The Integral Transform of N.I.Akhiezer
Victor Katsnelson

TL;DR
This paper investigates a specific integral transform introduced by N.I. Akhiezer, analyzing its properties and applications as presented in his textbook on integral transforms.
Contribution
It provides a detailed study of Akhiezer's integral transform, clarifying its form, properties, and potential uses in mathematical analysis.
Findings
Characterization of the transform's properties
Connections to classical integral transforms
Potential applications in solving integral equations
Abstract
We study the integral transform which appeared in a different form in Akhiezer's textbook "Lectures on Integral Transforms".
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Taxonomy
TopicsAlgebraic and Geometric Analysis · Mathematical and Theoretical Analysis · Matrix Theory and Algorithms
The Integral Transform of N.I. Akhiezer
Victor Katsnelson
Department of Mathematics
The Weizmann Institute
76100, Rehovot
Israel
[email protected]; [email protected]
(Date: July 1, 2016)
Abstract.
We study the integral transform which appeared in a different form in Akhiezer’s textbook ‘‘Lectures on Integral Transforms’’.
Key words and phrases:
Akhiezer integral transform, convolution operators
1991 Mathematics Subject Classification:
44A15; 44A35
1. The Akhiezer Integral Transforms: a formal definition.
In the present paper we consider the one-parametric family of pairs of linear integral operators. The parameter which enumerates the family can be an arbitrary positive number and is fixed in the course of our consideration. Formally the operators are defined as convolution operators according the formulas
[TABLE]
In (1.1), is vector-column,
[TABLE]
which entries are measurable functions, and , are matrices,
[TABLE]
where
[TABLE]
Here and in what follows, are hyperbolic functions.
For
[TABLE]
The operators are naturally decomposed into blocks:
[TABLE]
where and are convolution operators:
[TABLE]
In (1.6), is a -valued function.
The function is continuous and positive on . It decays exponentially as :
[TABLE]
Since
[TABLE]
the convolution kernel admits the estimate
[TABLE]
Definition 1.1**.**
The set as the set of all complex valued functions which are measurable, defined almost everywhere with respect to the Lebesgue measure on and satisfy the condition
[TABLE]
The set is the set of all columns such that and .
Lemma 1.2**.**
Let be a -valued function which belongs to the space . Then the integral in the right hand side of (1.6a) exists111That is the value of this integral is a finite complex number for every . for every .
We define the function by means of the equality (1.6a).
Remark 1.3*.*
For , the function is a continuous function well defined on the whole . Nevertheless the function may not belong to the space . The operator does not map the space into itself. (In other words, the operator considered as an operator in is unbounded.)
The situation with the integral in the right hand side of (1.6b) is more complicated. The function also decays exponentially as :
[TABLE]
However the function has the singularity at the point :
[TABLE]
where is a function which is continuous and bounded for . Thus the convolution kernel has a non-integrable singularity on the diagonal :
[TABLE]
Therefore the integral in the right hand side of (1.6b) may not exist as a Lebesgue integral. Given a function , the equality
[TABLE]
holds at every point which is a Lebesgue point of the function and . Nevertheless, under the condition (1.9) we can attach a meaning to the integral for almost every .
Lemma 1.4**.**
Let be a -valued function which belongs to the space . Then the principal value integral
[TABLE]
exists for almost every .
We define the function by means of the equality (1.6b), where the integral in the right hand side of (1.6b) is interpreted as a principal value integral.
Under the condition (1.9), the integral exists as a Lebesgue integral for every :
[TABLE]
This follows from the estimate (1.10). The assertion that the limit in (1.13) exists for almost every will be proved in section 4 using the Hilbert transform theory.
Remark 1.5*.*
Under the assumption (1.9), the function
[TABLE]
which is defined for almost every , is not necessary locally summable. It may happen that for every finite interval , .
Let us define the transforms and formally.
Definition 1.6**.**
For , we put
[TABLE]
where and are columns:
[TABLE]
with the entries
[TABLE]
The operators and are the same that appeared in Lemmas 1.2 and 1.4 respectively.
According to Lemmas 1.2 and 1.4, the values and are well defined for almost every .
