Mellin transform is the most popular integral transform in the analysis of algorithms. Its first occurrence is in a memoir of Riemann in which he used it to study the famous Zeta function. However, the transform gets its name from the Finish mathematician Hjalmar Mellin who did the first systematic study of the transform and its inverse. See [132] for a summary of his works. Nowadays, the Mellin transform is used in complex analysis, number theory, applied mathematics and analysis of algorithms. Apart from these applications in mathematics, the Mellin transform has also been applied in many different areas such as physics and engineering.
In the analysis of algorithms, the Mellin transform is mostly used to derive asymptotic expansions. The transform is defined in the following:
Definition 3.3.1. Let f (z) be a function which is locally Lebesque integrable
over (0, +∞). The Mellin transform of f(z) is defined byM [f(z); s] = f∗(s) =
∫ ∞
0
f (z)zs−1dz.
The largest open strip ⟨α, β⟩ in which the integral converges is called the fundamental strip. To determine the fundamental strip, we have the following lemma.
Lemma 3.3.2. If the function f (z) satisfies
f (z) ={ O(zu), z → 0+; O(zv), z → +∞,
then M [f(z); s] exists in the strip ⟨−u, −v⟩ and is analytic there.
f (z) f∗(s) < α, β >
F1 zνf (z) f∗(s + ν) < α− ν, β − ν >
F2 f (zρ) 1ρf∗(s/ρ) < ρα, ρβ > ρ > 0 f (1/z) −f∗(−s) <−β, −α >
F3 f (µz)∑ µ1sf∗(s) < α, β > µ > 0
kλkf (µkz) (∑
kλkµ−sk )f∗(s) < α, β >
F4 f (z) log z dsdf∗(s) < α, β >
F5 Θf (z) −sf∗(s) < α′, β′ > Θ = zdzd
d
dzf (z) −(s − 1)f∗(s− 1) < α′ − 1, β′− 1 >
∫z
0 f (t)dt −1sf∗(s + 1)
Table 3.1: Functional properties of Mellin transform
Before we start to develop a systematic theory of the Mellin transform, we give an easy example to show how the transform works.
Example 3.3.3. For f (z) = e
−z, it is obvious that f (z) ={ O(z0), z → 0+; O(z−b), z→ +∞,
where b is an positive real number which can be arbitrarily large. As a result, M [f(z); s] = Γ(s)
is defined in ⟨0, +∞⟩ and analytic there.
Simple changes of variables in the definition of Mellin transforms yields many useful functional properties summarized in Table3.1.
Similar to other integral transformations, the Mellin transform has an inverse. For a given function f (z) and its Mellin transform f∗(s), we let s = σ + 2πit and z = e−y. Then, the Mellin transform becomes a Fourier transform
f∗(s) =
∫ ∞
0
f (z)zs−1dz =
∫ ∞
−∞
f (e−y)e−σye−2πitydy =F [f(e−y)e−σy; t], where F [f(z); t] denotes the Fourier transform of f(z). As a result, the inversion theorem for the Mellin transform follows from the corresponding one for the Fourier transform. If ˆf (t) = F [f; t] is the Fourier transform, then the original function is recovered by
f (z) =
∫ ∞
−∞
f (t)eˆ 2πitzdt.
Thus,
f (e−y)e−σy =
∫ ∞
−∞
f∗(σ + 2πit)e2πitydt.
Changing the variables y back to z, we get f (z) = z−σ
∫ ∞
−∞
f∗(σ + 2πit)z−2πitdt.
Finally, by replacing σ + 2πit by s, we have f (z) = 1
2πi
∫ c+i∞
c−i∞
f∗(s)z−sds.
Theorem 3.3.4. Let f (z) be integrable with fundamental strip
⟨α, β⟩. If c is such that α < c < β and f∗(c + it) is integrable, then the equalityholds almost everywhere. Moreover, if f (z) is continuous, then the equality holds everywhere on (0, +∞).
As we mentioned before, the major application of the Mellin transform in the analysis of algorithms is to derive asymptotic expansions. This appli-cation comes from the correspondence between the asymptotic expansion of a function at 0 (and ∞), and poles of its Mellin transform in a left (resp.
right) half-plane. Before we explain the correspondence, we introduce the singular expansion of a function.
Definition 3.3.5. Let ϕ(s) be meromorphic with poles in Ω. The singular
expansion of ϕ(s) isϕ(s)≍ ∑
s0∈Ω
∆(s; s0),
where ∆(s, s0) is the principal part of the Laurent expansion of ϕ around the pole s = s0.
With the help of the singular expansion of a function, we now give the correspondence.
