Abstract

A scalar distribution function is called a spectral function for the Fourier transform (with respect to an interval ) if for each function with support in the Parseval identity holds. We show that in the case there exists a unique spectral function , in which case the above Parseval identity turns into the classical one. On the contrary, in the case of a finite interval , there exist infinitely many spectral functions (with respect to ). We introduce also the concept of the matrix-valued spectral function (with respect to a system of intervals ) for the vector-valued Fourier transform of a vector-function , such that support of lies in . The main result is a parametrization of all matrix (in particular scalar) spectral functions for various systems of intervals .

1. Introduction

Recall that according to the Plancherel theorem (see, e.g., [1, Ch.3.37]) the Fourier transformof a function satisfies the Parseval identityand the inverse Fourier transform is

As is known a number of classical problems (moment problems, eigenfunctions expansion of differential operators, and their various modifications) are reduced to searching for the distribution functions (“spectral functions”) such that the Stieltjes integral with a given family of functions satisfies certain preassigned conditions. The problem of such a kind for the Fourier transform (1) can be formulated as follows. Assume that , is an interval in and let be the set of all functions with support in . It is necessary to describe all distribution functions such that instead of (2) the following Parseval identity holds:A function satisfying (4) will be called a spectral function for the Fourier transform (1) (with respect to ); the set of all such spectral functions will be denoted by . If , then for each the inverse Fourier transform iswhere is the indicator of . Clearly formulas (1), (4), and (5) give a natural generalization of the classical Plancherel theorem. We have not seen such a generalization in the literature.

A parametrization of the set in the case of a bounded interval is given by the following theorem, which directly follows from Corollary 18 of the paper.

Theorem 1. Let , , be a bounded interval in . Then the equalitiesestablish a bijective correspondence between all holomorphic functions , with and all (scalar) spectral functions .

In the case the function is meromorphic with poles on , is a jump function, and equalities (1) and (5) give an expansion of a function into the Fourier series on . In the case one has and equality (5) turns into the classical inverse Fourier–Plancherel transform (3) of a function with support on . Moreover, according to Theorem 1 there exist infinitely many spectral functions . At the same time we show that in the case the set consists of the unique spectral function , for which equality (5) turns into the inverse Fourier–Plancherel transform (3) of a function . Similar situation takes place in the case (see Corollary 22).

Actually we study a more general object—a matrix-valued spectral function for the vector-valued Fourier transform. Namely, assume that is a system of intervals , . Denote by the set of all vector-functions such that and support of a coordinate function lies in . For each function equality (1) defines the vector-valued Fourier transform of . A matrix-valued distribution function will be called a spectral function for the vector-valued Fourier transform (1) (with respect to ) if the following Parseval identity holds: We denote the set of all such spectral functions by . If , then for each the inverse Fourier transform is where (see Proposition 12).

In the paper we parameterize spectral functions for various classes of . To illustrate the obtained results we specify below two theorems, which give a parametrization of the classes () and . These theorems are immediate from Corollary 19 and Theorem 21 of the paper.

Theorem 2. Assume that and let where is the upper half-plane of the complex plane. Then the equality together with formula (7) gives a bijective correspondence between all holomorphic contractive matrix functions , and all spectral functions .

Theorem 3. Let be the set of all complex-valued functions on admitting the representation with a holomorphic function defined on and satisfying , . Then the equalitiesgive a bijective correspondence between all functions and all spectral functions .

These theorems show that the sets of spectral functions differ significantly for systems formed by finite or infinite intervals. In particular, by Theorem 3 each spectral function is absolutely continuous with the matrix density defined by the first equality in (14).

In conclusion note that the results of the paper on spectral functions for the classical Fourier transform are obtained with the aid of the results from [24] concerning spectral functions for canonical differential systems.

2. Preliminaries

2.1. Notations and Definitions

In the following and are semiaxis of the real axis, is an upper half-plane of the complex plane, , is the Kronecker delta (i.e., if and if ), and is the set of all linear bounded operators from the Hilbert space to the Hilbert space , .