The integral transforms (1.1)-(1.3) are said to be the Akhiezer integral transforms.
In another form, these transforms appear in [1, Chapter 15]. (See Problems 3 and 4 to Chapter 15.) The matrix nature of the Akhiezer transforms was camouflaged there.
In what follows, we consider the Akhiezer transform in various functional spaces. We show that the operators and are mutually inverse in spaces of functions growing slower than .
2. The operators and in .
The Fourier transform machinery is an adequate tool for study convolution operators.
1. Studing the operators and by means of the Fourier transform technique, we deal with the spaces and . Both these spaces consist of measurable functions defined almost everywhere on the real axis with respect to the Lebesgue measure. The spaces are equipped by the standard linear operations and the standard norms. If , then
[TABLE]
The space consists of all such that . If , then
[TABLE]
The space consists of all such that . This space is equipped by inner product . If , , then
[TABLE]
2. The Fourier-Plancherel operator :
[TABLE]
where
[TABLE]
maps the space onto itself isometrically:
[TABLE]
The inverse operator is of the form
[TABLE]
where
[TABLE]
Lemma 2.1**.**
Let and . Then
- (1)
The integral
[TABLE]
exists as a Lebesgue integral (i.e. ) for almost every . 2. (2)
The function belongs to , and the inequality
[TABLE]
holds. 3. (3)
The Fourier-Plancherel transforms and are related by the equality
[TABLE]
where
[TABLE]
This lemma can be found in [2], Theorem 65 there. See also [3], Theorem 3.9.4.
3. Let us calculate the Fourier transforms of the functions and . The function belongs to . So its Fourier transform
[TABLE]
is well defined for every .
Lemma 2.2**.**
The Fourier transforms of the function is:
[TABLE]
The formula (2.14) can be found in [2], where it appears as (7.1.6).
4. The function does not belong to . This function has non-integrable singularity at the point . Therefore the integral does not exist as a Lebesgue integral. However So the integral
[TABLE]
exists as a Lebesgue integral for every . We define the Fourier transform as a principle value integral:
[TABLE]
Lemma 2.3**.**
The limit in (2.16) exists for every . The Fourier transforms of the function is:
[TABLE]
The difference
[TABLE]
satisfies the conditions
[TABLE]
The formula (2.17) can be found in [2], where it appears as (7.2.3).
5. In Section 1 we already have defined the functions and for from the space . The space is contained in . If , then
[TABLE]
According to Lemmas 1.2 and 1.4, if , then the function is defined for every and the function is defined for almost every . However for , we can obtain much more accurate results.
Lemma 2.4**.**
Let and , i.e.
[TABLE]
Then , and the Fourier-Plancherel transforms , of functions and are related by the equality
[TABLE]
where is determined by the equality (2.14).
Proof.
Lemma 2.4 is a direct consequence of Lemma 2.1.
∎
Lemma 2.5**.**
Let and , i.e.
[TABLE]
Then , and the Fourier-Plancherel transforms , of functions and are related by the equality
[TABLE]
where is determined by the equality (2.17).
Proof.
Since , Lemma 2.5 does not follow from Lemma 2.1 directly. Let
[TABLE]
The function belongs to for every . Let
[TABLE]
Applying Lemma 2.1 to , we conclude that for every and that the Fourier-Plancherel transforms , of the functions , are related by the equality
[TABLE]
where is defined by (2.15). According to Lemma 2.3,
[TABLE]
where
[TABLE]
and the family \{\textup{\LARGE\varrho}_{\omega}(\lambda,\varepsilon)\}_{0<\varepsilon<\infty} satisfies the conditions (2.19) and (2.20). From (2.19), (2.20), (2.30) and the Lebesgue Dominated Convergence Theorem it follows that
[TABLE]
In other words,
[TABLE]
where
[TABLE]
From (2.31) it follows that as , i.e.
[TABLE]
where . From the other side, for a.e. by Lemma 1.4. Hence , and . ∎
6. The equality
[TABLE]
plays a crucial role in this paper. This equation is a direct consequence of the explicite expressions (2.14) and (2.17) for and and the identity
[TABLE]
Lemma 2.6**.**
The operators and are contractive in the space . Moreover the equality
[TABLE]
holds.
