ν’1/2 ν’1/2

1

3 1 3

e’u du ≥

≥ ,

2 2 2

0

which ¬nishes the proof.

Finally, we derive upper bounds on K ν . We will need these bounds for complex-valued

ν. We start with their behavior if the argument tends to in¬nity.

Lemma 5.13 The modi¬ed Bessel function K ν , ν ∈ C, has the asymptotic behavior

2π ’r | (ν)|2 /(2r )

|K ν (r )| ¤ , r > 0.

ee (5.3)

r

Proof With b = | (ν)| we have

∞

1

e’r cosh t |eνt + e’νt |dt

|K ν (r )| ¤

2 0

1 ∞ ’r cosh t bt

[e + e’bt ]dt

¤ e

20

= K b (r ).

Furthermore, from et ≥ cosh t ≥ 1 + t 2 /2 for t ≥ 0 we can conclude that

∞

e’r (1+t /2) bt

2

K b (r ) ¤ e dt

0

∞

1

’r b2 /(2r )

e’s /2

2

=e e √ ds

√

r ’b/ r

√ 1

2πe’r eb /(2r )

2

¤ .

r

While the last lemma describes the asymptotic behavior of the modi¬ed Bessel functions

for large arguments, the next lemma describes the behavior in a neighborhood of the origin.

Nonetheless, in the case (ν) = 0 it holds for all r > 0.

Lemma 5.14 For ν ∈ C the modi¬ed Bessel functions satisfy, for r > 0,

§

⎨ 2| (ν)|’1 (| (ν)|)r ’| (ν)| , (ν) = 0,

|K ν (r )| ¤ 1 (5.4)

r

© ’ log , r < 2, (ν) = 0.

e 2

Proof Let us ¬rst consider the case (ν) = 0. Again, we set b = | (ν)| and know already

that |K ν (r )| ¤ K b (r ) from the proof of the last lemma.

54 Auxiliary tools from analysis and measure theory

From the de¬nition of the modi¬ed Bessel function, however, we can conclude for every

a > 0 that

∞

1

e’r cosh t ebt dt

K b (r ) =

2 ’∞

∞

1 ’t

e’r (e +e

t

= )/2 bt

e dt

2 ’∞

∞

1

= a ’b e(’r/2)(u/a+a/u) u b’1 du

2 0

by substituting u = aet . By setting a = r/2 we obtain

∞

K b (r ) = 2b’1r ’b e’u e’r /(4u) b’1

du ¤ 2b’1 (b)r ’b .

2

u

0

For (ν) = 0 we use cosh t ≥ et /2 to derive

∞ ∞

e’r cosh t dt ¤ e’ 2 e dt

r t

K 0 (r ) =

0 0

∞ ∞ 1

1 1

e’u du e’u du +

= ¤ du

u u

r/2 1 r/2

1 r

= ’ log .

e 2

5.2 Fourier transform and approximation by convolution

One of the most powerful tools in analysis is the Fourier transform. Not only will it help us

to characterize positive de¬nite functions, it will also be necessary in several other places.

Hence, we will dwell on this subject maybe at ¬rst sight longer than necessary. We start

with the classical L 1 theory.

De¬nition 5.15 For f ∈ L 1 (Rd ) we de¬ne its Fourier transform by

f (x) := (2π)’d/2 f (ω)e’i x ω dω

T

Rd

and its inverse Fourier transform by

f ∨ (x) := (2π)’d/2 f (ω)ei x ω dω.

T

Rd

We will always use this symmetric de¬nition. But the reader should note that there are

other de¬nitions on the market, which differ from each other and this de¬nition only by the

way in which the 2π terms are distributed.

For a function f ∈ L 1 (Rd ) the Fourier transform is continuous. Moreover, the following

rules are easily established. The overstrained reader might have a look at the book [180] by

Stein and Weiss.

5.2 Fourier transform and approximation by convolution 55

Theorem 5.16 Suppose f, g ∈ L 1 (Rd ); then the following is true.

(1) R d f (x)g(x)d x = R d f (x)g(x)d x.

(2) The Fourier transform of the convolution

f — g(x) := f (y)g(x ’ y) dy

Rd

is given by f — g = (2π )d/2 f g.