Theorem 3.3.7. (Direct mapping) Let f (z) be a function with transform
f∗(s) in the fundamental strip ⟨α, β⟩.(i) Assume that, as z → 0+, f (z) admits a finite asymptotic expansion of the form
f (z) = ∑
(ξ,k)∈A
cξ,kzξ(log z)k+O(zγ),
where −γ < −ξ ≤ α and the k′s are nonnegative. Then f∗(s) is continuable to a meromorphic function in the strip ⟨−γ, β⟩ where it admits a singular expansion
f∗(s)≍ ∑
(ξ,k)∈A
cξ,k (−1)kk!
(s + ξ)k+1, s∈ ⟨−γ, β⟩.
(ii) Similarly, assume that as z → +∞, f(z) admits a finite asymptotic expansion of the same form where now β ≤ −ξ < −γ. then f∗(s) is continuable to a meromorphic function in the strip ⟨α, −γ⟩ where
f∗(s)≍ − ∑
(ξ,k)∈A
cξ,k (−1)kk!
(s + ξ)k+1, s ∈ ⟨α, −γ⟩.
The direct mapping theorem gives a connection from the asymptotic ex-pansion of the function to the singular exex-pansion of the Mellin transform.
Thus, the natural question is whether or not there is a way to get the asymp-totic expansion from the singular expansion. The answer is yes, when the Mellin transform is small enough at ±i∞. More precisely, we have the fol-lowing result.
Theorem 3.3.8. (Converse mapping) Let f (z) be continuous in [0, +
∞) with Mellin transform f∗(s) having a nonempty fundamental strip ⟨α, β⟩.(i) Assume that f∗(s) admits a meromorphic continuation to the strip
⟨γ, β⟩ for some γ < α with a finite number of poles there, and is analytic on ℜ(s) = γ. Assume also that there exists a real number η∈ (α, β) such that
f∗(s) =O(|s|−r) with r > 1,
when|s| → ∞ in γ ≤ ℜ(s) ≤ η. If f∗(s) admits the singular expansion for ℜ(s) ∈ ⟨γ, α⟩,
f∗(s)≍ ∑
(ξ,k)∈A
dξ,k 1 (s− ξ)k, then an asymptotic expansion of f (z) at 0 is
f (z) = ∑
(ξ,k)∈A
dξ,k(−1)k−1
(k− 1)!z−ξ(log z)k−1+O(z−γ).
(ii) Similarly assume that f∗(s) admits a meromorphic continuation to
⟨α, γ⟩ for some γ > β and is analytic on ℜ(s) = γ. Assume also that
f∗(s) =O(|s|−r) with r > 1,
for η≤ ℜ(s) ≤ γ with η ∈ (α, β). If f∗(s) admits the singular expansion of the same form for ℜ(s) ∈ ⟨η, γ⟩, then an asymptotic expansion of f (z) at ∞ is
f (z) =− ∑
(ξ,k)∈A
dξ,k(−1)k−1
(k− 1)!z−ξ(log z)k−1+O(z−γ).
Note that to apply Theorem 3.3.8, we need the condition f∗(s) =O(|s|−r) with r > 1,
for s in a suitable strip. In other words, we need the Mellin transforms to be sufficiently small along an vertical line. It is well-known that the small-ness of a Mellin transform is directly related to the degree of ”smoothsmall-ness”
(differentiability, analyticity) of the original function. Here we introduce two theorems which are useful in determining the ”smallness” of the Mellin transforms.
Theorem 3.3.9. Let f (x)
∈ Cr with fundamental strip ⟨α, β⟩. Assume that f (x) admits an asymptotic expansion as x→ 0+ (x → ∞) of the formf (x) = ∑
(ξ,k)∈A
cξ,kxξ(log x)k+O(xγ) (3.7) where the ξ satisfy −α ≤ ξ < γ (γ < ξ ≤ −β). Assume also that each derivative f(j)(x) for j = 1, . . . , r satisfies an asymptotic expansion obtained by termwise differentiation of (3.7). Then the continuation of f∗(s) satisfies
f∗(σ + it) = o(|t|−r) as |t| → ∞
uniformly for σ in any closed subinterval of (−γ, β) ((α, −γ)).
Theorem 3.3.9 shows that smoothness implies smallness. The strongest possible form of smoothness for a function is analyticity. The following the-orem shows that the Mellin transform of an analytic function will decay exponentially in a quantifiable way.
Theorem 3.3.10. Let f (z) be analytic in a sector S
θ which is defined as Sθ ={z ∈ C|0 < |z| < ∞ and | arg(z)| ≤ θ} with 0 < θ < π.Assume that for f (z) =O(|z|−α) as |z| → 0 in Sθ, and f (z) = O(|z|−β) as
|z| → ∞ in Sθ. Then
f∗(σ + it) =O(e−θ|t|) (3.8) uniformly for σ in every closed subinterval of (α, β).
