On the Frame Properties of Degenerate System of Sines

Abstract

Systems of sines with degenerate coefficients are considered in this paper. Frame properties of these systems in Lebesgue spaces are studied.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Maria Isabel Berenguer

Institute of Mathematics and Mechanics of NAS of Azerbaijan, B. Vahabzade 9, 1141 Baku, Azerbaijan

Received 27 June 2012; Accepted 17 August 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Basis properties of classical system of exponents _{{ ^{e int} } n ∈ ...} ( ... is the set of all integers) in Lebesgue spaces _{L p} ( - π , π ) , 1 ...4;p < + ∞ , are well studied in the literature (see [ 1- 4]). Bari in her fundamental work [ 5] raised the issue of the existence of normalized basis in _{L 2} which is not Riesz basis. The first example of this was given by Babenko [ 6]. He proved that the degenerate system of exponents _{{ ^{| t | α e int} } n ∈ ...} with | α | < ( 1 / 2 ) forms a basis for _{L 2} ( - π , π ) but is not Riesz basis when α ...0;0 . This result has been extended by Gaposhkin [ 7]. In [ 8], the condition on the weight ρ was found which make the system _{{ ^{e int} } n ∈ ...} forms a basis for the weight space _{L p , ρ} ( - π , π ) with a norm _{|| f || p , ρ} = ^{( ∫ - π π ... | f ( t ) | p ρ ( t ) dt ) ( 1 / p )} . Basis properties of a degenerate system of exponents are closely related to the similar properties of an ordinary system of exponents in corresponding weight space. In all the mentioned works, the authors consider the cases when the weight or the degenerate coefficient satisfies the Muckenhoupt condition (see, e.g., [ 9]). It should be noted that the above stated is true for the systems of sines and cosines, too.

Basis properties of the system of exponents and sines with the linear phase in weighted Lebesgue spaces have been studied in [ 10- 12]. Those of the systems of exponents with degenerate coefficients have been studied in [ 13, 14]. Similar questions have previously been considered in papers [ 15- 18].

In this work, we study the frame properties of the system of sines with degenerate coefficient in Lebesgue spaces, when the degenerate coefficient, generally speaking, does not satisfy the Muckenhoupt condition.

2. Needful Information

To obtain our main results, we will use some concepts and facts from the theory of bases.

We will use the standard notation. ... will be the set of all positive integers; ∃ will mean "there exist(s)"; [implies] will mean "it follows"; ... will mean "if and only if"; ∃ ! will mean "there exists unique"; K ...1; ... or K ...1; ... will stand for the set of real or complex numbers, respectively; _{δ nk} is Kronecker symbol, _{δ k} = _{{ δ kn } k ∈ ...} .

Let X be some Banach space with a norm _{|| · || X} . Then ^{X *} will denote its dual with a norm _{|| · || ^{X *}} . By L [ M ] , we denote the linear span of the set M ⊂X , and M ¯ will stand for the closure of M .

System _{{ x n } n ∈ ...} ⊂X is said to be uniformly minimal in X if ∃ δ >0 [figure omitted; refer to PDF]

System _{{ x n } n ∈ ...} ⊂X is said to be complete in X if L [ _{{ x n } n ∈ ...} ] ¯ =X . It is called minimal in X if _{x k} ∉ L [ _{{ x n } n ...0;k} ] ¯ , for all k ∈ ... .

The following criteria of completeness and minimality are available.

Criterion 1 (Hahn-Banach theorem).

System _{{ x n } n ∈ ...} ⊂X is complete in X if f ( _{x n} ) =0 , for all n ∈ ... ,f ∈ ^{X *} [implies]f =0 .

Criterion 2 (see [ 19]).

System _{{ x n } n ∈ ...} ⊂X is minimal in X ... it has a biorthogonal system _{{ f n } n ∈ ...} ⊂ ^{X *} , that is, _{f n} ( _{x k} ) = _{δ nk} , for all n ,k ∈ ... .

Criterion 3.

Complete system _{{ x n } n ∈ ...} ⊂X is uniformly minimal in X ... _{sup n || x n || X || y n || ^{X *}} < + ∞ , where _{{ y n } n ∈ ...} ⊂ ^{X *} is a system biorthogonal to it.