Proof.
Let , and let , , be the Fourier-Plancherel transforms of the functions . According to Lemmas 2.4 and 2.5, the equalities
[TABLE]
hold. From (2.34) it follows that
[TABLE]
Integrating with respect to , we obtain the equality . In view of (2.6), the last equality is equivalent to the equality (2.36). ∎
3. The Akhiezer operators and in .
Definition 3.1**.**
The space is the set of all columns such that and . The set is equipped by the natural linear operations and by the inner product .
If and belong to , then
[TABLE]
The inner product (3.1) generates the norm
[TABLE]
Since222See (2.21). , also . Thus if , then the values and are defined by (1.16) for almost every . Using Lemmas 2.4 and 2.5, we conclude from (1.16) that the operators and are bounded operators in the space . In particular, the values and belong to .
Theorem 3.2**.**
Each of the operators and is an isometric operator in the space :
[TABLE]
Theorem 3.3**.**
The operators and are mutually inverse in the space :
[TABLE]
Proofs of Theorem 3.2.
Let us associate the matrix functions and with the operators and :
[TABLE]
where and are the same that in (2.14) and (2.17). Let
[TABLE]
and let , , where are the Fourier-Plancherel transforms of the functions respectively. According to the equality (1.16) and to Lemmas 2.4 and 2.5, the equality
[TABLE]
holds for almost every .
From the equality (2.34) it follows that the matrix is unitary for each :
[TABLE]
where is identity matrix. From (3.6) and (3.7) it follows that
[TABLE]
i.e.
[TABLE]
Integrating with respect to over and using the Parseval identity (2.6), we conclude that
[TABLE]
that is . The equality can be obtained analogously. ∎
Proof of Theorem 3.3.
Let
[TABLE]
Let , , , where are the Fourier-Plancherel transforms of the functions respectively. We already proved the equality (3.6). In the same way the equality
[TABLE]
can be established. From (3.6) and (3.8) it follows that
[TABLE]
From the equality (2.34) it follows that the matrices and are mutually inverse:
[TABLE]
where is identity matrix. From (3.9) and (3.10) we conclude that
[TABLE]
Finally .
The equality (3.4a) is proved. The equality (3.4b) can be proved in the same way. ∎
4. The Hilbert transform
Definition 4.1**.**
Let be a complex-valued function which is defined for almost every . We assume that the function satisfies the condition
[TABLE]
Then the integral
[TABLE]
exists for every and . For each , the function is a continuous function of for . The function is defined for those for which the value tends to a finite limit as :
[TABLE]
The function is said to be the Hilbert transform of the function .
Theorem. (A.I.Plessner.) Let be a function which is defined for almost every . If the function satisfies the condition (4.1), then its Hilbert transform exists for almost every .
Proof of this Plessner’s Theorem can be found in [2], Theorem 100 there.
If is a function from , then satisfies the condition (4.1). By Plessner’s theorem, the Hilbert transform exists for almost every .
Theorem. (E.C. Titchmarch.) Let be a function from . Then:
- (1)
Its Hilbert transform also belongs to , and the equality
[TABLE]
holds. 2. (2)
The equality
[TABLE]
holds for almost every .
This theorem means that the Hilbert transform, considered as an operator in , is an unitary operator which satisfies the equality
[TABLE]
where is the identity operator in .
Proof of Lemma 1.4..
We use the decomposition (1.11) of the kernel into the sum of the Hilbert kernel and the ‘regular’ kernel . Let be an arbitrary finite interval of the real axis. We split the function into the sum of two summands.
[TABLE]
So
[TABLE]
According to (1.11) and (4.7), the equality
[TABLE]
holds, where
[TABLE]
The function satisfies the condition (4.1). According Plessner’s Theorem, exists for almost every . Since the function is finitely supported and the kernel is continuous, exists for every . Since the function vanishes for and , exists for every . In view of (4.10), the limit in (1.13) exists for almost every . Since is an arbitrary finite interval, the limit in (1.13) exists for almost every . ∎
5. The operators and in .
In this section we consider the operators and acting in spaces of functions growing slower than as .