Note that a polynomial bound with positive power is enough for Theorem 3.3.8 while (3.8) provides an exponential bound. In fact, if the assumption
f∗(s) =O(|s|−r) with r > 1, in Theorem3.3.8 is replaced by
f∗(σ + it) =O(e−θ|t|),
we will get a stronger version of the converse mapping theorem as follows.
Theorem 3.3.11. Let f (z) be continuous in [0, +
∞) with Mellin transform f∗(s) having a nonempty fundamental strip ⟨α, β⟩.(i) Assume that f∗(s) admits a meromorphic continuation to the strip
⟨γ, β⟩ for some γ < α with a finite number of poles there, and is analytic on ℜ(s) = γ. Assume also that there exists a real number η∈ (α, β) such that
f∗(σ + it) =O(e−θ|t|), 0 < θ < π
when|t| → ∞ in γ ≤ ℜ(s) ≤ η. If f∗(s) admits the singular expansion for ℜ(s) ∈ ⟨γ, α⟩,
f∗(s)≍ ∑
(ξ,k)∈A
dξ,k 1 (s− ξ)k, then an asymptotic expansion of f (z) at 0 is
f (z) = ∑
(ξ,k)∈A
dξ,k(−1)k−1
(k− 1)!z−ξ(log z)k−1+O(z−γ) and the asymptotic expansion holds in the cone
Sθ ={z ∈ C|0 < |z| < ∞ and | arg(z)| ≤ θ} with 0 < θ < π.
(ii) Similarly assume that f∗(s) admits a meromorphic continuation to
⟨α, γ⟩ for some γ > β and is analytic on ℜ(s) = γ. Assume also that
f∗(s) =O(|s|−r) with r > 1,
for η≤ ℜ(s) ≤ γ with η ∈ (α, β). If f∗(s) admits the singular expansion of the same form for ℜ(s) ∈ ⟨η, γ⟩, then an asymptotic expansion of f (z) at ∞ is
f (z) =− ∑
(ξ,k)∈A
dξ,k(−1)k−1
(k− 1)!z−ξ(log z)k−1+O(z−γ) and the asymptotic expansion holds in the cone
Sθ ={z ∈ C|0 < |z| < ∞ and | arg(z)| ≤ θ} with 0 < θ < π.
The advantage of asymptotic expansions holding in a cone in the complex plane is that asymptotic expressions of the derivative are obtained by term-by-term differentiation (the same is not true for asymptotic expansions which just hold on the real line). The justification of this follows from a useful theorem due to J. Ritt [163].
Theorem 3.3.12. Let f (z) be analytic in an annular sector
SR which is defined asSR={z ∈ C|R < |z| < ∞ and θ1 < arg(z)≤ θ2} for some θ1, θ2 and 0≤ R.
If for some fixed real number p, we have
f (z) =O(zp) (or f (z) = o(xp)) as z→ ∞ in SR, then
f(m)(z) =O(zp−m) ( or f(m)(z) = 0(xp−m)) as z→ ∞ in any closed annular sector properly interior to SR.
Now, with the knowledge of the Mellin transform, we can handle the functional equation (3.5). From the assumptions of Theorem 3.2.3 and the fact that P0 = P1 = 0, we have that
f (z) =˜
{ O(z2), as z→ 0+, O(z1+ϵ), as z → ∞.
Now, applying the Mellin transform on (3.5), we obtain forℜ(s) ∈ ⟨−2, −1−
ϵ⟩
f˜∗(s) = −Γ(s + 1)
1− p−s− q−s. (3.9)
For the sake of simplicity, we assume that p = q = 1/2, then (3.9) become f˜∗(s) = −Γ(s + 1)
1− 2s+1 .
We let χk = 2kπi/ log 2 for k∈ Z. By a simple computation, we get f˜∗(s)≍ 1
(s + 1)2 1
log 2 − 1 s + 1
( γ log 2 +1
2 )
− 1
log 2
∑
k∈Z\{0}
Γ(χk) s + 1− χk
.
From [54], we have that the gamma function admits a bound
|Γ(σ + it)| = O(
|t|σ−1/2e−π|t|/2)
, as|t| → ∞.
Thus, we can apply Theorem3.3.8. This plus the result from depoissonization in (3.6) yields
E(Pn) = ˜f (n) = n log2n + n ( γ
log 2 +1
2 + P (log2n) )
+ o(n), where P (t) is a 1-periodic function with the Fourier expansion given by
P (t) = ∑
k∈Z\{0}
Γ(−χk) log 2 e2kπit.
With the mean of the external path length of symmetric tries solved, let us turn our attention back to the internal path length of symmetric DSTs.
Apart from the Rice method, P. Flajolet and B. Richmond proposed another method to handle such problems in [66]. The Flajolet-Richmond method is a combination of the Euler transform, the Mellin transform and the singularity analysis. Before we explain their approach, we introduce singularity analysis.