Singular value decomposition of fractional integration operators in L2 ...

Singular value decomposition of fractional integration operators in L2–spaces with weights Rudolf Gorenflo Department of Mathematics and Informatics, Free University of Berlin Vu Kim Tuan∗ Department of Numerical Analysis, Hanoi Institute of Mathematics

Dedicated to the memory of Academician A. N. Tikhonov

Abstract For the fractional integration operators of any positive order on the half-line and on a bounded interval we explicitly present a singular value decomposition, thereby considering these operators as acting on and into spaces of functions square integrable with appropriate weights. The results obtained amount to an interpretation of properties of systems of orthogonal polynomials. Keywords: Fractional Integration Operators, Singular Value Decomposition, Orthogonal Polynomials. 1991 Mathematics Subject Classifications: 45C05, 45E10, 45P05, 47A75, 47B06, 65R30

1. Introduction As is well known, the inverse of a compact linear operator, acting on and into an infinite-dimensional Hilbert space (there may be two Hilbert spaces involved), is not continuous. Hence, in numerical treatment of the corresponding operator equation, one has to cope with ill-posedness, the more so if the data function (as is common in applications) is given only approximately. One way of dealing with the arising difficulties consists in using filtering techniques via singular value decomposition – either directly or in devising a regularization scheme or in analyzing such a scheme theoretically. See [2], [7], [9]. Let U and V be infinite-dimensional Hilbert spaces, and let A : U → V be an injective compact linear operator. Then there exist ∗

Supported by the Alexander von Humboldt Foundation

1

(i) an orthonormal basis {uj }j∈IN0 in U, (ii) an orthonormal system {vj }j∈IN0 in V, (iii) a non-increasing sequence {σj }j∈IN 0 of positive numbers with limit 0 for j → ∞, such that j ∈ IN0 .

Auj = σj vj ,

(1)

Here by IN0 we denote the set {0, 1, 2, 3, . . .} of non-negative integers. We call the system {σj , uj , vj }j∈IN 0 a “singular system of the operator A”. For the operator A then the following ”singular value decomposition” At =

∞ X

σj (t, uj )U vj ,

t ∈ U,

(2)

j=0

is valid. We shall consider, for α > 0, the operators of fractional integration, x

(Iaα u)(x)

1 Z u(t)(x − t)α−1 dt , = Γ(α) a

α u)(x) (Ib−

1 Z = u(t)(x − t)α−1 dt , Γ(α) x

−∞ ≤ a < x < b ≤ ∞,

(3)

−∞ ≤ a < x < b ≤ ∞,

(4)

b

in weighted L2 –spaces U, V on the interval Ω = (a, b) corresponding weight functions ϕ, ψ and scalar products (f, g)U =

Z

(f, g)V =

f (t)g(t)ϕ(t)dt,

Ω

Z

(= IR+ or (−1, 1)) with the

f (x)g(x)ψ(x)dx.

Ω

For general information on these operators, one may consult [12], [10], [6]. In [2] and [9] one finds as an example the singular value decomposition of the operator I01 , acting from L2 (0, 1) into L2 (0, 1). One has uj (t) =

√

2 cos((j + 1/2)πt), vj (x) =

1 Evidently, σj ∼ πj

√

2 sin((j + 1/2)πx), σj =

1 , j ∈ IN0 . π(j + 1/2)

as j → ∞. 1/2

In [3] the operator I−1 : U → V with U = L2 ((−1, 1), ϕ), ϕ(t) =

(1 − t)α , (1 + t)β

V = L2 ((−1, 1), ψ), ψ(x) =

(1 − x)α−1/2 , (1 + x)β+1/2 2

α ≥ 0,

0 ≤ β < 1/2.

is considered and the singular values σn =

√



π

Γ(n + α + 1/2)Γ(n + β + 1) Γ(n + α + 1)Γ(n + β + 3/2)

