faculty of Mathematics and Mechanics, Saratov State University, Russia. Abstract In the previous work, we have studying an affine system of Walsh type generated by a periodic function in connection with multishift in Hilbert space. In this paper, we give a new method for characterization of Bessel system. This method is based on consideration the question: under which necessary and sufficiently condition on the function f an affine system of functions of the Walsh type {f n } n 0 to be Bessel system in the space L 2 0, 1. Finally, some examples are given to explain our representation method. AMS subject classification: Keywords: Affine system of Walsh type, Rademacher system, Walsh-Paley system, Riesz bases, Bessel system. 1. Introduction The new notion of affine system of Walsh type was introduced, studied and proved results about orthogonalizing and completion with preservation of structure of affine system by Terekhin P.A. [1]. Our results work in [2] about affine system of Walsh type can be classified in to three sections results: first, on the basis of the functional analytic structure of a multishift in a Hilbert space, which is a generalized analogue of the operator simple, one - side shift and closely related to the representations of the Cuntz C -algebra, the definition of an affine system of functions of the Walsh type was given, second, various criteria and signs of the completeness of affine systems of functions were given, finally, the minimality of the affine system is established as well as an explicit form of the biorthogonally conjugate system of functions was indicated and its completeness was established. Mironov V.A., Sarsenbi A.M., and Terekhin P.A., [10] were studied an affine Bessel sequences in connection with the spectral theory and the multishift structure in Hilbert space. They constructed a non-besselian affine system

2 254 Al-jourany Khalid Hadi Hameed {u n x} n=0 generated by continuous periodic function ux. Their results were based on Nikishin s example concerning convergence in measure, also they showed that affine systems {u n x} n=0 generated by any Lipchitz function ux are Besselian. In this paper, we will give the necessary and sufficient condition for the function f an affine system of functions of the Walsh type {f n } n 0 to be Bessel system in the space L 2 0, 1. Definition 1.1. Let H be a Hilbert space, and W 0,W 1 : H H isometric operators operating in space H. Let s say that the two of isometrics W 0 and W 1 defines the structure of multishifts, if there is a vector e H such that: W α1...w αk 1 e, α v {0, 1}, 0 v k 1,k 0 Form an orthonormal basis of the space H. The concepts of multishift introduced and studied in many works [3 5]. Consider an example of the structure of a multishift in a Hilbert space H = L 2 0 0, 1, consisting of all 1-periodic functions fx, x R, such that f L 2 0, 1 and For f L 2 0 0, 1 we set W 0 fx= f2x, W 1 fx= rxf2x, 1 0 fxdx = 0. where, rx = 1 for each x [m, m+ and rx = 1 for each x [m+,m+1, m Z, is a periodic Haar-Rademacher-Walsh function. Theorem 1.2. [2] The operators W 0 and W 1 are isometric, their images are orthogonal. The space H = L 2 0 0, 1 decomposes into a direct orthogonal sum H = R W 0 H W 1 H, where, R =[r] is the one-dimensional subspace spanned by the function r. We denote A = {0, 1} k the family of all finite sequences α = α 0,...,α k 1, consisting of zeros and ones: α ν {0, 1},0 ν k 1including the empty sequence for k = 0. We put α the length of a sequence α = α 0,...,α k 1 A, i.e. α =k for α = α 0,...,α k 1 the length of any empty sequence is set to zero. Let αβ, the concatenation of sequences α, β A: ifα = α 1,...,α k and β = β 1,...,β l, then αβ = α 1,...,α k,β 1,...,β l. Let us point out the natural one-to-one correspondence between the set of natural numbers N and the family A. Consider the binary expansion k 1 n = α ν 2 ν + 2 k ν=0