System _{{ x n } n ∈ ...} ⊂X is said to be a basis for X if for all x ∈X , ∃ ! _{{ λ n } n ∈ ...} ⊂K :x = ^{∑ n =1 ∞} ... _{λ n x n} .

If system _{{ x n } n ∈ ...} ⊂X forms a basis for X , then it is uniformly minimal.

Definition 2.1 (see [ 20, 21]).

Let X be a Banach space and ...A6; a Banach sequence space indexed by ... . Let _{{ f k } k ∈ ...} ⊂X , _{{ g k } k ∈ ...} ⊂ ^{X *} . Then ( _{{ g k } k ∈ ...} , _{{ f k } k ∈ ...} ) is an atomic decomposition of X with respect to ...A6; if

(i) _{{ g k (f ) } k ∈ ...} ∈ ...A6; , for all f ∈X ;

(ii) ∃A ,B >0 : [figure omitted; refer to PDF]

(iii) : f = ^{∑ k =1 ∞} ... _{g k} ( f ) _{f k} , for all f ∈X .

Definition 2.2 (see [ 20, 21]).

Let X be a Banach space and ...A6; a Banach sequence space indexed by ... . Let _{{ g k } k ∈ ...} ⊂ ^{X *} and S : ...A6; [arrow right]X be a bounded operator. Then ( _{{ g k } k ∈ ...} ,S ) is a Banach frame for X with respect to ...A6; if

(i) _{{ g k (f ) } k ∈ ...} ∈ ...A6; , for all f ∈X ;

(ii) ∃A ,B >0 : [figure omitted; refer to PDF]

(iii) : S [ _{{ g k (f ) } k ∈ ...} ] =f , for all f ∈X .

It is true the following.

Proposition 2.3 (see [ 20, 21]).

Let X be a Banach space and ...A6; a Banach sequence space indexed by ... . Assume that the canonical unit vectors _{{ δ k } k ∈ ...} constitute a basis for ...A6; and let _{{ g k } k ∈ ...} ⊂ ^{X *} and S : ...A6; [arrow right]X be a bounded operator. Then the following statements are equivalent:

(i) ( _{{ g k } k ∈ ...} ,S ) is a Banach frame for X with respect to ...A6; .

(ii) ( _{{ g k } k ∈ ...} , _{{ S ( δ k ) } k ∈ ...} ) is an atomic decomposition of X with respect to ...A6; .

More details about these facts can be found in [ 20- 23].

3. Completeness and Minimality

We consider a system of sines [figure omitted; refer to PDF] with a degenerate coefficient ν [figure omitted; refer to PDF] where 0 = _{t 0} < _{t 1} < ... < _{t r} = π are points of degeneration and ^{{ _{α k} } 0 r} ⊂R .

The notation f ~g , t [arrow right]a , means that the inequality 0 < δ ...4; | f ( t ) / g ( t ) | ...4; ^{δ -1} < + ∞ holds in sufficiently small neighborhood of the point t =a with respect to the functions f and g . Thus, it is clear that sin nt ~t , t [arrow right]0 and sin nt ~ π -t , t [arrow right] π for all n ∈ ... . Proceeding from these relations, we immediately obtain that the inclusion _{{ ^{S n ν} } n ∈ ...} ⊂ _{L p} ( 0 , π ) , 1 ...4;p < + ∞ , is true if and only if the following relations hold [figure omitted; refer to PDF] In what follows, we will always suppose that this condition is satisfied. Assume that the function f ∈ _{L q} ( 0 , π ) ( ( 1 / p ) + ( 1 / q ) =1 ) is otrhogonal to the system _{{ ^{S n ν} } n ∈ ...} , that is [figure omitted; refer to PDF] where ( · ¯ ) is a complex conjugate. By _{C 0} [ 0 , π ] , we denote the Banach space of functions which are continuous on [ 0 , π ] with a sup-norm and vanish at the ends of the interval [ 0 , π ] . It is absolutely clear that νf ∈ _{L 1} ( 0 , π ) ⊂ ^{C 0 *} [ 0 , π ] . As the system of sines _{{ sin nt } n ∈ ...} is complete in _{C 0} [ 0 , π ] , we obtain from the relations ( 3.4) that ν ( t ) f ( t ) =0 a.e. on ( 0 , π ) , and, consequently, f ( t ) =0 a.e. on ( 0 , π ) . This proves the completeness of system ( 3.3) in _{L p} ( 0 , π ) .