We denote by the set of all holomorphic operator-functions such that (the latter means that the function is contractive). Moreover, we put and (that is, is the set of all scalar contractive holomorphic functions on ). If , then according to the Fatou’s theorem there exists the limitWe denote by the set of all functions admitting the representation (15) with some function . Clearly, each function is contractive and Borel measurable.

Let , be an open interval in . We denote by the Hilbert space of Borel measurable functions such that .

Recall that a nondecreasing left-continuous operator-function with is called a distribution function. For each distribution function there exists a Borel measure on and an operator-function such that (a.e. on ) and , . Moreover, a linear space of all Borel measurable functions satisfying is a Hilbert space with respect to the inner productdefined for all (see, e.g., [5, Ch.3.15]).

A distribution function is called absolutely continuous if there exists a locally integrable operator-function (the density of ) such that . As is known, in this case (a.e. on ).

Remark 4. Let be the set of all -matrices with complex entries. As is known each operator can be identified with its matrix in the standard bases of and . Hence a (holomorphic) operator-function with values in can be identified with a (holomorphic) matrix function with values in , which implies that the class is in fact the well-known Schur class of holomorphic in contractive matrix functions with values in . Nevertheless it is more convenient to use the operator terminology in our considerations (of course, all final results of the paper can be readily translated into the matrix language).

Let be a closed densely defined symmetric operator in with the domain and let , , be the defect subspace of . We will need the following definitions.

Definition 5 (see [6]). A collection consisting of a Hilbert space and linear mappings , , is called a boundary triplet for if the mapping is surjective and the following Green’s identity holds:

Definition 6 (see [7, 8]). The operator-function defined byis called the Weyl function of the boundary triplet for .

2.2. Nevanlinna Functions and Nevanlinna Pairs

As is known a holomorphic operator-function is called a Nevanlinna function (and is referred to the class ) if . Recall also (see, e.g., [9]) that a pair of holomorphic operator-functions , is called a Nevanlinna pair if and . Two Nevanlinna pairs and are said to be equivalent if , with some holomorphic operator-function , such that the operator is invertible for all . Clearly, the set of all Nevanlinna pairs falls into nonintersecting classes of equivalent pairs (the equivalence classes). In the following we do not distinguish equivalent Nevanlinna pairs. Moreover, we denote by the equivalence class generating by the Nevanlinna pair and by the set of all such equivalence classes. Identifying of a function with a pair enables one to consider the class as a subclass of .

Definition 7 (see [3]). Let be a subspace in and let . A pair is referred to the class if .

Clearly, .

A parametrization of the class in terms of contractive operator-functions is given by the following proposition.

Proposition 8 (see [3]). Let be a subspace in and let .Then the equalities give a bijective correspondence between all operator-functions with the block representation and all pairs .

2.3. Canonical Systems

As is known (see, e.g., [10, 11]) a canonical differential system of an order on the semiaxis is of the formwhere is an operator satisfying and is a Borel measurable operator-function integrable on each bounded interval and satisfying (a.e. on ). In the following we assume also that is invertible a.e. on .

Denote by the Hilbert space of all Borel measurable functions satisfying and by the set of all functions ) such that a.e. on , where depends on . Since is invertible a.e. on , system (20) gives rise to the maximal operator and the minimal operator in defined as follows. Let be the set of all functions , which are absolutely continuous on each compact subinterval . Then the domain of is and . The minimal operator is the closure of the operator defined as a restriction of onto the set of all functions with compact support. It turns out that is a closed densely defined symmetric operator, , and there exists the limit(see, e.g., [12]). Clearly, equality (21) defines a skew-Hermitian bilinear form on .

For each there exists a unique locally absolutely continuous operator-function such that (a.e. on ) and . The operator-function is called the matrizant of system (20).

In the following we denote by the deficiency indices of the system (20), i.e., deficiency indices of . Clearly, , where is the linear space of all vector solutions of (20) belonging to . Hence . Moreover, if , then and hence for any the equalitydefines a linear operator .