Definition 5.1**.**
For , the space is the space of all functions which are measurable, defined almost everywhere with respect to the Lebesgue measure and satisfy the condition , where
[TABLE]
The space is equipped by the standard linear operations and by the norm (5.1).
It is clear that the space which appeared in section 2 is the space , that is with .
In Section 1 we already have defined the functions and for from the space . For , the space is contained in . If , then
[TABLE]
According to Lemmas 1.2 and 1.4, if , then the function is defined for every and the function is defined for almost every .
In Section 2 we obtained that if , than and . Moreover we proved that the operators and are contractive in : see Corollary 2.6. In this section we show that if and , than and . Moreover we show that the operators and are bounded in the space .
Lemma 5.2**.**
Assume that . Let , and is related to by means of the formula (2.22), i.e. . Then , and
[TABLE]
where is a value which does not depend on and .
Proof.
Let
[TABLE]
Since , . The equality (2.22) can be rewritten as
[TABLE]
Let us estimate the kernel
[TABLE]
For the inequality
[TABLE]
holds. Hence
[TABLE]
From this inequality and from the expression (1.4) for we conclude that
[TABLE]
where
[TABLE]
For , the function belongs to and
[TABLE]
where
[TABLE]
The integral in (5.11) can be calculated explicitly:
[TABLE]
Thus
[TABLE]
Since for , the inequality (5.14) implies the inequality
[TABLE]
From (5.5) and (5.9) we obtain the inequality , and
[TABLE]
where
[TABLE]
According to Lemma 2.1, , and the inequality
[TABLE]
holds. The inequality
[TABLE]
is a consequence of the equalities (5.16), (5.18) and (5.15). According to (5.4), , . So the inequality (5.3) holds with . ∎
Lemma 5.3**.**
Assume that . Let , and is related to by means of the formula (2.24), i.e. . Then , and
[TABLE]
where is a value which does not depend on and .
Proof.
Let , be defined according to (5.4). Since , . The equality (2.24) can be rewritten as
[TABLE]
We present as
[TABLE]
where
[TABLE]
[TABLE]
Let us estimate the kernel
[TABLE]
From the inequalities , from (5.7) and from the expression (1.4) for we conclude that
[TABLE]
where
[TABLE]
The function belongs to , and
[TABLE]
where is the same that in (5.12). The integral in (5.27) can be calculated explicitly:
[TABLE]
Thus the inequality
[TABLE]
holds. From (5.23), (5.24) and (5.25) it follows that
[TABLE]
where
[TABLE]
According to Lemma 2.1, , and the inequality
[TABLE]
holds. From (5.29), (5.32) and (5.30) we conclude that
[TABLE]
According to Lemma 2.6, the inequality
[TABLE]
holds. From (5.21), (5.34) and (5.33) we derive the inequality
[TABLE]
with . ∎
6. The Akhiezer operators and in .
Definition 6.1**.**
The space is the set of all columns such that and , where was defined in Definition 5.1. The set is equipped by the natural linear operations and by the norm
[TABLE]
Since333See (5.2). , also . Thus if , then the values and are defined by (1.16) for almost every . From Lemmas 5.2 and 5.3 we derive
Lemma 6.2**.**
We assume that . Let and let be defined by (1.16). Then , , and the estimates hold
[TABLE]
where is a value which does not depend on .
The following theorem is a main result of this paper.
Theorem 6.3**.**
We assume that . Then for every the equalities
[TABLE]
hold.
Proof.
From Lemma 6.2 it follows that the operators and are bounded linear operators in the space . The set is a dense subset of the space . By Theorem 3.3, the equalities (6.4) holds for every . By continuity, the equalities (6.4) can be extended from to . ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] N.I. Akhiezer. Lectures on Integral Transforms. Amer. Math. Soc., Providence, RI. 1985.
- 2[2] E.C. Titchmarch. Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford 1937. x+394pp. Third Edition. Chelsea Publishing Co., New York 1986.
- 3[3] V.I. Bogachev. Measure Theory. Vol.1 . Springer-Verlag, Berlin-Heidelberg-New York, 2007.
