MATH 590: Meshfree Methods Chapter 5: Completely Monotone and Multiply Monotone Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 – Chapter 5 1
Outline Completely Monotone Functions 1 Multiply Monotone Functions 2 fasshauer@iit.edu MATH 590 – Chapter 5 2
In Chapter 3 we saw that translation invariant (“stationary” in the statistics literature) strictly positive definite functions can be characterized via Fourier transforms. Since Fourier transforms are not always easy to compute, we now present two alternative criteria that allow us to decide whether a function is strictly positive definite and radial on R s (“isotropic” in the statistics literature): complete monotonicity (for the case of all s ), and multiple monotonicity (for only limited choices of s ). fasshauer@iit.edu MATH 590 – Chapter 5 3
Completely Monotone Functions Definition A function ϕ : [ 0 , ∞ ) → R that is in C [ 0 , ∞ ) ∩ C ∞ ( 0 , ∞ ) and satisfies ( − 1 ) ℓ ϕ ( ℓ ) ( r ) ≥ 0 , r > 0 , ℓ = 0 , 1 , 2 , . . . , is called completely monotone on [ 0 , ∞ ) . fasshauer@iit.edu MATH 590 – Chapter 5 5
Completely Monotone Functions Example The following are completely monotone on [ 0 , ∞ ) : ϕ ( r ) = ε , ε ≥ 0, ϕ ( r ) = e − ε r , ε ≥ 0, since for ℓ = 0 , 1 , 2 , . . . ( − 1 ) ℓ ϕ ( ℓ ) ( r ) = ε ℓ e − ε r ≥ 0 , 1 ϕ ( r ) = ( 1 + r ) β , β ≥ 0, since for ℓ = 0 , 1 , 2 , . . . ( − 1 ) ℓ ϕ ( ℓ ) ( r ) = ( − 1 ) 2 ℓ β ( β + 1 ) · · · ( β + ℓ − 1 )( 1 + r ) − β − ℓ ≥ 0 . fasshauer@iit.edu MATH 590 – Chapter 5 6
Completely Monotone Functions Properties of completely monotone functions (see [Cheney and Light (1999), Feller (1966), Widder (1941)]) A non-negative finite linear combination of completely monotone 1 functions is completely monotone. The product of two completely monotone functions is completely 2 monotone. If ϕ is completely monotone and ψ is absolutely monotone (i.e., 3 ψ ( ℓ ) ≥ 0 for all ℓ ≥ 0), then ψ ◦ ϕ is completely monotone. If ϕ is completely monotone and ψ is a positive function such that 4 its derivative is completely monotone, then ϕ ◦ ψ is completely monotone. fasshauer@iit.edu MATH 590 – Chapter 5 7
Completely Monotone Functions Remark 1 ϕ ( r ) = e − ε r and ϕ ( r ) = ( 1 + r ) β , β ≥ 0 are reminiscent of Gaussians and inverse multiquadrics (subject to transformation r �→ r 2 ). Question Is there a connection between completely monotone functions and strictly positive definite radial functions? Possible Answer Find an integral characterization of completely monotone functions. fasshauer@iit.edu MATH 590 – Chapter 5 8
Completely Monotone Functions Just as we recalled Fourier transforms and Fourier-Bessel transforms earlier, we now need to remember a third integral transform. In the following, the Laplace transform will be important: Definition Let f be a piecewise continuous function that satisfies | f ( t ) | ≤ M e at for some constants a and M . The Laplace transform of f is given by � ∞ f ( t ) e − st d t , L f ( s ) = s > a . 0 Similarly, the Laplace transform of a Borel measure µ on [ 0 , ∞ ) is given by � ∞ e − st d µ ( t ) . L µ ( s ) = 0 The Laplace transform is continuous at the origin if and only if µ is finite. fasshauer@iit.edu MATH 590 – Chapter 5 9
Completely Monotone Functions Theorem (Hausdorff-Bernstein-Widder) A function ϕ : [ 0 , ∞ ) → R is completely monotone on [ 0 , ∞ ) if and only if it is the Laplace transform of a finite non-negative Borel measure µ on [ 0 , ∞ ) , i.e., ϕ is of the form � ∞ e − rt d µ ( t ) . ϕ ( r ) = L µ ( r ) = 0 Remark The HBW-Theorem shows that the functions ϕ ε ( r ) = e − ε r can be viewed as the fundamental completely monotone functions. fasshauer@iit.edu MATH 590 – Chapter 5 10
Completely Monotone Functions Proof. Widder’s proof of this theorem can be found in [Widder (1941), p. 160], where he reduces the proof of this theorem to another theorem by Hausdorff on completely monotone sequences. A detailed proof can also be found in the books [Cheney and Light (1999), Wendland (2005a)]. fasshauer@iit.edu MATH 590 – Chapter 5 11