With each function one associates the function (the generalized Fourier transform of ) given by

Definition 9. A distribution function is called a spectral function of system (20) if the following Parseval identity holds:

Below within this subsection we assume that (hence ) and the operator in (20) isAssume that system (20) has the maximal deficiency index and let . Then , and by [3, Lemma 4.1] there exists an operatorsuch that(here is the orthoprojector in onto ).

A description of all spectral functions of system (20) is given by the following theorem, which directly follows from Theorem 4.17 and formulas in [3].

Theorem 10. Let system (20), (25) satisfies , let , let , let be the operator (26) satisfying (27), (28), and let be the operator (22). Moreover, let be an operator-function (the monodromy matrix) defined by , let

be the block representation of , and letThen the equalitiesestablish a bijective correspondence between all pairs and all spectral functions of the system.

3. Spectral Functions for the Fourier Transform

3.1. Definitions and Auxiliary Results

Letbe a system of intervals , , . Denote by the set of all functions with a.e. on and by the set of all functions with the following property: if is an infinite interval, then there exists a finite interval (depending on ) such that a.e. on . Since is unitarily equivalent to the Hilbert space , it follows that is a closed subspace in . Moreover, is a linear manifold dense in .

With each function one associates the function (the classical vector-valued Fourier transform of ) given byAssume that is an operator-valued distribution function such that for each the following Parseval identity holds:Then for each the integral in (37) converges in the norm of and the equality , defines an isometry from to .

Definition 11. A -valued distribution function such that (38) holds will be called a spectral function for the classical Fourier transform (37) (with respect to the system of intervals ).

In what follows the set of spectral functions in the sense of Definition 11 will be denoted by . Moreover, in the case when and the system consists of a unique interval , we let .

Proposition 12. Let be system of intervals (36), let be the indicator of an interval , let , and let . Then the inverse to (37) transform iswhere the integral converges in the norm of . If in addition is absolutely continuous and is the density of , then Parseval identity (38) and the inverse transform (39) can be written as

Proof. Since is an isometry, to prove (39) it is sufficient to show thatwhere is the set of all functions with compact support.
Let , let be given by (for and see assertions before (16)), and let . Let us show that for each finite interval Indeed, let . Then by using the Fubini theorem and (16) one gets which proves (44). Since (a.e. on ), it follows from (44) that (a.e. on ) and hence (42) holds.
The last statement of the proposition is obvious.

Remark 13. If and system consists of a unique interval , then equality (37) defines the classical -valued Fourier transform of a scalar function with compact support belonging to (we denote the set of such functions by ). A spectral function for this transform is a scalar distribution function such that the Parseval identity holds; moreover, the inverse Fourier transform is

Assume that , , , and is a system of intervals given by(here ). With the system one associates canonical differential systemwhereIn (50) , and , are continuous functions on such that , , , andOne can easily verify that the matrizant of system (49) iswhereAssume that , , is the standard basis in and letSince by (53) and are columns of the matrizant , the system is a basis in the space of all solutions of system (49). Moreover, the immediate checking shows thatfor andfor . Therefore deficiency indices of system (49) areNow assume that is system of intervals (48) with and let , so that Moreover, let and be mutually adjoint unitary operators in defined byTogether with canonical system (49), (50) we associate with system of intervals canonical system (20), where is of the form (25) andClearly, admits the representation and the immediate checking shows thatIn the following we denote by and the maximal operators induced by systems (20) and (49), respectively. It follows from (63) and (64) that the equalitydefines a unitary operator from onto such that andMoreover, the matrizant of system (20) iswhere is the matrizant (53) of system (49). Observe also that systems (20) and (49) have the same deficiency indices and by (60) deficiency indices of the system (20) are

Proposition 14. Let be system of intervals (48). Then the set coincides with the set of spectral functions of system (49), (50). If in addition , then the equalitygives a bijective correspondence between all spectral functions of system (20), (63) and all spectral functions .

