Integral approximation of the characteristic function of an interval by trigonometric polynomials
Tóm tắt
We prove that the value E
n−1(χ
h
)
L
of the best integral approximation of the characteristic function χ
h
of an interval (−h, h) on the period [−π,π) by trigonometric polynomials of degree at most n − 1 is expressed in terms of zeros of the Bernstein function cos {nt − arccos[(2q − (1 + q
2) cost)/(1 + q
2 − 2q cost)]}, t ∈ [0, π], q ∈ (−1,1). Here, the parameters q, h, and n are connected in a special way; in particular, q = sech − tanh for h = π/n.
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