Favard Constants

The th Favard constant , for a positive integer , is the smallest constant for which, whenever is a -periodic function that is real-valued and whose st derivative is absolutely continuous and whose th derivative satisfies almost everywhere, there exists a trigonometric polynomial , for a nonnegative integer , of the form

(1)

The trigonometric polynomial has real coefficients and satisfies

(2)

for all .

The sum formula for , also used to define , is

(3)

which can be written in terms of the Lerch transcendent as

(4)

These can be expressed by

(5)

where is the Dirichlet lambda function and is the Dirichlet beta function. Explicitly,

			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

(OEIS A050970 and A050971).