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User:Mkelly86/The Beurling-Selberg Extremal Problem

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The Beurling–Selberg Extremal Problem is a problem in harmonic analysis that is principally motivated by its applications in number theory. Loosely speaking the problem asks: given a function , find a function with the properties that: is an entire function with controlled growth, is real whenever is real, and the area between the graphs of and is as small as possible. Oftentimes an additional restriction on is: or , i.e. either majorizes or minorizes .

The subject was initiated with the unpublished work of Arne Beurling in the late 1930's and continued with Atle Selberg in the mid 1970's who used his results to prove a sharp form of the large sieve.[1][2][3][4] Other notable applications include: improved bounds of the Riemann zeta function in the critical strip[5], Erdös–Turán inequalities[6][7], estimates of Hermitian forms[8], and a simplified proof of Montgomery and Vaughan's version of Hilbert's inequality [9].

The statement of the problem

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The Beurling-Selberg extremal problem can be formulated in several different ways. Here we include several common formulations.

Majorant/minorant/best approximation

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The Problem of Best Approximation Given and a function , find an analytic function such that

  1. is real when .
  2. and are entire functions of exponential type at most .
  3. is minimized with respect to the norm.

Such a function is then called a best approximation to . If in addition for every , then the problem is the majorant problem and the function is an extremal majorant of . Similarly, if , then the problem is the minorant problem and the function is an extremal minorant of . We call any of the solutions to the above problems Beurling–Selberg extremal functions of . In applications it is often desirable to solve the majorant and minorant problem simultaneously, but simultaneous solutions need not exist.[10] For instance, has a known extremal majorant, but no extremal minorant of exists because it would necessarily have a pole at zero.

Reformulation in a de Branges space

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Let be an entire function of bounded type in the upper half plane, and for every and . Every such function has an associated de Branges Space which we will denote with norm . In this formulation one wishes to obtain a majorant and minorant of some prescribed exponential type . Generally speaking, the function one wishes to majorize or minorize is not analytic, so in contrast to the above problem, one (roughly) seeks to minimize the difference of the majorant and minorant with respect to . Of course, such a minimization can only occur if the difference is analytic.
Here is the formulation of the problem:

Given a function , determine functions and such that

  1. and are of entire functions of exponential type at most
  2. for every
  3. is as small as possible.[11]

A Simple Application

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To demonstrate the utility of the Beurling-Selberg extremal functions, we consider the problem of estimating

where and is an almost periodic function

where are real numbers such that when and are complex numbers.
Let and let and be the Beurling-Selberg extremal majorant and minorant of of exponential type . To simplify notation let and , then

for every real . Observe

but after writing and rearranging we get

But

By the Paley-Wiener theorem if , thus

.

By repeating the same argument with we obtain the estimate

.

Now using the fact that and the estimate can finally be rewritten as

where . This identity was also obtained by Montgomery and Vaughan from a generalization of Hilbert's inequality.[12][13]

The Problem for the Signum function

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The Interpolation Approach

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is shown here in red, in black, and the extremal minorant in blue.

In the late 1930's Beurling considered the majorant problem for the signum function:

for which he obtained the solution:

Furthermore he showed that is unique in the sense that if is another entire function of exponential type , and , then

is shown here in blue, is in black.

with equality if and only if .

Observe that the odd part of is given by

and the even part of is given by

is shown here in blue.

which is Fejér's Kernel for .[14] It can be shown that

and is the extremal minorant.
The solution for the problem of best approximation is also known and is given by:[15]

Minimization in a de Branges space

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If is an entire function of Bounded Type in the upper half plane, and for every and , the Beurling-Selberg extremal functions (with the minimization taking place in ) for are known. [16]
Let be a real number such that . If is the reproducing kernel for define by

Corresponding to is an associated function which is initially defined in a strip, but can be shown to extend to an entire function by analytic continuation, given by

where is the unique Borel probability measure that satisfies

in an open vertical strip that contains 0. The functions and can be shown to satisfy

and if and are functions of exponential type less than or equal to twice the exponential type of that satisfy , then

with equality if and only if

and

The problem in several variables

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Compared to what is known in the single variable case, relatively little is known about the Beurling-Selberg extremal problem for several variables. Selberg developed a procedure to majorize and minorize a box in Euclidean space whose sides are parallel to to the coordinate axis. It is easy to construct a majorant of such a function by multiplying the known majorants of characteristic functions of intervals. A minorant is less simple and can be obtained as the combination of majorants and minorants[17][citation needed], the periodic case is treated in the paper of Barton, Montgomery, Vaaler.

Characteristic function of a ball in Euclidean space

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The only known function for which the Beurling-Selberg extremal problem has been solved in several variables is the characteristic function of the ball of radius and center 0 in , which we will denote:

In particular, for fixed , , and , they find an explicit majorant and minorant that have exponential type at most and minimize the value of the integral

We will let denote the minimum value of this integral. In order to solve the problem in they first solve the problem in and radially extend the 1-dimensional solutions (which they show are extremal). Let be the normalized characteristic function of the interval :

then

.

Using the solution to the above problem for signum, the authors obtain the majorant and minorant as a linear combination of the majorant and minorant of the problem for the signum function:

.

The minimization occurs in a de Branges space that is in sympathy with radial extensions: a (de Branges) homogeneous space[18][19] where and

and

The following identity makes this choice of de Branges space clear:

where and is a Bessel function of the first kind.

