In 1937, Boas gave a smart proof for an extension of the Bernstein theorem for trigonometric series. It is the purpose of the present note (i) to point out that a formula which Boas used in the proof is related with the Shannon sampling theorem; (ii) to present a generalized Parseval formula, which is suggested by the Boas’ formula; and (iii) to show that this provides a very smart derivation of the Shannon sampling theorem for a function which is the Fourier transform of a distribution involving the Dirac delta function. It is also shows that, by the argument giving Boas’ formula for the derivative of a function , we can derive the corresponding formula for , by which we can obtain an upperbound of . Discussions are given also on an extension of the Szegö theorem for trigonometric series, which Boas mentioned in the same paper.
Pólya and Szegö has taken up the Bernstein theorem for trigonometric series in their famous book . In  ([ Vol. II, p. 11]), the theorem is given as follows.
Let be a trigonometric polynomial of order , which is expressed as follows:
, and hold for all . Then , with equality if and only if is of the form for and .
Here and denote the sets of all real numbers and all integers, respectively, and , and for and satisfying . We use also which denotes the set of all complex numbers.
Boas  gave generalizations of this theorem and a related Szegö theorem for trigonometric series, which are Theorems 2 and 9 given below. The generalized theorems are concerned with a function which can be expressed as follows:
Here and throughout the present paper, is a complex-valued function of bounded variation, and .
The generalized Bernstein theorem is the following.
Let a function be given by (2), , and hold for all . Then .
In a later paper , Boas said that the proofs in  are lengthy and complicated, and gave a very smart proof for this theorem. That proof was based on the following formula, which we shall call Boas’ formula:
Proof of Theorem 2
By using and the well-known summation formula:
in (3), we obtain . A derivations of (4) is given in Remark 4 in Section 3.1. ■
When we see the formula (3), we expect that there must exist a sampling theorem which is applicable to the function , and (3) must be obtained by its differentiation. This is the motivation of the present paper. To achieve this object, we present the following sampling theorem.
Let a function be given by (2) in terms of which is continuous in a neighbourhood of as well as of . Then is expressed as follows:
By taking the term-by-term differentiation of (5) with respect to x, and then putting , we obtain (3).
In the formulas (3) and (5), and are expressed in terms of an enumerable set . We shall call such a formula a sampling formula (S-formula).
We can use the argument deriving Boas’ formula (3) to derive the corresponding formula for . We prove the following theorem in Section 3.2.
Let the assumption in Theorem 2 be satisfied. Then .
In Section 2, we give a generalized Parseval formula and the lemmas that provide the conditions under which the formula holds. By using these, we show that it readily provides a very simple derivation of the S-formulas (3) and (5) and of S-formulas for some functions defined similarly to (2), in Section 3. Some comments are given on the derivation of the Boas’ formula (3), in Section 3.1. In Section 4, discussions are given on the generalized Szegö theorem. Concluding remarks are presented in Section 5.
Here we note that a function expressed as (2) is continuous and bounded. In fact, if we denote the total variation of by T, then (2) shows for all .
2. Generalized Parseval Formula
In the present paper, we are concerned with integrals of the form:
Here we assume that is continuous in , and is integrable in and has the Fourier series, so that is expressed as follows:
for at which is continuous.
When is absolutely continuous in , and its derivative is defined by , I is expressed as . If the squares of and of are integrable in , we have the Parseval formula:
where are the coefficients in (7) and are given by .
We now present two lemmas which guarantee the validity of the formula (8) for the I defined by (6), assuming that are defined by:
Let the Fourier series (7) of converge uniformly, and let be defined by (9). Then the formula (8) holds for defined by (6).
Substituting (7) in (6), we obtain (8) by term-by-term integration, which is allowed, since the convergence of the Fourier series (7) is uniform and is of bounded variation. ■
If is of bounded variation and continuous in and satisfies , then its Fourier series (7) converges uniformly, by Theorem in  (Vol. I, p. 58) or by the Fejér theorem  (Vol. I, p. 89).
