The Bessel function of order zero can be written as (a) Show that | 0 ( )| ≤ 1, | ’ 0 ( )| ≤ 1, | ” 0 ( )| ≤ 1, and in fact, for any positive integer ,                                 In what follows,...

The Bessel function of order zero can be written as

(a) Show that |
₀()| ≤ 1, |’₀()| ≤ 1, |”₀
()| ≤ 1, and in fact, for any positive integer
,

In what follows, determine how many interpolation points over the interval 0 ≤
≤ 10 are needed so the error is no more than 10⁻⁶.

(b) Using piecewise linear interpolation.

(c) Using a clamped cubic spline.

(d) Using Chebyshev interpolation.

(e) The Bessel function of order
 can be defined as

How do your answers in parts (b)–(d) change for this function?