Now consider the minimality of system ( 3.3) in _{L p} ( 0 , π ) . It is clear that _{{ ^{S n ν -1} } n ∈ ...} ⊂ _{L q} ( 0 , π ) if and only if [figure omitted; refer to PDF] It is easily seen that the system _{{ ( 2 / π ) ^{S n ν -1} } n ∈ ...} is biorthogonal to _{{ ^{S n ν} } n ∈ ...} . So the following theorem is true.

Theorem 3.1.

System _{{ ^{S n ν} } n ∈ ...} is complete in _{L p} ( 0 , π ) , 1 ...4;p < + ∞ , if the relations ( 3.3) hold. Besides, it is minimal in _{L p} ( 0 , π ) if both ( 3.3) and ( 3.5) hold. Consequently, system _{{ ^{S n ν} } n ∈ ...} is complete and minimal in _{L p} ( 0 , π ) if the following relations hold: [figure omitted; refer to PDF]

It is known that (see, e.g., [ 10, 11]) if ^{{ _{α k} } 0 r} ⊂ ( - ( 1 / p ) , ( 1 / q ) ) , then system _{{ S n ( ν ) } n ∈ ...} forms a basis for _{L p} ( 0 , π ) , 1 <p < + ∞ . Let β ∈ [ ( 1 / q ) , ( 1 / q ) +1 ) , where either β = _{α 0} or β = _{α r} . In the sequel, we will suppose that the condition ( 3.6) is satisfied for ^{{ _{α k} } 1 r -1} . We have [figure omitted; refer to PDF] On the other hand [figure omitted; refer to PDF] where c >0 is some constant (in what follows c will denote constants that may be different from each other), δ >0 is such that [ 0 , δ ] does not contain the points ^{{ _{α k} } 1 r -1} . Let us show that _{inf n ∈ ... || ^{S n ν} || p} >0 . We have ( α = _{α 0} p +1 ) [figure omitted; refer to PDF] where _{M n} ...1; { k ...5;0 : ( k +1 ) π - ( π / 4 ) ...4;n δ } . Thus [figure omitted; refer to PDF] It is absolutely clear that [figure omitted; refer to PDF] Taking into account this relation, we obtain [figure omitted; refer to PDF] where _{λ n} ...5; ( n δ / π ) -2 . Consequently [figure omitted; refer to PDF] It follows immediately that _{inf n || ^{S n ν} || p} >0 .

Regarding biorthogonal system we get [figure omitted; refer to PDF] Choose [straight epsilon] >0 as small as the interval [ 0 , [straight epsilon] ] does not contain the points ^{{ _{α k} } 1 r -1} . Consequently [figure omitted; refer to PDF] where m >0 is some constant. We have [figure omitted; refer to PDF] First we consider the case _{α 0} ∈ ( ( 1 / q ) , ( 1 / q ) +1 ) . In this case, for sufficiently great n , we have [figure omitted; refer to PDF] Let _{α 0} = ( 1 / q ) . Consequently [figure omitted; refer to PDF] and, as a result [figure omitted; refer to PDF] where _{c i} are some constants. So we obtain that for β ∈ [ ( 1 / q ) , ( 1 / q ) +1 ) , _{sup n || ^{S n - ν} || q} = + ∞ . Consequently, in this case we have [figure omitted; refer to PDF] Then it is known that (see, e.g., [ 22]) the system _{{ S n ( ν ) } n ∈ ...} is not uniformly minimal and, besides, does not form a basis for _{L p} .