3 On Characterization of Bessel System 255 of the number n.the collection α = α 1,...,α k A is assigned to the natural number n. By using the sequence α = α 1,...,α k A, it is convenient to write the product of operators W α = W α0...w αk 1, α = α 0,...,α k 1, Denote the product of the operators: the operator W αk 1 acts first, W α0 acts last for k = 0 the empty product is equal to the identity operator I. For any function f L 2 0 and for each n N, we have: k 1 f n x = f α x = W α fx= r α 0 x...r α k 1 2 k 1 xf2 k x = f2 k x r α ν ν x, where, r k x = r2 k x, k 0, Rademacher functions. In addition, let f 0 x 1. Definition 1.3. The sequence of functions {f n x} n=0 is called an affine system of Walsh functions, generated by the function fx. If the generating function select wx = rx, then the system {w n } n=0 will the classical system of Walsh-Paley system. Walsh functions without constant w 0 t 1: k 1 w n x = w α x = W α wx = W α0...w αk 1 wx = r k x r α v v x n N Form an orthonormal basis of the space H = L 2 0 0, 1, therefore according to the definition 1 operators: W 0 and W 1, define the structure of multishift. Definition 1.4. [6]The Walsh-Paley system, w = w n,n N is defined as: if n = n k 2 k N {0} has binary coefficient n k,k N {0}, then Where rx = w n = { 1, x 0, 1, x, 1 v=0 ν=0 r n k k 1.1 r x + k = r x, x 0, 1, k N and r k x = r2 k x, x R, k N {0}, where rx is the Rademacher function. Definition 1.5. A system sequence {f n } n N in Hilbert space H is called a Bessel system, if there exists a positive constant B for which g, f n 2 B g 2 g H 1.2 n=1

4 256 Al-jourany Khalid Hadi Hameed Definition 1.6. [7] Let n 0, the Cuntz algebra n is the C -algebra generating by some isometries S i i Zn satisfying the Cuntz relations: Where i, j Z n. S i S j = δ ij I, i Z n S i S j = I 1.3 It should be noted that, the extensions of operators {W 0,W 1 } to the space L 2 0, 1 of periodic function fxare defined by: V 0 fx= f2x, V 1 fx= rxf2x 1.4 From equation 4, we have representation of the Cuntz algebra 2, which satisfy the Cuntz relations: Vi V j = δ ij I V 0 V 0 + V 1V 1 = I Thus, the operators structure of the multishift {W 0,W 1 } is a restriction to the subspace L 2 0, 1 of the representation {V 0,V 1 } in the space L 2 0, 1 of the Banach C -algebra of Cuntz The Main Results with Examples Lemma 2.1. For all α, β A, we have: wα,f β = { wα,f, ifα = βγ 0, o.w. Proof. Write the Fourier-Walsh series of the function f as: f = γ A f, wγ wγ Also, we have: f, wγ W β w γ = f, wγ wβγ f β = W β f = γ A γ A On other hand f β = γ A fβ,w α wβα, α = βγ The coefficient of the Fourier-Walsh series are uniqueness. Also, if α = βγ for some γ A, then fβ,w α = f, wγ

5 On Characterization of Bessel System 257 It should be noted that, if α can not be expressed as βγ, γ A, then fβ,w α = 0 Theorem 2.2. Let f L 2 0, 1, suppf [0, 1], 2 k 1 f, w kj 2 j=0 1 0 fxdx = 0. If the inequality: = c Then the affine system of Walsh type {f n } n 0 is Bessel system with Bessel constant B = max {1,c} 2. Proof. Write the Fourier-Walsh series of the function f as: f = α A x α w α and write the polynomial of affine system {f n } n 1 finite sum as: P = c β f β We consider for k = 0, 1,..., the Walsh-Paley polynomials can be represented as: P k = x α c β w βα The system { w βα : α = kk fixed,β A } is orthogonal system. w βα = w β α, α = k, β α A,α = α,β = β,β A. Now, if βα = β α, then: α + β = α β α β +, α = and β =,α = α andβ = β. We can count: P k =, x α c β 2 = x α 2 c β 2

6 258 Al-jourany Khalid Hadi Hameed And P k = c β 2 We calculate: P,wγ = c β fβ,w γ =. α,β:γ =βα x α 2 c β f, w α = α,β:γ =βα By using Lemma 1. P k,w γ = Pk,w γ = x α c β wβα,w γ = since w βα,w γ = δβα,γ Now: From the above, we have the following induction: P = we have: Where, P P k = P k! x α 2 c β f β f cβ 2 f = x α 2. It is equivalent to Bessel inequality: g, f β 2 f g 1 2 g, f n 2 gtdt + g, f β 2 0 cβ 2 g, f n 2 B g 2, B = max { 1, f } 2 n=0 x α c β α,β:γ =βα x α c β max { 1, f }. g 2

