A note on harmonic continuation of characteristic function
aa r X i v : . [ m a t h . C A ] S e p A note on harmonic continuation of characteristic function
Saulius Norvidas
Institute of Data Science and Digital Technologies, Vilnius University,Akademijos str. 4, Vilnius LT-04812, Lithuania(e-mail: [email protected])
Abstract
We propose a necessary and sufficient condition for a real-valued function on the real line to be a charac-teristic function of a probability measures. The statement is given in terms of harmonic functions and completelymonotonic functions.
Keywords : Characteristic functions; positive definite functions; completely monotonic functions; harmonic func-tions; the Dirichlet problem for harmonic functions.
Mathematics Subject Classification : 31A05 - 42A82 - 60E10 If σ is a probability measure on the real line R , then θ ( x ) = Z ∞ − ∞ e ixt d σ ( t ) is called the characteristic function of the probability measure σ . We will call θ the characteristicfunction for short. Note that a characteristic function is real-valued if and only if it is even on R .The motivation of the present paper stems from Egorov’s paper [2, p. 567], where the followingcriterion for real-valued infinitely differentiable characteristic functions is given: Theorem 1 . Let ϕ ( t ) be an infinitely differentiable absolutely integrable even function on R with real values. Let ϕ ( t ) satisfy the condition ϕ ( m ) ( t ) t − = O ( ) as t → ∞ , m = , , . . . , and letthe Fourier transform b ϕ belong to L ( R ) . Then ϕ ( t ) is a characteristic function if and only if1) for all < δ < ∞ and all p = , , , ... ( − ) p Z ∞ δ + t ϕ ( p ) ( t ) dt ≥ , ( − ) p + Z ∞ t δ + t ϕ ( p + ) ( t ) dt ≥ ϕ ( ) = . We claim that some conditions of Theorem 1 are unnecessary. Furthermore, this criterion cannot be applied to the simplest infinitely differentiable characteristic functions ϕ ( t ) ≡ ϕ ( t ) = os at , where a ∈ R (since they are not absolutely integrable on R ). Therefore, we will follow herea different approach to a similar criterion associated with the theory of harmonic functions. Thisapproach allow us to simplify and strengthen the statement of Theorem 1; compare Corollary 3below.Let us start by introducing some notation and basic facts that will be used throughout thispaper. Let CB ( R ) denote the Banach space of bounded continuous functions f : R → R with theusual uniform norm. If f ∈ CB ( R ) , then, as is well known (see, for example [5, Chapter II]), theDirichlet problem for the upper half plane R + = { ( x , y ) ∈ R : y > } has only unique solutionin CB ( R + ) , where R + is the closure of R + . More precisely, given f ∈ CB ( R ) , there exists afunction u f on R + such that u f is bounded and continuous in R + ,lim y → u f ( x , y ) = f ( x ) (1.1)for each x ∈ R , and u f is harmonic in R + . This means that u f is infinitely differentiable in R + and there satisfies (cid:18) ∂ ∂ x + ∂ ∂ y (cid:19) u f = . We call u f the harmonic continuation of f into R + . The solution of the Dirichlet problem for R + can be obtained by using the Poisson kernel P ( x , y ) = π yx + y . (1.2)Namely, the harmonic continuation u f