Spectral Functions for the Vector-Valued Fourier Transform
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 .
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 [2–4] concerning spectral functions for canonical differential systems.
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 ). 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., ) 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 ). Let be a subspace in and let . A pair is referred to the class if .
A parametrization of the class in terms of contractive operator-functions is given by the following proposition.
Proposition 8 (see ). 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., ). 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 .
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