User:Mkelly86/The Beurling-Selberg Extremal Problem

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The subject was initiated with the unpublished work of Arne Beurling in the late 1930's and continued with Atle Selberg in the mid 1970's who used his results to prove a sharp form of the large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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