1/2

n ∈ IN0 ,

,

as well as the singular functions un (t) and vn (x) as appropriate multiples of (1 + t)β Pn(α,β) (t) and (1 + x)β+1/2 Pn(α−1/2,β+1/2) (x) are obtained. For these Jacobi polynomials Pn(α,β) (t) and later for other systems of orthogonal polynomials see [1], [8]. By Stirling’s formula one finds √ σn ∼ πn−1/2 . The exact singular values σn of Iaα in L2 (a, b), −∞ < a < b < ∞, for arbitrary positive α are not known. But from [5], [4] one can conclude that they decay like n−α . In this paper the singular value decompositions of the operator Iaα for some special weighted L2 –spaces are obtained, and it is interesting to notice that for some cases the singular values decay like n−α/2 , that means the corresponding operator equation in such weighted spaces is less ill–posed. 2. Fractional integration operators on the half–line Take U = L2 (IR+ , ϕ), V = L2 (IR+ , ψ) with weight functions ϕ, ψ still to be specified, and the operator I0α : U → V . By exploiting properties of the generalized Laguerre polynomials we shall find a singular system for the operator I0α . The generalized Laguerre polynomials are defined as L(γ) n (x) =

n X

n + γ xk , n − k k! !

(−1)k

k=0

γ > −1, n ∈ IN0 ,

(5)

and obey the orthogonality relations (see [1], p. 775), with δmn the Kronecker symbol, Z∞

(γ) e−x xγ L(γ) m (x)Ln (x)dx =

0

Γ(γ + n + 1) δmn , n!

m, n ∈ IN0 .

(6)

Furthermore, there is Koshlyakov’s formula (see [11], p. 462, formula (2))    

1 Rx(x − t)α−1 tβ L(β) (t)dt = Γ(β + n + 1) xα+β L(α+β) (x), n n Γ(α) 0 Γ(α + β + n + 1)

   β > −1,

α > 0,

(7)

x > 0.

Putting now, for n ∈ IN0 , un (t) = vn (x) =

n! Γ(β + n + 1)

n! Γ(α + β + n + 1)

ϕ(t) = t−β e−t ,

!1/2

!1/2

xα+β L(α+β) (x), n

ψ(x) = x−(α+β) e−x , 3

tβ L(β) n (t),

β > −1 ,

(8) (9)

we see that {un }n∈IN0 and {vn }n∈IN0 are orthonormal systems in U and V , respectively. Taking (7) into account we observe that (I0α un )(x) = σn vn (x) with Γ(n + β + 1) Γ(n + α + β + 1)

σn =

!1/2

.

(10)

Now by Stirling’s formula we have σn ∼ n−α/2

as n → ∞,

(11)

that means in such a pair of spaces the operator equation considered here is less ill–posed than the similar ones in L2 (a, b) considered in [4], [5]. The simplest case of (9) is β = 0. We then have ϕ(t) = e−t ,

ψ(x) = x−α e−x .

Theorem 1. Let α > 0, β > −1, ϕ(t) = t−β e−t , ψ(x) = x−(α+β) e−x , I0α : L2 (IR+ , ϕ) → L2 (IR+ , ψ). Then the singular value decomposition {σn , un , vn }n∈IN0 of I0α is given by 

σn =

Γ(n + β + 1) Γ(n + α + β + 1) 

un (t) =

n! Γ(n + β + 1)

1/2

,

1/2



n! vn (x) = Γ(n + α + β + 1)

tβ L(β) n (t), 1/2

xα+β L(α+β) (x). n

and σn /n−α/2 → 1 as n → ∞. α We are looking now for a singular system of the Weyl operator I−α = I∞− , b = ∞ in formula (4),

I α : L2 (IR+ ; ϕ) → L2 (IR+ ; ψ) , where ϕ(t) = tλ et ,

α > 0,

ψ(x) = xλ−α ex , s

n! e−t L(λ) n (t) , Γ(λ + n + 1)

(12)

n! e−x L(λ−α) (x) . n Γ(λ − α + n + 1)

(13)

un (t) = s

vn (x) =

λ > α − 1. Put

4

Using the formula (6) we have Z∞

un (t)um (t)ϕ(t)dt = δmn ,

n, m ∈ IN0 ,

(14)

vn (x)vm (x)ψ(x)dx = δmn ,

n, m ∈ IN0 .