Let be of bounded variation and piecewise continuous in , and let be continuous at every discontinuous point of , as well as at and when is discountinuous at , or both, or when , and let be defined by (9). Then the formula (8) holds for defined by (6).
We use the notations that:
and is the sum of neighbourhoods of the points at which is not continuous. There exists such a that and for , since the partial sums of the Fourier series are uniformly bounded  (Vol. I, p. 90, Theorem (3.7)). For an arbitrary , we choose such that the the total variation of in is less than ϵ, and then choose such that outside . This is possible since converges uniformly outside , as seen in Remark 2. If we denote the total variation of by T, we have the inequality . This shows that as . ■
The two generalized theorems in Boas’ paper  are proved below with the aid of Lemma 1 and Remark 2.
3. Generalized Sampling Theorem
We consider four functions of for , which are:
We define four functions of for , by:
By (2), . We now define the function by:
We then note that the derivatives of and are expressed as follows:
We confirm that the exchange of integration and differentiation in each of these relations and (17) given below is allowed, with the aid of the method presented in  (Section 4.2).
We define by for and . They are listed in the second column of Table 1. For and , we define by:
The following lemma is easily confirmed.
Let . Then .
Since depends on l and a, its Fourier coefficients depend on l and a, and hence we express them by , and the Fourier series (7) as:
In the third column of Table 1, satisfying (16) are given for four functions . We note that for , and hence:
Because of (9) and (2), when , is given by .
Now the Parseval formula (8) for (6) gives:
for (15). By using this in Lemma 3, we obtain:
for arbitrary value of .
We call the formula (19) the sampling formula (S-formula) for the function .
The formula (5) for and the corresponding S-formulas for , and , are obtained by using and , for , 2, 3 and 4, respectively, ofTable 1, in (19), where we put .
Fourier coefficients of function in , and satisfying (12).
Fourier coefficients of function in , and satisfying (12).
Proof of Theorem 3
Lemma 4 shows that the formula (5) for takes the form (19) which is (8) for the present case, and hence is proved by using Lemma 2. ■
Let be continuous at and . Then the S-formulas for , and obtained in Lemma 4, are valid. In the case of , is required to be continuous also at .
The proof follows to the proof of Theorem 3 given above. ■
Let or . Then if the formula (19) is valid when , and if satisfies the condition for it in Lemma 2 for , then we have:
We use (17). ■
Formula (20) is obtained from (8) by term-by-term differentiation with respect to a. The , and given in rows for and in Table 1 are obtained from those in the rows for and , respectively, by differentiation with respect to a or to x, and then dividing by R.
As a consequence of Remark 3, we have the following lemma.
The S-formulas for and are obtained by term-by-term differentiation of the corresponding S-formulas for and , respectively.
3.1. Derivation of Boas’ Formula (3)
In this section, we put and .
Boas’ formula (3) is derived by putting in the S-formula for obtained in Lemma 4.
Boas’ formula (3) is valid, without the additional assumptions on given in Theorem 3.
Lemmas 4 and 7 show that (3) takes the form of (19), and hence of the form (8), with . For this , the validity of (3) is concluded by Lemma 1, with the aid of Remark 2 or (21) given below, without invoking Lemma 2. ■
When (3) is proved in the proof of Lemma 7, (16) is expressed as follows:
By putting in (21) , we obtain the summation formula (4), which was used in the proof of Theorem 2.
Proof of Proposition 1
This is a consequence of Lemmas 6 and 7. ■
We can use the steps in Proposition 1 to derive (3) from (5). In the course of the steps, it is assumed that is continuous at and . But in the final form (3) , the proof of Theorem 6 shows that Lemma 1 applies, and the additional assumption on is not necessary.
3.2. Proof of Theorem 4
Let , and be defined by:
where , and . Then (19) and (16) for are valid.
We put and . Then , , , , and
Now (19) and (16) are:
By comparing (26) with (4), we have the well-known formula:
By using the S-formulas for , and obtained in Lemma 4, in (32) , we obtain:
From the rows for and in Table 1, the formula (19) for given by (32), becomes (33), and the corresponding Fourier series (16) is given by:
Writing this formula (35) with for and for , we derive the summation formulas:
4.1. Proof of Theorem 9
In this section, we put and .