can be obtained as a convolution with respect to x of f and P ( x , y ) u f ( x , y ) = (cid:16) f ∗ x P (cid:17) ( x , y ) = π Z ∞ − ∞ y ( x − t ) + y f ( t ) dt . (1.3)where ( x , y ) ∈ R + .We recall that a function ω : [ , ∞ ) → R is called completely monotonic if it is infinitelydifferentiable on [ , ∞ ) and ( − ) n ω ( n ) ( x ) ≥ x ∈ ( , ∞ ) and all n = , , , . . . . The following elementary functions are immediate examplesof completely monotonic functions, which is verified directly: e − ax , ( α x + β ) p , and ln (cid:16) b + cx (cid:17) , where a ≥ α ≥ β ≥
0, and p ≥ α and β not both zero and b ≥ c > Theorem 2 . Suppose that f ∈ CB ( R ) is an even function and f ( ) = . Then f is a characteris-tic function if and only if the restriction of u f to the imaginary axis, i.e., the function y → u f ( , y ) ,y ∈ [ , ∞ ) , is completely monotonic. orollary 3 . Let f ∈ CB ( R ) . Suppose(i) f is infinitely differentiable on R and f ( k ) ∈ CB ( R ) for all k = , . . . ;(ii) f is even and f ( ) = .Then f is characteristic function if and only if Z ∞ − ∞ ℑ (cid:16) t − iy (cid:17) f ( t ) dt ≥ , (1.5) ( − ) n + Z ∞ − ∞ ℜ (cid:16) ( t − iy ) (cid:17) f ( n ) ( t ) dt ≥ and ( − ) n Z ∞ − ∞ ℑ (cid:16) ( t − iy ) (cid:17) f ( n + ) ( t ) dt ≥ for y > and all n = , , , . . . . A complex-valued function ϕ on R is said to be positive definite if n ∑ i , j = ϕ ( x i − x j ) c i c j ≥ x , . . . , , x n ∈ R , for every choice of complex numbers c , . . . , c n ∈ C , andall n ∈ N . The Bochner theorem (see [3, p. 150]) characterizes continuous positive definitefunctions: a continuous function ϕ : R → C is positive definite if and only if there exists a finitenon-negative measure µ on R such that ϕ ( x ) = Z ∞ − ∞ e ixt d µ ( t ) . Note that ϕ is a characteristic function if and only if ϕ is continuous positive definite and ϕ ( ) =
1. Any characteristic function ϕ satisfies:(i) ϕ is uniformly continuous on R ;(ii) ϕ is bounded on R , i.e., ϕ ( x ) | ≤ ϕ ( ) = x ∈ R ;(iii) The real part of θ ( x ) is also an even characteristic function.In the case of completely monotonic functions, the Bernstein-Widder theorem (see [7, p. 161])asserts that a function f : [ , ∞ ) → R is completely monotonic if and only if it is the Laplacetransform of a finite non-negative measure η supported on [ , ∞ ) , i.e., ϕ ( x ) = Z ∞ e − xt d η ( t ) (2.1)for x ∈ [ , ∞ ) .Let f ∈ CB ( R ) . It is well known that harmonic continuation (1.3) has harmonic conjugate.Recall that if U is a harmonic function in R + , then another function V harmonic in R + is called armonic conjugate of U , provided U + iV is analytic in C + = { z = x + iy ∈ C : y > } . Theharmonic conjugate V is unique, up to adding a constant. Let v f denote a harmonic conjugate of u f . We can choose (see [4, p.p. 108-109]) v f ( x , y ) = π Z ∞ − ∞ (cid:16) x − t ( x − t ) + y + tt + (cid:17) f ( t ) dt , (2.2)where x ∈ R and y >