Completely Monotone Functions The following connection between positive definite radial and completely monotone functions was first pointed out by Schoenberg in 1938: Theorem A function ϕ is completely monotone on [ 0 , ∞ ) if and only if Φ = ϕ ( � · � 2 ) is positive definite and radial on R s for all s. Remark Note that the function Φ is now defined via the square of the norm. This differs from our earlier definition of radial functions. fasshauer@iit.edu MATH 590 – Chapter 5 12
Completely Monotone Functions Proof We prove only one direction ( ϕ completely monotone = ⇒ Φ positive definite and radial on any R s ). Details for the other direction are in [Wendland (2005a)]. The HBW theorem implies � ∞ e − rt d µ ( t ) ϕ ( r ) = 0 with a finite non-negative Borel measure µ . Therefore, Φ( x ) = ϕ ( � x � 2 ) has the representation � ∞ e −� x � 2 t d µ ( t ) . Φ( x ) = 0 fasshauer@iit.edu MATH 590 – Chapter 5 13
Completely Monotone Functions Now look at the quadratic form � ∞ N N N N c j c k e − t � x j − x k � 2 d µ ( t ) . � � � � c j c k Φ( x j − x k ) = 0 j = 1 k = 1 j = 1 k = 1 Since we saw earlier that the Gaussians are strictly positive definite and radial on any R s it follows that the quadratic form is non-negative, and therefore Φ is positive definite on any R s . � Remark One could also have used a change of variables to combine Schoenberg’s characterization of functions that are positive definite and radial on any R s with the HBW characterization of completely monotone functions. fasshauer@iit.edu MATH 590 – Chapter 5 14
Completely Monotone Functions Up to now we only have a connection between completely monotone functions and positive definite functions, but not with strictly positive definite ones! We can see from the previous proof that if the measure µ is not concentrated at the origin, then Φ is even strictly positive definite and radial on any R s . This condition on the measure is equivalent with ϕ not being constant. With this additional restriction on ϕ we can connect completely monotone function with the scattered data interpolation problem. fasshauer@iit.edu MATH 590 – Chapter 5 15
Completely Monotone Functions The following interpolation theorem already appears in [Schoenberg (1938a), p. 823]. It provides a very simple test for verifying the well-posedness of many scattered data interpolation problems. Theorem A function ϕ : [ 0 , ∞ ) → R is completely monotone but not constant if and only if ϕ ( � · � 2 ) is strictly positive definite and radial on R s for any s. Remark Schoenberg only showed “completely monotone and not constant = ⇒ strictly positive definite and radial”. A proof that the converse also holds can be found in [Wendland (2005a)]. fasshauer@iit.edu MATH 590 – Chapter 5 16
Completely Monotone Functions Example Gaussians 1 ϕ ( r ) = e − ε r , ε > 0, is completely monotone on [ 0 , ∞ ) and not constant. The Schoenberg interpolation theorem tells us that Gaussians Φ( x ) = ϕ ( � x � 2 ) = e − ε 2 � x � 2 are strictly positive definite and radial on R s for all s . Inverse multiquadrics 2 ϕ ( r ) = 1 / ( 1 + r ) β , β > 0, is completely monotone on [ 0 , ∞ ) and not constant. The Schoenberg interpolation theorem tells us that inverse multiquadrics Φ( x ) = ϕ ( � x � 2 ) = 1 / ( 1 + � x � 2 ) β are strictly positive definite and radial on R s for all s . Remark Not only is the test for complete monotonicity simpler than the Fourier transform, but we also are able to verify strict positive definiteness of the inverse multiquadrics without any dependence of s on β . fasshauer@iit.edu MATH 590 – Chapter 5 17
Completely Monotone Functions Remark For radial (or “isotropic”) strictly positive definite functions complete monotonicity is a simple test. As long as we have translation invariant (or “stationary”) strictly positive definite functions we can use Fourier transforms. If we don’t have either property, then we need to use the definition of general positive definite kernels: Definition A complex-valued continuous function K : R s × R s → C is called positive definite on R s if N N � � c j c k K ( x j , x k ) ≥ 0 (1) j = 1 k = 1 for any N pairwise different points x 1 , . . . , x N ∈ R s , and c = [ c 1 , . . . , c N ] T ∈ C N . fasshauer@iit.edu MATH 590 – Chapter 5 18
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