(15)

0

Z∞ 0

Therefore, {un }n∈IN0 and {vn }n∈IN0 are orthonormal complete systems of the spaces L2 (IR+ ; ϕ) and L2 (IR+ ; ψ) respectively. Applying the formula R∞ (t − x)α−1 x

Γ(α)

−x (λ−α) (x) , e−t L(λ) n (t)dt = e Ln

λ > α − 1 , α > 0, n ∈ IN0

(see [11], p. 463, formula (7) ), we obtain v u u Γ(λ − α + n + 1) α (I un )(x) = t vn (x).

(16)

Γ(λ + n + 1)

From (14), (15) and (16) we obtain Theorem 2. Let ϕ(t) = tλ et , ψ(x) = xλ−α ex , λ + 1 > α > 0, I α : L2 (IR+ ; ϕ) → L2 (IR+ ; ψ) . Then {σn , un , vn }n∈IN0 with v u u Γ(λ − α + n + 1) σn = t ∼ n−α/2 , n → ∞,

Γ(λ + n + 1)

(17)

and un , vn as in (12), (13), is a singular value decomposition of the operator I α . 3. Fractional integration operators on an interval α We are looking for a singular system of the operator I1− : U → V for α = k/2 , k ∈ IN , where U and V are suitably weighted L2 – spaces over the interval (0, 1). We shall succeed by aid of the Gegenbauer (“ultraspherical”) polynomials (defined for α > 0)

Cn(α) (x)

X Γ(α + n − j) 1 [n/2] (−1)j (2x)n−2j , = Γ(α) j=0 j!(n − 2j)!

(18)

and an integral representation formula, listed in [1] under number (22.10.11): θ

Cn(α) (cos θ)

Z 21−α Γ(n + 2α) cos((n + α)τ )dτ 1−2α = (sin θ) , 2 n!(Γ(α)) (cos τ − cos θ)1−α 0

5

0 < θ < π, α > 0. (19)

By substituting x := cos θ, t := cos τ we obtain 1

Cn(α) (x)

Z 21−α Γ(n + 2α) 2 1/2−α = (1 − x ) (t − x)α−1 yn (t)dt, 2 n!(Γ(α)) x

(20)

yn (t) = (1 − t2 )−1/2 cos((n + α) arccos t). We have, fixing now α to be a multiple of 1/2, Z1 √

t2

1−

yn (t)ym (t)dt =

−1

Zπ

cos((m + α)t) cos((n + α)t)dt

0

=

1 2

Zπ

{cos((m − n)t) + cos((m + n + 2α)t)} dt =

0

π δmn . 2

q

2 y (t) are orthonormal on (−1, 1) with respect to the Hence the functions un (t) = π n √ weight function ϕ(t) = 1 − t2 . From the orthogonality relations ([1], p. 774) Z1

(α) (1 − x2 )α−1/2 Cn(α) (x)Cm (x)dx =

−1

π21−2α Γ(n + 2α) δmn , n!(n + α)(Γ(α))2

m, n ∈ IN0 ,

(21)

we obtain Z1

ψ(x)vn (x)vm (x)dx = δmn

−1

with the weight function ψ(x) = (1 − x2 )1/2−α and v u u n!(n + α) vn (x) = t Γ(α)2α−1/2 (1 − x2 )α−1/2 Cn(α) (x).

πΓ(n + 2α)

Now formula (19) becomes α (I1− un )(x)

=

n! (n + α)Γ(n + 2α)

!1/2

vn (x).

As we naturally expect, we have σn =

n! (n + α)Γ(n + 2α)

!1/2

∼ n−α as n → ∞.

