In mathematical analysis, Haar's Tauberian theorem named after Alfréd Haar, relates the asymptotic behaviour of a continuous function to properties of its Laplace transform. It is related to the integral formulation of the Hardy–Littlewood Tauberian theorem.
Simplified version by Feller William Feller gives the following simplified form for this theorem: Suppose that f ( t ) {\displaystyle f(t)} is a non-negative and continuous function for t ≥ 0 {\displaystyle t\geq 0} , having finite Laplace transform
F ( s ) = ∫ 0 ∞ e − s t f ( t ) d t {\displaystyle F(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt}
for s > 0 {\displaystyle s>0} . Then F ( s ) {\displaystyle F(s)} is well defined for any complex value of s = x + i y {\displaystyle s=x+iy} with x > 0 {\displaystyle x>0} . Suppose that F {\displaystyle F} verifies the following conditions: 1. For y ≠ 0 {\displaystyle y\neq 0} the function F ( x + i y ) {\displaystyle F(x+iy)} (which is regular on the right half-plane x > 0 {\displaystyle x>0} ) has continuous boundary values F ( i y ) {\displaystyle F(iy)} as x → + 0 {\displaystyle x\to +0} , for x ≥ 0 {\displaystyle x\geq 0} and y ≠ 0 {\displaystyle y\neq 0} , furthermore for s = i y {\displaystyle s=iy} it may be written as
F ( s ) = C s + ψ ( s ) , {\displaystyle F(s)={\frac {C}{s}}+\psi (s),}
where ψ ( i y ) {\displaystyle \psi (iy)} has finite derivatives ψ ′ ( i y ) , … , ψ ( r ) ( i y ) {\displaystyle \psi '(iy),\ldots ,\psi ^{(r)}(iy)} and ψ ( r ) ( i y ) {\displaystyle \psi ^{(r)}(iy)} is bounded in every finite interval; 2. The integral
∫ 0 ∞ e i t y F ( x + i y ) d y {\displaystyle \int _{0}^{\infty }e^{ity}F(x+iy)\,dy}
converges uniformly with respect to t ≥ T {\displaystyle t\geq T} for fixed x > 0 {\displaystyle x>0} and T > 0 {\displaystyle T>0} ; 3. F ( x + i y ) → 0 {\displaystyle F(x+iy)\to 0} as y → ± ∞ {\displaystyle y\to \pm \infty } , uniformly with respect to x ≥ 0 {\displaystyle x\geq 0} ; 4. F ′ ( i y ) , … , F ( r ) ( i y ) {\displaystyle F'(iy),\ldots ,F^{(r)}(iy)} tend to zero as y → ± ∞ {\displaystyle y\to \pm \infty } ; 5. The integrals
∫ − ∞ y 1 e i t y F ( r ) ( i y ) d y {\displaystyle \int _{-\infty }^{y_{1}}e^{ity}F^{(r)}(iy)\,dy} and ∫ y 2 ∞ e i t y F ( r ) ( i y ) d y {\displaystyle \int _{y_{2}}^{\infty }e^{ity}F^{(r)}(iy)\,dy}
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