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<creators>
<creator>
<creatorName></creatorName>
<affiliation>Саратовский государственный университет им. Н.Г. Чернышевского, Rusia</affiliation>
</creator>
</creators>
<titles>
<title xml:lang='en'>On Characterization of Bessel System</title>
</titles>
<publisher>Instrumentul Bibliometric National</publisher>
<publicationYear>2017</publicationYear>
<relatedIdentifier relatedIdentifierType='ISBN' relationType='IsPartOf'> 978-9975-76-247-2</relatedIdentifier>
<dates>
<date dateType='Issued'>2017</date>
</dates>
<resourceType resourceTypeGeneral='Text'>Conference Paper</resourceType>
<descriptions>
<description xml:lang='en' descriptionType='Abstract'><p>The new notion of ane system of Walsh type was introduced , studied and proved results about orthogonalizing and completion with preservation of structure of ane system in[1] .In [2] , we study an ane system of Walsh type generated by a periodic function in connection with multishifte in Hilbert space as well as we gave a necessary sucient conditions on an ane system of Walsh type to be complete sequence in the space L2[0; 1], nally we showed that an ane system is a fundamental sequence.In this work , we will give a necessary and suciently conditions on the function &#39; an ane system of Walsh type f&#39;ngn0 to be Bessel system in the space L2(0; 1). De nition(1):Let H be a Hilbert space,and W0;W1 : H ! H isometric operators operating in space H .Let&#39;s say that the two of isometrics W0 and W1 de nes the structure of multishifts , if there is a vector e 2 H such that : W 1 : : :W k?1 e; v 2 f0; 1g ; 0  v  k ? 1; k  0 Forms an orthonormal basis of the space H. The concepts of multishift introduced and studied in the works [3]-[5]. Suppose that , the function &#39;(s) , s 2 Re , ( where Re is a real number space ) , satis ed the condition : &#39;(s) 2 L2[0; 1]; R 1 0 &#39;(s)ds = 0; &#39;(s + 1) = &#39;(s) and let L20 = L20 (0; 1) be a space such functions (where, L20 is the space of square - integral and having a zero integral ), as well as , we denote a linear operators in this space as : W0&#39;(s) = &#39;(2s);W1&#39;(s) = r(s)&#39;(2s) (1) Where r(s) is the periodic function : Haar-Rademacher-Walsh. For any n 2 N , using the binary representation , n = Pk?1 v=0 v2v + 2k we set : &#39;n(s) = &#39; (s) = &#39;kj(s) = Wn&#39;(s) = W &#39;(s) = W 1 : : :W k&#39;(s) Where , k = 0; 1; :::; j = 0; 1; :::; 2k?1; = ( 1; : : : ; k) 2 ; &nbsp;= S1 k=0 f0; 1gk.Be sides , we set &#39;0(s)  1, W 1 : : :W k Denote the product of the operators : the operator W k acts rst , W 1 acts last , and the empty product is set the equal to the identity operator I .For any function &#39; 2 L20 , we have : &#39; (s) = W &#39;(s) = W 0 : : :W k?1&#39;(s) = &#39;(2ks)r k?1(2k?1s) : : : r 0 (s) = &#39;(2ks) Qk?1 v=0 r v v (s) Where , rk(s) = r(2ks); k = 0; 1; ::: is Rademacher system . De nition(2):The system f&#39;ngn0 = fW &#39;g is the ane system of Walsh type of the function &#39; without the constant &#39;0(s)  1 . If the generating function select !(s) = r(s) , then the system f!ng1 n=0 will the classical system of Walsh - Paley system .Walsh functions (without constant !0(s)  1): !n(s) = ! (s) = W !(s) = W 0 : : :W k?1!(s) = rk(s) Qk?1 v=0 r v v (s) Forms an orthonormal basis of the space H = L20 (0; 1) , therefore according to the de nition(1) operators :W0&#39;(s) = &#39;(2s);W1&#39;(s) = r(s)&#39;(2s) , de ne the structure of multishift , [2]. Theorem(1):Let &#39; 2 L2(0; 1) , supp&#39;  [0; 1] , R 1 0 &#39;(s)ds = 0 . If the inequality : P1 k=0 P2k?1 j=0 j(&#39;; !kj)j2 1=2 = c  1 Then , the ane system of Walsh type f&#39;ngn0 is Bessel system with Bessel constant B = max f1; cg2.</p></description>
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