6

(22)

Theorem 3. Let k ∈ IN, ϕ(t) =

√

1 − t2 , ψ(x) = (1 − x2 )

1−k 2

,

k/2

I1− : L2 ((−1, 1), ϕ) → L2 ((−1, 1), ψ). k/2

Then the singular value decomposition {σn , un , vn }n∈IN 0 of I1− is given by 1/2



n! σn = ∼ n−k/2 , n → ∞, (n + k/2)Γ(n + k) q 2 (1 − t2 )−1/2 cos((n + k ) arccos t), un (t) = π 2 

n!(n + k/2) vn (x) = πΓ(n + k)

1/2

Γ(k/2)2

k−1 2

(1 − x2 )

k−1 2

(k)

Cn 2 (x).

q

2 (1 − t2 )−1/2 cos((n + k/2) arccos t) Remark: The strange looking functions un (t) = π are in fact algebraic and can be expressed via Chebyshev polynomials of first and second kind (see [1], p. 776, formulas (22.3.15) and (22.3.16) ). With t = cos τ, 0 ≤ τ ≤ π, we sin(mτ ) have, for m ∈ IN0 , cos(mτ ) = Tm (t), sin τ = Um−1 (t), U−1 (t) ≡ 0. Hence, s

un (t) =

2 (1 − t2 )−1/2 Tn+ k (t) , 2 π

o 1 n un (t) = √ (1 − t)−1/2 Tn+ k−1 (t) − (1 − t)1/2 Un+ k−3 (t) , 2 2 π

k ∈ IN, 2 k if 6∈ IN. 2

if

We are now looking for a singular system of the operator α : L2 (Ω, ϕ) → L2 (Ω, ψ) , I−1

t where Ω = (−1, 1), ϕ(t) = 11 − +t λ + 1/2 > α > 0 . 

α > 0, λ−1/2

,

ψ(x) = (1 − x)λ−α−1/2 (1 + x)−λ−α+1/2 ,

Put, for n ∈ IN0 ,

un (t) =

 s   22λ−1 (n + λ)n!   Γ(λ)(1 + t)λ−1/2 Cnλ (t) , λ 6= 0,   πP (n + 2λ)   √n (1 + t)−1/2 Cn0 (t) ,    2π    −1/2 −1/2

π

(1 + t)

,

λ = 0,

λ = 0,

n 6= 0,

(23)

n = 0.

Applying (21) we obtain Z1

un (t)um (t)ϕ(t)dt = δmn ,

n, m ∈ IN0 ,

(24)

−1

that means the system {un }n∈IN0 is orthonormal in the space L2 (Ω; ϕ) . Using the formula (valid for α > 0 , λ > −1/2 , n ∈ IN0 , x ∈ Ω) 7

Rx (x − t)α−1

Γ(α)

−1

(t + 1)λ−1/2 Cnλ (t)dt =

Γ(2λ + n)Γ(λ + 1/2) (1 + x)α+λ−1/2 Pn(λ−α−1/2,λ+α−1/2) (x) , Γ(2λ)Γ(λ + α + n + 1/2) (see [11], p. 518, formula 9), where the Pn(α,β) (x) are orthogonal Jacobi polynomials ([1], p. 779; [8]) Pn(α,β) (x)

n+α m

n P

−n

=2

m=0

!

!

n+β (x − 1)n−m (x + 1)m , n ∈ IN0 , n−m

we have v u u Γ(n + λ − α + 1/2) α vn (x) , (I−1 un )(x) = t

Γ(n + λ + α + 1/2)

n ∈ IN0 , x ∈ Ω,

(25)

where vn (x) = cn (1 + x)(α+λ−1/2) Pn(λ−α−1/2,λ+α−1/2) (x) , v u u cn = t

(2n + 2λ)n!Γ(n + 2λ) . 2 Γ(n + λ − α + 1/2)Γ(n + λ + α + 1/2) 2λ

(26) (27)

Since R1 −1

(1 − x)α (1 + x)β Pn(α,β) (x)Pm(α,β) (x)dx =

Γ(n + α + 1)Γ(n + β + 1) 2α+β+1 δ , α, β > −1 , n, m ∈ IN0 , (2n + α + β + 1) Γ(n + 1)Γ(n + α + β + 1) mn we obtain Z1

vn (x)vm (x)ψ(x)dx = δmn ,

m, n ∈ IN0 ,

(28)

