Lidstone series - Alchetron, The Free Social Encyclopedia

In mathematics, a Lidstone series, named after George James Lidstone, is a kind of polynomial expansion that can expressed certain types of entire functions.

Let ƒ(z) be an entire function of exponential type less than (N + 1)π, as defined below. Then ƒ(z) can be expanded in terms of polynomials A_n as follows:

f ( z ) = ∑ n = 0 ∞ [ A n ( 1 − z ) f ( 2 n ) ( 0 ) + A n ( z ) f ( 2 n ) ( 1 ) ] + ∑ k = 1 N C k sin ⁡ ( k π z ) .

Here A_n(z) is a polynomial in z of degree n, C_k a constant, and ƒ⁽ⁿ⁾(a) the nth derivative of ƒ at a.

A function is said to be of exponential type of less than t if the function

h ( θ ; f ) = lim sup r → ∞ 1 r log ⁡ | f ( r e i θ ) |

is bounded above by t. Thus, the constant N used in the summation above is given by

t = sup θ ∈ [ 0 , 2 π ) h ( θ ; f )

with

N π ≤ t < ( N + 1 ) π .

References

Lidstone series Wikipedia

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