We obtain the S-formula for by using (33) and (34) in (31). We then put and in the obtained S-formula for . Then we obtain:
where . When for all , we obtain from (37) with (38), by using the first summation formula in (36).
Before we put and , the S-formula for is valid when an additional assumption on given in Theorem 3 is assumed. But the coefficients in the final form (37) are such that the series (16) converges absolutely and uniformly, and hence Lemma 1 applies. As the result, the additional assumption on is not required in the validity of (37).
Here we note that given by (31) with (33) and (34) is expressed as (15), if we put
When y is chosen to be , so that , is continuous as a function of t in , and satisfies , and hence we obtain (37) by using the Fourier coefficients of the Fourier series of , with the aid only of Lemma 1 and Remark 2.
5. Concluding Remarks
5.1. Concluding Remark 1
In the present paper, the function is expressed in the form of the Stieltjes integral as (2), and hence the sampling theorem presented here applies to the cases when the spectrum is discrete as well as continuous.
If is assumed to be expressed in terms of an integrable function in , by
then a very simple proof of the sampling theorem was presented by Boas  and Pollard and Shisha . In this case, for are the Fourier coefficients of , and hence
Substituting this into the right-hand side of (40) and performing the term-by-term integration, we obtain the Shannon sampling theorem (5) for . The term-by-term integration in this case is justified by the Lebesgue theorem  (p. 37).
It is recalled here that Campbell  presented the sampling theorem for the case when is a distribution. In that paper, the author mentioned that the Shannon sampling theorem (5) for is valid when is a Dirac’s delta function.
For the case when (40) applies, an extensive review of works related on the Shannon sampling theorem is found in the book . In recent papers [12,13], extensions of the sampling theorem to the Hilbert and the Banach space are discussed.
5.2. Concluding Remark 2
In this section, we denote the scalar product of two functions and which are integrable in , by . The quantity calculated in (40) is expressed as for and . When the squares of both of these functions are integrable in , and the Fourier series of the two functions in the interval are expressed as follows:
where , then (5) for is the result of the Parseval formula .
We now note that the Parseval formula is achieved by using only one of the Fourier series either of or and integrating term by term, as
with the aid of the Lebesgue theorem  (p. 37). In fact, Boas  and Pollard and Shisha  proposed to use this calculation using the Fourier series of which is assumed to be integrable in , without assuming that the square of is integrable in , in deriving the Shannon sampling theorem, as stated above.
In the present paper, may be expressed as a sum of an absolutely continuous function and a step function, is a distribution involving Dirac’s delta function and hence we have to use the Fourier series of .
5.3. Concluding Remark 3
At the end of Section 1, it was mentioned that , for the function expressed as (2), where T is the total variation of . If we use the first equation in (14) for , we can confirm that . In Remark 7, defined by (29) is expressed as (15) by using (39). By confirming that by (39), we see that .
Hence Theorems 2 and 9 are proved in this way, if .
5.4. Concluding Remark 4
In recent papers [14,15], a generalized Shannon sampling theorem is applied to non-limited-band signal, on the basis of an inequality given in a Boas’s paper . The inequality is given as in the following lemma.
Let and . Then
We put in (45), and then we obtain
Let and . Then
In confirming this, we note that .
We find that this lemma is a part of Theorem 3.12 in the book .
When , and hence
In , the case of is studied. We now recommend to use the inequality (46) in the study.
The authors are grateful to the reviewers for helpful comments. With the help of them, the authors could improve the descriptions in this paper.
Ken-ichi Sato showed Boas’ formula to Morita and asked the possibility of deriving it from the sampling formula which appears in the Shannon sampling theorem. Answering this question, Tohru Morita proposed such a modified sampling formula that its derivative gives Boas’ formula, and wrote a manuscript to explain it. Tohru Morita and Ken-ichi Sato kept revising the manuscript to the present form.
Conflicts of Interest
The author declares no conflict of interest.
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