7 On Characterization of Bessel System 259 Then, we have: if f = x α 2 Then the affine system of Walsh type {f n } n 0 is Bessel system. Remark 2.3. Theorem 2 in this paper is an analog of some results obtained by the authors in [8 10]. We are going to give some examples to apply theorem 2. These examples are based on consideration that: H is the Banach algebra of analytic functions on the open unit disk and GH is the group of invertible elements of the algebra H. Note that for ζ to be belong to GH, it is necessary and sufficient that the function ζz be analytic on the disk z 1 and that the following inequalities be valid: 0 inf ζz,sup ζz Let f RH, where RH is the space of Rademacher. Let RH = span[r k ]be linear closure of the span Rademacher system {r k }. The space RH invariant with respect to W 0 and the multishift operator RH as: r k = W0 k r, k = 0, 1,... And fx= We assign the analytic function: a k r k, a k φz = a k z k 2.2 In the unite disk D = z 1 of Hardy space H 2 D, with the coefficient a k from equation 5. This mapping is an isometric isomorphism of RH on to Hardy space H 2 D, and the restriction of W 0 to RH is unitary equivalent by this mapping to the operator of multiplication by z, i.e. is a shift operator. Theorem 2.4. [11] Let {w n } n 0 be the Walsh system, {r k } k 0 be the Rademacher system and f = a k r k, a k 2 If the analytic function φz = a k z k, z 1

8 260 Al-jourany Khalid Hadi Hameed belong to GH, then the affine system of Walsh type {f n } n 0 is Riesz bases in L 2 0, 1. Theorem 2.5. Affine system of Walsh type {f n } n 0 is Riesz bases iff 0 c 1 φz c 2 where, is analytic function. φz = a k z k, z 1 Example 2.6. The analytic function φz = a k z k, z 1 has no zero in the closed unit circle, then affine system of Walsh type {f n } n 0 form Riesz bases and since any Riesz bases is Bessel system, then affine system of Walsh type {f n } n 0 form Bessel system too. Indeed of Wiener theorem an absolutely convergent series Tayler follows that φ GH. Example 2.7. Let fx = 1 2x, 0 x 1 and satisfy the following condition 1 0 fxdx = 0.f RH, this meaning that, the function f can be representation as Rademacher system: f = a k r k = r k 2 k+1 Then the corresponding analytic function as: φz = a k z k z k = 2 k+1 = 1 2 z It is observe that, φ GH, then, we have the affine system of Walsh type {f n } n 0 form Riesz bases and Bessel system too. References [1] Terekhin, P. A., Affine Systems of Walsh Type. Orthogonalization and Completion // Izv. Saratov Univ. N.S., Ser. Math. Mech. Inform., 2014, vol. 14, no. 4, pp in Russia [2] Khalid, H.H., Mironov V.A., and Terekhin P.A., Affine System of Walsh type. Completeness and Minimality // Izv Saratov UniversityN.S. Ser. Math., Mech., Inform., 2016, vol. 16, no. 3, pp in Russia

9 On Characterization of Bessel System 261 [3] Terekhin, P.A., On representation properties of a system of contractions and shift of functions on an interval // Izv Tul sk.gos. Univ., Ser. Math., Mech., Inform., 1998, vol. 4, no. 1, pp in Russia [4] Terekhin, P.A., On the multiplicative structure of the centeralizer of a multishift on a Hilbert space // Mathem., Mech.: Collection of Scientific Papers, Saratov, Saratov University., Press 2000, 2, pp in Russia [5] Terekhin, P.A., Multishifts in Hilbert spaces // Functional Analysis and Its Applications, 2005, vol. 39, no. 1, pp DOI: /faa32. [6] Schipp, F. W. R., Simon P., Pal J. Walsh series: an introduction to dyadic harmonic analysis // Bristol; N. Y. : Adam Hilger, [7] Joachim, C., Simple C -algebras generated by isometries // Comm. Math. Phys., 1977, vol. 57, no. 2, pp [8] Terekhin, P. A., Riesz bases generated by contractions and translations of a function on an interval // Mathematical Notes, 2002, vol. 72, no. 4, pp DOI: /mzm444. [9] Sarsenbi, A. M., Terekhin, P. A., Riesz basicity for general systems of functions // Journal of Function Spaces, V Article ID PP article ID DOI: /2014/ [10] Mironov, V. A., Sarsenbi, A. M., Terekhin, P. A., Affine Bessel sequences and Nikishin s example // Filomat 31:41, P DOI /FIL M. [11] Khalid, H.H., Terekhin, P.A., On construction of Riesz bases using Walsh type affine systems in the space L 2 0, 1 // Mathem., Mech.: Collection of Scientific Papers, Saratov, Saratov University. Press 2016, vol. 18, pp in Russia