Proof. Let , , and . Moreover, let , , and , . Clearly, () is a strictly increasing absolutely continuous function mapping bijectively onto (resp. ). With each vector-functionwe associate the vector-functiongiven by (here and ). Since and , it follows that and hence Thus, the equality gives a bijective correspondence between all functions and all functions . Moreover, .
Next assume that and are vector-functions (70) and (71) and let be the corresponding Fourier transforms (23) and (37) (in (23) and should be replaced with and , respectively). Then by (53) and (50) and, consequently, . This yields the first statement of the proposition.
If , then by (63) and (67) the equality (69) gives a bijective correspondence between all spectral functions of system (20), (63) and all spectral functions of system (49), (50). This and the first statement of the proposition yield the second statement.

Below unless otherwise stated we assume that is system of intervals (48) with and . This means thatand according to (68) deficiency indices of the respective system (20) are

Lemma 15. Let , be the standard basis in , let and letwhere and are given by (54). Then and the equalitiesdefine linear mappings and such that the mapping is surjective and the identityis valid. Moreover, ifis the block representation of , then the equalitiestogether with (26) define the operator such that (27) and (28) hold.

Proof. LetThen by (55) , , and in view of (56) and (58) . Since and , it follows thatwhere is the unitary operator (65). Hence .
Clearly,This and (66), (87) yield Since by (51) and (52) , , and , , it follows thatNext, denote by the kernel of the form . According to [13, Proposition 5.5] the indices of inertia of the form are and (78) yields . Therefore by (90) and (91) system forms a basis in and hence each admits a unique representation where and . Let and be the operators given byThen by (90) and (91) for any one has Moreover, by (93) the operator is surjective. Note also that by (90) and (91) , , and hence (93) admits the representation (80). This proves the first statement of the lemma.
Now assume that and are the operators (83)–(85), is given by (26), and is the orthoprojector in onto . Then which in view of surjectivity of yields (27). Finally, for any one has This proves (28).

3.2. Parametrization of Spectral Functions

A parametrization of all spectral functions is given by the following theorem.

Theorem 16. Assume that , , , , and is a system of intervals (77). Let , let and be operator (matrix) functions given byand let(in formulas (97)–(104) ). Then the equalitiesestablish a bijective correspondence between all pairs and all spectral functions .

Proof. First by using Theorem 10 we parameterize spectral functions of system (20) with of the form (25) and defined by (63) and (50).
Let be the operator defined by (83)–(85) and (26) and let be the matrizant of system (20). Since (see (78)) and by Lemma 15 satisfies (27) and (28), the equality , , defines the monodromy matrix . To calculate we first calculate the operators and (here and are operators (80)). Let be given by (86) and let and be operators defined by Moreover, let be the matrizant (53) of system (49), (50), let be the respective operator (22), and let , . Since and (see (55)) are columns of the matrizant (53), it follows that Moreover, by (88) And, consequently, Hencewhere It follows from (87) and (66) that , where is the unitary operator (65). This and (67) imply that , , and, consequently, (here are entries of the block representation (82) of ). Therefore by (83)–(85) the monodromy matrix is whereand the equalities (30)–(33) take the formAccording to Theorem 10 equalities (34) and (35) with of the form (119)–(122) give a bijective correspondence between all pairs and all spectral functions of system (20). Combining of this statement with Proposition 14 implies that equality (105) with , and the equality (106) establish a bijective correspondence between all pairs and all spectral functions . Moreover, the immediate calculation shows that are of the form (101)–(104).

In the case spectral functions can be parameterized in a somewhat other form. Namely, the following theorem is valid.

Theorem 17. Assume that , , , , andwhere . Let and let (in formulas (125)–(128) ). Then the equalitiesestablish a bijective correspondence between all contractive operator-functions and all spectral functions .

Proof. Clearly, system of intervals (123) is system (77) with and . Therefore the operator-functions (see (97)–(100)) are , , and hence where are given by (101)–(104). Moreover, the subspace in Theorem 16 coincides with . Therefore and according to Proposition 8 the equalitiesgive a bijective correspondence between all operator-functions and all pairs . Substituting (132) into (105) one gets Thus (105) admits the representation (129), which in view of Theorem 16 yields the required statement.