0. Then E f ( z ) = E f ( x , y ) = u f ( x , y ) + iv f ( x , y ) (2.3)is analytic in C + .We note that usually the following simpler integral e v f ( x , y ) = π Z ∞ − ∞ x − t ( x − t ) + y f ( t ) dt (2.4)chosen as the harmonic conjugate of u f . In that case the analytic function u f + i e v f coincides withthe usual Cauchy type integral ( u f + i e v f )( z ) = i π Z ∞ − ∞ f ( t ) z − t dt for z ∈ C + . However, the integral in (2.4) converges as long as Z ∞ − ∞ | f ( t ) | + | t | dt < ∞ . For example, this condition is satisfied if f ∈ L p ( R ) , 1 ≤ p < ∞ , but not in the case of an arbitrary f ∈ CB ( R ) . Proof of Theorem 2 . Suppose that f is a real-valued characteristic function. By the Bochnertheorem, there exists a probability measure σ on R such that f ( t ) = Z ∞ − ∞ e ixt d σ ( x ) . (2.5)For each y > x on R . Hence, bythe Fubini theorem, it follows from (1.3) and (2.5) that u f ( , y ) = π Z ∞ − ∞ (cid:16) Z ∞ − ∞ P ( t , y ) e ixt dt (cid:17) d σ ( x ) . Since Z ∞ − ∞ P ( t , y ) e ixt dt = e −| x | y , it follows that u f ( , y ) = Z ∞ − ∞ e −| x | y d σ ( x ) . (2.6)For each y > n = , , , . . . , there exists 0 < a ( y , n ) < ∞ such thatsup x ∈ R ∂ n ∂ y n e −| x | y ≤ a ( y , n ) . herefore, if we recall that σ is a finite measure, then we have that the partial derivatives of u f ( , y ) in y can be obtained by differentiation under the integral sign in (2.6) (see, for example,[1, p. 283]). Now the positiveness σ implies that ( − ) n (cid:16) ∂ n ∂ y n u f (cid:17) ( , y ) = Z ∞ − ∞ | x | n e −| x | y d σ ( x ) ≥ y > n = , , , . . . . Note that the harmonic function u f ( x , y ) is continuous in R + .Therefore, by the Bernstein-Widder theorem, we obtain that u f ( , y ) is completely monotonic on [ , ∞ ) .Suppose u f ( , y ) is completely monotonic for y ∈ [ , ∞ ) . By the Bernstein-Widder theorem,there exists a finite non-negative measure η supported on [ , ∞ ) such that u f ( , y ) = Z ∞ e − yt d η ( t ) , (2.7) y ∈ [ , ∞ ) . According to (1.1), we getlim y → u f ( , y ) = f ( ) = . (2.8)This means that η is a probability measure. Set K η ( z ) = Z ∞ e izt d η ( t ) (2.9)for z ∈ C + = { z = x + iy ∈ C : y ≥ } . We can consider the measure η as tempered distributionon R . Then K η is the distributional Laplace transform of η (see [6, p. 127]). Therefore, since η is supported on [ , ∞ ) , we have that K η is analytic function in R + i ( , ∞ ) = C + and the followingdifferentation formula in C + holds d n d z n K η ( z ) = ( i ) n Z ∞ e ixt (cid:16) t n e − yt (cid:17) d η ( t ) , [6, p.p. 127-128].We claim that K η and (2.3) coincide in C + . Indeed, since f is even on R , it follows from (2.2)that v f ( , y ) = y >
0. Hence E f ( iy ) = u f ( , y ) (2.10)for y >
0. On the other hand, using (2.7) and (2.9), we have K η ( iy ) = u f ( , y ) , y >
0. If we combine this with (2.10) and use the uniqueness theorem for analytic functions, weobtain that E f = K η in C + .By (2.9), we have that for any fixed y ∈ ( , ∞ ) the function x → K η ( x + iy ) is positive definitefor x ∈ R . Therefore the function x → ℜ (cid:2) K η ( x + iy ) (cid:3) also is positive definite for x ∈ R . Since E f = K η in C + , it follows from (2.3) that u f ( x , y ) = ℜ (cid:2) E f ( x , y ) (cid:3) = ℜ (cid:2) K η ( x + iy ) (cid:3) or x ∈ R . Thus each function x → u f ( x , y ) , where y ∈ ( , ∞ ) , is positive definite on R . Sincethe pointwise limit of positive definite functions also is positive definite function, (1.1) impliesthat f is positive definite on R . Finally, since f is continuous on R , it follows from (2.8) and theBochner theorem that f is a characteristic function. This proves Theorem 2. Proof of Corollary 3 . By Theorem 2 and (1.4), a necessary and sufficient condition for f tobe a characteristic function is that ( − ) k h ∂ k ∂ y k u f ( , y ) i ≥ k = , , , . . . , and y >