Consider the case β ∈ ( - ( 1 / p ) -1 , - ( 1 / p ) ] . Without limiting the generality, we will suppose that β = _{α 0} . In this case, with regard to the biorthogonal system we have [figure omitted; refer to PDF] Taking sufficiently small δ >0 , we obtain [figure omitted; refer to PDF] As _{α 0} q <0 , then, in the absolutely same way as in the previous case, we get [figure omitted; refer to PDF] On the other hand, [figure omitted; refer to PDF] where _{α 0} p ...4;1 . Similarly to the previous case again, we get [figure omitted; refer to PDF] As a result we obtain [figure omitted; refer to PDF] Thus, the following theorem is true.

Theorem 3.2.

Let { _{α 0} ; _{α r} } ⊂ ( - ( 1 / p ) -1 , - ( 1 / p ) +2 ) ; { _{α 0} ; _{α r} } _... ... ^{M p ( 0 )} ...0; ∅ , and _{A r} ⊂ ( - ( 1 / p ) , ( 1 / q ) ) , where ^{M p ( 0 )} ...1; ( - ( 1 / p ) -1 , - ( 1 / p ) ] _... ... [ - ( 1 / p ) +1 , - ( 1 / p ) +2 ) , _{A r} ...1; ^{{ _{α k} } 1 r -1} . Then the system _{{ ^{S n ν} } n ∈ ...} is complete and minimal in _{L p} ( 0 , π ) , 1 ...4;p < + ∞ , but does not form a basis for it.

4. Defective Case

Here, we consider the defective system of sines _{{ ^{S n ν} } n ∈ ... ( k 0 )} , where _{... ( k 0 )} ...1; ... \ { _{k 0} } , _{k 0} ∈ ... is some number. It follows directly from Theorem 3.2that if the condition [figure omitted; refer to PDF] holds, then the system _{{ ^{S n ν} } n ∈ ... ( k 0 )} is minimal but not complete in _{L p} ( 0 , π ) , 1 ...4;p < + ∞ . Assume ^{M p ( 1 )} ...1; [ ( 1 / q ) +1 , ( 1 / q ) +3 ) . Let _{α 0} ∈ ^{M p ( 1 )} . Consider the completeness of system _{{ ^{S n ν} } n ∈ ... ( k 0 )} in _{L p} ( 0 , π ) . Suppose that f ∈ _{L q} ( 0 , π ) is orthogonal to the system, that is, [figure omitted; refer to PDF] As νf ∈ _{L 1} ⊂ ^{C 0 *} [ 0 , π ] and system _{{ sin nt } n ∈ ...} is complete and minimal in _{C 0} [ 0 , π ] , from ( 4.2) we get [figure omitted; refer to PDF] It is clear that ^{ν -1} ( t ) ~ ^{t - _{α 0}} , sin _{k 0} t ~t as t [arrow right]0 . Consequently, f ~ ^{t - _{α 0} +1} , t [arrow right]0 . As q ( - _{α 0} +1 ) ...4; -1 , then f ∈ _{L q} ( 0 , π ) if and only if c =0 , and, consequently, f =0 . The similar result is true for _{α r} ∈ ^{M p ( 1 )} . Thus, if _{α 0} ; _{α r} > - ( 1 / p ) -1 , and max { _{α 0} ; _{α r} } ∈ ^{M p ( 1 )} , then the system _{{ ^{S n ν} } n ∈ ... ( k 0 )} is complete in _{L p} ( 0 , π ) . Now we consider the minimality of this system. Let [figure omitted; refer to PDF] We have [figure omitted; refer to PDF] Let us show that _{{ [vartheta] n } n ∈ ... ( k 0 )} ⊂ _{L q} ( 0 , π ) . In fact [figure omitted; refer to PDF] and, consequently [figure omitted; refer to PDF] From these relations, we immediately find that _{[vartheta] n} ( t ) ~ ^{t 3 - _{α 0}} , t [arrow right]0 . As a result, _{{ [vartheta] n } n ∈ ... ( k 0 )} ⊂ _{L q} ( 0 , π ) . Then the relations ( 4.5) imply the minimality of system _{{ ^{S n ν} } n ∈ ... ( k 0 )} in _{L p} ( 0 , π ) . Similar result is true for _{α r} ∈ ^{M p ( 1 )} . In the end, we obtain that if max { _{α 0} ; _{α r} } ∈ ^{M p ( 1 )} , then the system _{{ ^{S n ν} } n ∈ ...} has a defect equal to 1.

