Favard Constants
المؤلف:
Finch, S. R.
المصدر:
"Achieser-Krein-Favard Constants." §4. 2 in Mathematical Constants. Cambridge, England: Cambridge University Press
الجزء والصفحة:
...
23-2-2020
1026
Favard Constants
Let 
be an arbitrary trigonometric polynomial
![T_n(x)=1/2a_0+<span style=]() {sum_(k=1)^n[a_kcos(kx)+b_ksin(kx)]} " src="http://mathworld.wolfram.com/images/equations/FavardConstants/NumberedEquation1.gif" style="height:45px; width:274px" /> |
(1)
|
with real coefficients, let 
be a function that is integrable over the interval ![[-pi,pi]](http://mathworld.wolfram.com/images/equations/FavardConstants/Inline3.gif)
, and let the 
th derivative of 
be bounded in ![[-1,1]](http://mathworld.wolfram.com/images/equations/FavardConstants/Inline6.gif)
. Then there exists a polynomial 
for which
 |
(2)
|
for all ![x in [-pi,pi]](http://mathworld.wolfram.com/images/equations/FavardConstants/Inline8.gif)
, where 
is the smallest constant possible, known as the 
th Favard constant.
can be given explicitly by the sum
![K_r=4/pisum_(k=0)^infty[((-1)^k)/(2k+1)]^(r+1),](http://mathworld.wolfram.com/images/equations/FavardConstants/NumberedEquation3.gif) |
(3)
|
which can be written in terms of the Lerch transcendent as
 |
(4)
|
These can be expressed by
![K_r=<span style=]() {4/pilambda(r+1) for r even; 4/pibeta(r+1) for r odd, " src="http://mathworld.wolfram.com/images/equations/FavardConstants/NumberedEquation5.gif" style="height:82px; width:177px" /> |
(5)
|
where 
is the Dirichlet lambda function and 
is the Dirichlet beta function. Explicitly,
(OEIS A050970 and A050971).
REFERENCES:
Finch, S. R. "Achieser-Krein-Favard Constants." §4. 2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 255-257, 2003.
Kolmogorov, A. N. "Zur Grössenordnung des Restgliedes Fourierscher reihen differenzierbarer Funktionen." Ann. Math. 36, 521-526, 1935.
Sloane, N. J. A. Sequences A050970 and A050970 in "The On-Line Encyclopedia of Integer Sequences."
Zygmund, A. G. Trigonometric Series, Vols. 1-2, 2nd ed. New York: Cambridge University Press, 1959.
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