For every and , satisfies the following inequality[20]

where is the surface area of n-sphere and equality occurs if and only if

.

The Problem in the Periodic Case

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The Beurling-Selberg extremal problem has a natural analogue for periodic functions. The best approximation problem is:
Given a function that is periodic with period 1, find an entire function such that

  • is real whenever
  • is a trigonometric polynomial of degree at most
  • is as small as possible in the -norm.

If in addition for all , the problem is the majorant problem. If for all , the problem is the minorant problem.
Periodic analogues of problems on can intuitively be approached by periodization of the non-periodic problem and then an application of the Poisson summation formula. While this idea is oftentimes in the background, there are some technicalities. For instance, Montgomery (1994) provides a method of solving the problem for the sawtooth function:

( is the fractional part of )that avoids using the Poisson summation formula as was used in Vaaler (1985). The technicality in this case is the analogue for in is which is not absolutely integrable, so the Fourier transform is not immediately defined. Vaaler worked around the issue by writing (defined above) and computing the Fourier transforms of and .

Functions for which the Beurling-Selberg Functions are known

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There are several papers where it is shown how to produce the Beurling-Selberg extremal functions for a large class of functions. For instance, Vaaler & Graham (1981) took steps for majorizing and minorizing integrable functions with some additional regularity. In Vaaler (1985) it is shown how to majorize and minorize a function of bounded variation, and in Carneiro & Vaaler (2010) it is shown how to solve the problem of best approximation of functions of the form

where is a Borel measure that satisifies

Examples of such functions include: , and where .

The following table contains functions for which the Beurling-Selberg extremal problem has been worked out, and is far from complete. The references in the following table may not be the reference in which the functions were introduced, but rather serve as a source to find the functions explicitly.[21]

Function References
Beurling (unpublished)
Vaaler (1985)
Holt and Vaaler (1996)

When
Selberg (lectures in mid-70's)(collected works - 1991)
Holt and Vaaler (1996)

When
Logan(1977)

Where
Holt and Vaaler (1996)
Lerma (1998 - Phd. Dissertation)
Carniero, Vaaler (2010)
for Carniero, Vaaler (2010)
for Carniero, Littmann, Vaaler (2010)
for Carniero, Littmann, Vaaler (2010)
for and Carniero, Littmann, Vaaler (2010)
for and Carniero, Littmann, Vaaler (2010)
for Carniero, Littmann, Vaaler (2010)
for and Carniero, Vaaler (2010)
Carniero, Littmann, Vaaler (2010)
[note for the minorant problem]
for Carniero, Littmann, Vaaler (2010)
for Carniero, Littmann, Vaaler (2010)

Graphs of Some Known extremal functions

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The following extremal functions have exponential type

See Also

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Notes

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  1. ^ Montgomery (1978)
  2. ^ Selberg (collected works)
  3. ^ Vaaler (1985)
  4. ^ It should be noted that Selberg and Beurling's work on the problem was carried out independently (Selberg had no prior knowledge of Beurling's work). At the time Selberg worked on the problem both men were faculty at The Institute for Advanced Study.
  5. ^ Carnerio and Chandee (2011)
  6. ^ Vaaler (1985)
  7. ^ Drmota and Tichy (1997)
  8. ^ Vaaler and Holt (1996)
  9. ^ Vaaler (1985)
  10. ^ Carneiro,Littmann,Vaaler (2010)
  11. ^ It is worth noting that the function is non-negative on the real axis and is of exponential type. Thus, by a generalization of Fejér–Riesz theorem(see Boas), for some analytic function of exponential type. Hence
    Such an expression can be bounded below by knowledge of the reproducing kernel of and the Cauchy–Schwarz inequality.
  12. ^ Vaaler (1985)
  13. ^ Montgomery and Vaughan (1974)
  14. ^ The function is the extremal majorant for the Dirac delta function.
  15. ^ Vaaler (1985)
  16. ^ Vaaler & Holt (1996)
  17. ^ This approach yields majorants and minorants, but the extremal functions are not known.
  18. ^ de Branges (1968)
  19. ^ Vaaler & Holt (1996)
  20. ^ Scaling properties of the function make this formula sufficient, see Vaaler & Holt (1996)
  21. ^ The characteristic functions appearing in this table are normalized, i.e. .

References

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  • Barton, Jeffrey; Montgomery, Hugh; Vaaler, Jeffrey (2001). "Note on a Diophantine inequality in several variables". Proc. Amer. Math. Soc. 129 (2): 337–345 (electronic). doi:10.1090/S0002-9939-00-05795-6. ISSN 0002-9939. S2CID 118982870.
  • Boas, Jr., Ralph Philip (1954). Entire functions. Academic Press Inc.. pp. 124-132.
  • Carneiro, Emmanuel; Chandee, Vorrapan (2011). "Bounding in the Critical Strip". J. Number Theory (N.S.). 131 (3): 363–384. doi:10.1016/j.jnt.2010.08.002. S2CID 119591029.
  • Carniero, E.; Littmann, F.; Vaaler, J. (2010). "Gaussian subordination for the Beurling-Selberg extremal problem". Transactions of the American Mathematical Society. v1. 365 (7): 3493–3534. arXiv:1008.4969. doi:10.1090/S0002-9947-2013-05716-9. S2CID 50680870.


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