Consider the case when max { α 0 ; α r } = α 0 ∈ M p ( 2 ) , where M p ( 2 ) ...1; [ ( 1 / q ) +3 , ( 1 / q ) +5 ) . We look at the system { S n ν } n ∈ ... ( k 1 ; k 2 ) , where ... ( k 1 ; k 2 ) ...1; ... \ { k 1 ; k 2 } , k 1 ; k 2 ∈ ... , k 1 ...0; k 2 , are some numbers. Let f ∈ L q ( 0 , π ) cancel this system out, that is, [figure omitted; refer to PDF] Using the previous reasoning, we find that for some constants c 1 , c 2 , the following is true: [figure omitted; refer to PDF] Using representations [figure omitted; refer to PDF] we obtain [figure omitted; refer to PDF] ( c ...0;0 is some constant), where it can be easily seen that g 2 ∈ L q and [figure omitted; refer to PDF] Thus, f ∈ L q if and only if t - α 0 g 1 ( t ) ∈ L q . Assume b 1 = c 1 k 1 + c 2 k 2 ; b 2 = - ( 1 / 6 ) ( c 1 k 1 3 + c 2 k 2 3 ) . As ( 1 - α 0 ) q ...4; -1 ( 3 - α 0 ) q ...4; -1 , it is clear that t 1 - α 0 ∉ L q and t 3 - α 0 ∉ L q . Suppose b 1 ...0;0 . We have [figure omitted; refer to PDF] It follows directly that for sufficiently small δ >0 , we have [figure omitted; refer to PDF] where c δ >0 is some constant depending only on δ and b 2 . As a result, f ∉ L q .

Consequently, _{b 1} =0 . Moreover, it is not difficult to derive that _{b 2} =0 . Thus, we obtain the following system for _{c 1} and _{c 2} : [figure omitted; refer to PDF] It is clear that det | _{k 1 k 2} ^{k 1 3 k 2 3} | ...0;0 . And, consequently, _{c 1} = _{c 2} =0 . As a result, f =0 , which, in turn, implies that the system _{{ ^{S n ν} } n ∈ ... ( k 1 ; k 2 )} is complete in _{L p} . Let us show that it is also minimal in _{L p} . Assume ^{γ n ( k )} = ( 1 / 6 ) ( ^{k 2} - ^{n 2} ) . Consider the system ^{[vartheta] n ( _{k 1} ; _{k 2} )} ( t ) ...1; [ ^{a n ( _{k 1} ; _{k 2} )} sin nt - ( 1 / _{k 1} ^{γ n ( _{k 1} )} ) sin _{k 1} t + ( 1 / _{k 2} ^{γ n ( _{k 2} )} ) sin _{k 2} t ] ^{ν -1} ( t ) , for all n ∈ _{... ( k 1 ; k 2 )} , where [figure omitted; refer to PDF] Simple calculations give the following representation: [figure omitted; refer to PDF] where _{c n} ...0;0 , for all n ∈ _{... ( k 1 ; k 2 )} are some constants. We obtain directly from this representation that _{{ ^{[vartheta] n ( k 1 ; k 2 )} } n ∈ ... ( k 1 ; k 2 )} ⊂ _{L q} . On the other hand, [figure omitted; refer to PDF] Thus, if _{α 0} ∈ ^{M p ( 2 )} , then the system _{{ ^{S n ν} } n ∈ ... ( k 1 ; k 2 )} is complete and minimal in _{L p} , and, as a result, the system _{{ ^{S n ν} } n ∈ ...} has a defect equal to 2. It is easy to see that the similar result is true if _{α r} ∈ ^{M p ( 2 )} with _{α 0} ...4; _{α r} . Continuing this way, we obtain that if β =max { _{α 0} ; _{α r} } = _{α 0} ∈ ^{M p ( k )} , where ^{M p ( k )} ...1; [ - ( 1 / p ) +2k , - ( 1 / p ) +2 ( k +1 ) ) , then the system _{{ ^{S n ν} } n ∈ ... { n - k }} is complete and minimal in _{L p} , where { _{n - k} } = { _{n 1} ; ... ; _{n k} } ⊂ ... , _{n i} ...0; _{n j} with i ...0;j .