0. In the case k =
0, the condition (2.11) is u f ( , y ) = π Z ∞ − ∞ yt + y f ( t ) dt = π Z ∞ − ∞ ℑ (cid:16) t − iy (cid:17) f ( t ) dt ≥ . (2.12)Let us calculate the partial derivatives in (2.11). To this end, we recall that if F is a analyticfunction, then the Cauchy-Riemann equations imply ddz F ( z ) = ∂∂ y (cid:16) ℑ F ( z ) (cid:17) − i ∂∂ y (cid:16) ℜ F ( z ) (cid:17) . Hence, in the case if F is the function (2.3), then we get d n dz n E f ( z ) = ( − ) n h ∂ n ∂ y n u f ( x , y ) + i ∂ n ∂ y n v f ( x , y ) i for n = , , . . . , and d n + dz n + E f ( z ) = ( − ) n h ∂ n + ∂ y n + v f ( x , y ) − i ∂ n + ∂ y n + u f ( x , y ) i for n = , , , . . . , where z ∈ C + . Therefore ∂ n ∂ y n u f ( x , y ) = ( − ) n ℜ h d n dz n E f ( z ) i (2.13)for n = , , . . . , and ∂ n + ∂ y n + u f ( x , y ) = ( − ) n + ℑ h d n + dz n + E f ( z ) i (2.14)for n = , , , . . . .On the other hand, by (1.3), (2.2) and (2.3), we have E f ( z ) = i π Z ∞ − ∞ (cid:16) z − t + tt + (cid:17) f ( t ) dt . Hence d k dz k E f ( z ) = ( − ) k i k ! π Z ∞ − ∞ f ( t )( z − t ) k + dt = ( − ) k i π Z ∞ − ∞ f ( k − ) ( t )( z − t ) dt , where k = , , . . . . Therefore, (2.13) and (2.14) imply ∂ n ∂ y n u f ( , y ) = ( − ) n ℜ h d n dz n E f ( z ) i ( z = iy ) = ( − ) n ℜ h i π Z ∞ − ∞ f ( n − ) ( t )( iy − t ) dt i = ( − ) n + π Z ∞ − ∞ ℑ h ( t − iy ) i f ( n − ) ( t ) dt (2.15) or n = , , . . . , and ∂ n + ∂ y n + u f ( , y ) = ( − ) n + ℑ h d n + dz n + E f ( z ) i ( z = iy )= ( − ) n + ℑ h − i π Z ∞ − ∞ f ( n ) ( t )( iy − t ) dt i = ( − ) n π Z ∞ − ∞ ℜ h ( t − iy ) i f ( n ) ( t ) dt (2.16)where n = , , , . . . .Finally, (2.12), (2.15), and (2.16) show that (2.11) is equivalent to the conditions (1.5)–(1.7). References [1] T.M. Apostol,
Mathematical analysis , 2nd ed. Addison-Wesley Publishing Co. Mass.-London-Don Mills. 1974.[2] A.V. Egorov, On the theory of characteristic functions. (English. Russian original) Russ.Math. Surv. (3), 567-568 (2004); translation from Usp. Mat. Nauk (3), 167-168 (2004).[3] Y. Katznelson, An introduction to harmonic analysis . 3nd ed. Cambridge Mathematical Li-brary. Cambridge University Press. Cambridge 2004.[4] P. Koosis,
Introduction to H p spaces , 2nd ed. Cambridge Tracts in Mathematics . Cam-bridge University Press. Cambridge 1998.[5] E.M. Stein and G. Weiss, Introduction to Fourier analysis in Euclidean spaces . PrincetonMathematical Series, No. . Princeton University Press. Princeton 1971.[6] V.S. Vladimirov, Methods of the theory of generalized functions . Taylor&Francis. London2002.[7] D.V. Widder,
The Laplace Transform , Princeton Mathematical Series, v. . Princeton Uni-versity Press, Princeton, N. J. 1941.. Princeton Uni-versity Press, Princeton, N. J. 1941.