3.3. Particular Cases

In this subsection by using Theorems 16 and 17 we obtain parametrization of spectral functions in certain particular cases.

Corollary 18. Assume that , andThen the equalitytogether with (130) gives a bijective correspondence between all operator-functions and all spectral functions .

Proof. Clearly, system of intervals (134) is system (123) with , , and hence the corresponding operator-functions (see (125)–(128)) areLet and let be given by (129) with instead of . It follows from (137) and (138) that whereTherefore , where is given by (136) with of the form (140).
Let and be operators in defined by Then and according to [14] equality (140) gives a bijection of the set onto itself. This and Theorem 17 yield the desired statement.

Corollary 19. Assume that , , and(). Let and be given by (124) and let be the operator-functions defined by the block representations Then statement of Theorem 17 holds with the equalityinstead of (129).

Proof. Since system of intervals (142) is system (123) with , the operator-functions (see (125)–(128)) are Therefore equality (129) admits the representation (144), which proves the required statement.

Proposition 20. Assume that , ,, and is given by (135). Then the equalities together with (130) establish a bijective correspondence between all operator-functions with the block representation , and all spectral functions .

Proof. Clearly, system of intervals (146) is system (77) with and . Therefore for this system Theorem 16 holds with , , , and where . Moreover, the subspace in Theorem 16 is and according to Proposition 8 the equalitiesgive a bijective correspondence between all operator-functions with the block representation and all pairs .
It follows from (152) and (150), (151) that and substitution of these equalities into (105) gives where is of the form (147) and Thus is of the form (149), which in view of Theorem 16 proves the required statement.

Theorem 21. Assume that and is a system of intervals given byThen the equalitiesgive a bijective correspondence between all operator-functions and all spectral functions .

Proof. Clearly, system of intervals (156) is system (48) with and . Therefore , , and (62) takes the formConsider canonical system (20) associated with (recall that and in this system are given by (25) and (63), respectively). Clearly, ; that is, system (20) is a Hamiltonian system. Moreover, by (68) deficiency indices of this system are . Therefore according to [15] the equalities define a boundary triplet for (in (159) and are components of the representation , ).
Let be the Weyl function of the triplet . Since is a densely defined operator, it follows from [16, Corollary 4.8] that the equality () together with (35) gives a bijective correspondence between all pairs and all spectral functions of system (20). Therefore by Proposition 14 the equality together with (106) gives a bijective correspondence between all pairs and all spectral functions .
Let us calculate . Assume that , , , be vector-functions defined by the second equality in (55) and let (this means that , where is the unitary operator (65)). It follows from (57) that is a basis in . Let be the operator solution of system (20) given by Clearly, and the equality , , defines a bijective linear mapping from onto . It follows from (159) and (163) that Hence , and by definition (18) of the Weyl function one hasMoreover, according to Proposition 8 the equalities give a bijective correspondence between all operator-functions and all pairs . Substituting (165) and (166) into (160) one obtains and the immediate calculation with taking (158) and (161) into account givesThus equalities (168) and (130) give a bijective correspondence between all operator-functions and all spectral functions . Next, in view of (168) one has Since for each finite interval the function is bounded on , it follows from the Lebesgue theorem that admits the representation , where is given by the first equality in (157) with . Letting now , we obtain the required statement.

Corollary 22. Let either or . Then the set consists of the unique spectral function , .

Proof. First assume that . Let , let (that is, is given by the right hand side of (156)), and let in (157) . Then by Theorem 21 the equality defines a spectral function . This implies that .
Conversely, let . Then the equality defines a spectral function . Clearly, with (see (157)) and hence . This proves the required statement for . In the case the statement of the corollary is implied by the Plancherel theorem and the obvious inclusion .

Remark 23. The fact that is a spectral function of the classes and directly follows from the Plancherel theorem. Corollary 22 claims that there are no other spectral functions in these classes.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.