Consider the basicity of system _{{ ^{S n ν} } n ∈ ... { n - 1 }} (i.e., the case of k =1 ) in _{L p} . Similar to the case of ^{M p ( 0 )} , it can be proved that [figure omitted; refer to PDF] Concerning biorthogonal system, we have [figure omitted; refer to PDF] where the interval [ 0 , [straight epsilon] ] ( [straight epsilon] >0 ) does not contain the points ^{{ _{t k} } 1 r -1} . As ( sin _{n 1} x / _{n 1} ) -x ~ ^{x 3} , for x [arrow right]0 , it is clear that ( -q _{α 0} +3 > -1 ) [figure omitted; refer to PDF] Taking this circumstance into account, we have [figure omitted; refer to PDF] Consider the case of _{α 0} ∈ ( ( 1 / q ) +1 , ( 1 / q ) +3 ) : [figure omitted; refer to PDF] where α = - _{α 0} q +q +1 <0 . Consequently, _{sup n || ^{S n - ν} || q} = + ∞ . Let _{α 0} = ( 1 / q ) +1 . In this case we have [figure omitted; refer to PDF] and, consequently _{sup n || ^{S n - ν} || q} = + ∞ . As a result, we get that for _{α 0} ∈ ^{M p ( 1 )} , the system _{{ ^{S n ν} } n ∈ ... { n - 1 }} does not form a basis for _{L p} . Assume that in this case, the system _{{ ^{S n ν} } n ∈ ...} is a frame in _{L p} , that is any function from _{L p} can be expanded with respect to this system. As it does not form a basis for _{L p} , zero has a non trivial decomposition, that is, [figure omitted; refer to PDF] where ∃ _{n 0} ∈ ... : _{a n 0} ...0;0 . As the system _{{ ^{S n ν} } n ∈ ... { n - 1 }} is complete and minimal in _{L p} , it is clear that _{a n 1} ...0;0 . Consequently, ^{S _{n 1} ν} = _{∑ n ...0; n 1} ... ( _{a n} / _{a n 1} ) ^{S n ν} . It follows directly that the arbitrary element can be expanded with respect to the system _{{ ^{S n ν} } n ∈ ... { n - 1 }} . But this is impossible. Similar result is true for max { _{α 0} ; _{α r} } ∈ ^{M p ( 1 )} .

Proceeding in an absolutely similar way as we did in the previous case, we can prove that for max { _{α 0} ; _{α r} } ∈ ^{M p ( k )} , the system _{{ ^{S n ν} } n ∈ ... { n - k }} is complete and minimal in _{L p} , but does not form a basis for it. Consequently, system _{{ ^{S n ν} } n ∈ ...} has a defect equal to ( k ) . The fact that it is not a frame in _{L p} in this case too is proved as follows. Let k =2 : { _{n - k} } ...1; { _{n 1} ; _{n 2} } . Assume that the system _{{ S n } n ∈ ...} is a frame in _{L p} . Then zero has a non trivial decomposition: 0 = ^{∑ n =1 ∞} ... _{a n} ^{S n ν} . It is clear that | _{a n 1} | + | _{a n 2} | >0 , and let _{a n 1} ...0;0 . It follows directly that the system _{{ S n } n ...0; n 1} is a frame in _{L p} . The further reasoning is absolutely similar to the case of k =1 . This scheme is applicable for for all k ∈ ... . Thus, we have proved the following main theorem.

Theorem 4.1.

Let the following necessary condition be satisfied [figure omitted; refer to PDF]

Then the system _{{ ^{S n ν} } n ∈ ...} is a frame (basis) in _{L p} if and only if _{α 0} ; _{α r} ∈ ( - ( 1 / p ) , ( 1 / q ) ) . Moreover, for max { _{α 0} ; _{α r} } ∈ ^{M p ( k )} , k ∈ ... , it has a defect equal to ( k ) , where ^{M p ( k )} ...1; [ ( 1 / q ) +2k , ( 1 / q ) +2 ( k +1 ) ) .

Acknowledgment

The authors are thankful to the referees for their valuable comments.

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