Does a Paley-Wiener type function have existing boundary values on to the real line?
Let $f : mathbb C^+ to mathbb C$ (where $mathbb C^+$ denotes the upper half plane) be a holomorphic function with the property that $$sup_{y>0} int_{-infty}^infty lvert f(x+iy) rvert^2 , dx = C < infty. qquad (1)$$ The Paley-Wiener theorem gives a classification of this type of function, i.e. there exists $Fin L^2(0,infty) $ such that $$f(z) = int_0^infty F(t) e^{itz}, dt$$
for $zin mathbb C^+$ and $int_0^infty lvert F(t) rvert , dt = C$ (see Big Rudin for a proof). I now found a similar type of theorem:
Let $f: mathbb C^+ to mathbb C$ be a holomorphic function with the condition that for all $epsilon > 0$ there is $C_epsilon > 0 $ such that $$lvert f(z) rvert leq C_epsilon exp(epsilon lvert z rvert)$$ for all $zin mathbb C^+$ and assume that for almost every $xin mathbb R$ $f$ has boundary values $$f_0(x) = lim_{substack{yto 0\ y > 0}}f(x+iy)$$ with the property that $$int_{mathbb R} lvert f_0(x) rvert^2<
infty.$$
Then there exists $Fin L^2(0,infty)$ such that$$f(z) = int_0^infty e^{itz}F(t) , dt$$
for all $zin mathbb C^+$.
My question: Are these two theorems or conditions imposed on the function $f$ equivalent?
Explanation: The theorems seem to be closely linked because they both give the same function as an output. So to me the question arose whether one of the condition implies the other or if they are even equivalent.
complex-analysis functional-analysis analysis fourier-analysis
asked 2 days ago
bavor42bavor42
30419
add a comment |
Let $f : mathbb C^+ to mathbb C$ (where $mathbb C^+$ denotes the upper half plane) be a holomorphic function with the property that $$sup_{y>0} int_{-infty}^infty lvert f(x+iy) rvert^2 , dx = C < infty. qquad (1)$$ The Paley-Wiener theorem gives a classification of this type of function, i.e. there exists $Fin L^2(0,infty) $ such that $$f(z) = int_0^infty F(t) e^{itz}, dt$$
for $zin mathbb C^+$ and $int_0^infty lvert F(t) rvert , dt = C$ (see Big Rudin for a proof). I now found a similar type of theorem:
Let $f: mathbb C^+ to mathbb C$ be a holomorphic function with the condition that for all $epsilon > 0$ there is $C_epsilon > 0 $ such that $$lvert f(z) rvert leq C_epsilon exp(epsilon lvert z rvert)$$ for all $zin mathbb C^+$ and assume that for almost every $xin mathbb R$ $f$ has boundary values $$f_0(x) = lim_{substack{yto 0\ y > 0}}f(x+iy)$$ with the property that $$int_{mathbb R} lvert f_0(x) rvert^2<
infty.$$
Then there exists $Fin L^2(0,infty)$ such that$$f(z) = int_0^infty e^{itz}F(t) , dt$$
for all $zin mathbb C^+$.
My question: Are these two theorems or conditions imposed on the function $f$ equivalent?
Explanation: The theorems seem to be closely linked because they both give the same function as an output. So to me the question arose whether one of the condition implies the other or if they are even equivalent.
complex-analysis functional-analysis analysis fourier-analysis
asked 2 days ago
bavor42bavor42
30419
add a comment |
Let $f : mathbb C^+ to mathbb C$ (where $mathbb C^+$ denotes the upper half plane) be a holomorphic function with the property that $$sup_{y>0} int_{-infty}^infty lvert f(x+iy) rvert^2 , dx = C < infty. qquad (1)$$ The Paley-Wiener theorem gives a classification of this type of function, i.e. there exists $Fin L^2(0,infty) $ such that $$f(z) = int_0^infty F(t) e^{itz}, dt$$
for $zin mathbb C^+$ and $int_0^infty lvert F(t) rvert , dt = C$ (see Big Rudin for a proof). I now found a similar type of theorem:
Let $f: mathbb C^+ to mathbb C$ be a holomorphic function with the condition that for all $epsilon > 0$ there is $C_epsilon > 0 $ such that $$lvert f(z) rvert leq C_epsilon exp(epsilon lvert z rvert)$$ for all $zin mathbb C^+$ and assume that for almost every $xin mathbb R$ $f$ has boundary values $$f_0(x) = lim_{substack{yto 0\ y > 0}}f(x+iy)$$ with the property that $$int_{mathbb R} lvert f_0(x) rvert^2<
infty.$$
Then there exists $Fin L^2(0,infty)$ such that$$f(z) = int_0^infty e^{itz}F(t) , dt$$
for all $zin mathbb C^+$.
My question: Are these two theorems or conditions imposed on the function $f$ equivalent?
Explanation: The theorems seem to be closely linked because they both give the same function as an output. So to me the question arose whether one of the condition implies the other or if they are even equivalent.
complex-analysis functional-analysis analysis fourier-analysis
asked 2 days ago
bavor42bavor42
30419
Let $f : mathbb C^+ to mathbb C$ (where $mathbb C^+$ denotes the upper half plane) be a holomorphic function with the property that $$sup_{y>0} int_{-infty}^infty lvert f(x+iy) rvert^2 , dx = C < infty. qquad (1)$$ The Paley-Wiener theorem gives a classification of this type of function, i.e. there exists $Fin L^2(0,infty) $ such that $$f(z) = int_0^infty F(t) e^{itz}, dt$$
for $zin mathbb C^+$ and $int_0^infty lvert F(t) rvert , dt = C$ (see Big Rudin for a proof). I now found a similar type of theorem:
Let $f: mathbb C^+ to mathbb C$ be a holomorphic function with the condition that for all $epsilon > 0$ there is $C_epsilon > 0 $ such that $$lvert f(z) rvert leq C_epsilon exp(epsilon lvert z rvert)$$ for all $zin mathbb C^+$ and assume that for almost every $xin mathbb R$ $f$ has boundary values $$f_0(x) = lim_{substack{yto 0\ y > 0}}f(x+iy)$$ with the property that $$int_{mathbb R} lvert f_0(x) rvert^2<
infty.$$
Then there exists $Fin L^2(0,infty)$ such that$$f(z) = int_0^infty e^{itz}F(t) , dt$$
for all $zin mathbb C^+$.
My question: Are these two theorems or conditions imposed on the function $f$ equivalent?
Explanation: The theorems seem to be closely linked because they both give the same function as an output. So to me the question arose whether one of the condition implies the other or if they are even equivalent.
complex-analysis functional-analysis analysis fourier-analysis
complex-analysis functional-analysis analysis fourier-analysis
asked 2 days ago
bavor42bavor42
30419
asked 2 days ago
bavor42bavor42
30419
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