−1

that means the system of functions {vn }n∈IN0 is orthonormal in the space L2 (Ω; ψ). Since {Cnλ (t)}n∈IN0 is a complete basis in the space L2 (Ω; (1 − t2 )λ−1/2 ), the family {un }n∈IN0 is complete in the corresponding space L2 (Ω; ϕ). Therefore {un }n∈IN0 is an orthonormal basis of the space L2 (Ω; ϕ). From (24), (25) and (28) we have t Theorem 4. Let ϕ(t) = 11 − +t λ + 1/2 > α > 0, and 

λ−1/2

, ψ(x) = (1 − x)λ−α−1/2 (1 + x)−λ−α+1/2 ,

α : L2 ((−1, 1); ϕ) → L2 ((−1, 1); ψ) . I−1

Then {σn , un , vn }n∈IN0 with 8

s

σn =

Γ(n + λ − α + 1/2) ∼ n−α , n → ∞, Γ(n + λ + α + 1/2)

α and un , vn as in (23), (26) is a singular value decomposition of the operator I−1 . α In particular, suppose λ = 1/2. Then I−1 , 0 < αq< 1, boundedly maps the space 2 −α L2 (Ω) into the space L2 (Ω; (1 − x ) ), with the norm Γ(1 − α)/Γ(1 + α) and the sin-

gular value decomposition (26) λ is replaced by 1/2.

q

Γ(n + 1 − α)/Γ(n + 1 + α) , un , vn

 n∈IN0

where in (23) and

Analogously (argue by symmetry), the system s

!

Γ(n + λ − α + 1/2) , u (t), vn (x) Γ(n + λ + α + 1/2) n

, n∈IN0

where s

un (t) =

22λ−1 (n + λ)n! Γ(λ)(1 − t)λ−1/2 Cnλ (t) , πΓ(n + 2λ)

vn (x) = cn (1 − x)α+λ−1/2 Pn(λ+α−1/2,λ−α−1/2) (x) ,

n ∈ IN0 ,

cn as in (27), is a singular value decomposition of the operator α : L2 ((−1, 1); ϕ) → L2 ((−1, 1); ψ) I1−

with +t ϕ(t) = 11 − t 

λ−1/2

,

ψ(x) = (1−x)−λ−α+1/2 (1+x)λ−α−1/2 , λ+1/2 > α > 0,

Acknowledgement: For valuable discussions the authors express their gratitude to Dr. Norbert Gorenflo (Berlin), Prof. Dr. Bernd Hofmann (Chemnitz) and Dipl.–Math Christine Kutsche (Berlin).

References [1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover Publications, New York, 1965. [2] J. Baumeister. Stable Solution of Inverse Problems. Vieweg und Sohn, Braunschweig/Wiesbaden, 1987. [3] M Cuer. “Problem III, an illustrative problem on Abel’s integral equation.” In: G. T. Herman, et. al. (Eds.) Basic Methods of Tomography and Inverse Problems, p. 643-667. Adam Hilger, Bristol and Philadelphia, 1987. 9

[4] Dostani´c M.R. Asymptotic behavior of the singular values of fractional integral operators. J. Math. Anal. & Appl. 175(1993), p. 380–391. [5] Faber V. and Wing G.M. Singular values of fractional integral operators: A unification of theorems of Hille, Tamarkin, and Chang. J. Math. Anal. & Appl. 120(1986), p. 745–760. [6] R. Gorenflo and S. Vessella. Abel Integral Equations. Lecture Notes in Mathematics No. 1461, Springer–Verlag, Berlin, 1991. [7] B. Hofmann. Regularization for Applied Inverse and Ill-Posed Problems. Teubner, Leipzig, 1986. [8] N. N. Lebedev. Special Functions and Their Applications. Translated from the Russian. Dover Publications, New York, 1972. [9] A. K. Louis. Inverse und schlecht gestellte Probleme. Teubner, Stuttgart, 1989. [10] K. Nishimoto. Fractional Calculus. Descartes Press, Koriyama, 1991. [11] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev. Integrals and Series, Vol. 2: Special Functions. Gordon and Breach Scientific Publishers, Philadelphia, 1990. [12] S. G. Samko, A. A. Kilbas and O. I. Marichev. Fractional Integrals and Derivatives. Theory and Applications. Translated from the Russian. Gordon and Breach Scientific Publishers, Philadelphia, 1993.

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