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Exponential type
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<h2 id="basic-idea">Basic idea</h2> <p>A function f ( z ) {\displaystyle f(z)} defined on the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> is said to be of exponential type if there exist real-valued constants M {\displaystyle M} and τ {\displaystyle \tau } such that </p> | f ( r e i θ ) | ≤ M e τ r {\displaystyle \left|f\left(re^{i\theta }\right)\right|\leq Me^{\tau r}} <p>in the limit of r → ∞ {\displaystyle r\to \infty } . Here, the <a href="/facts/Complex_variable/hgCoQlH0">complex variable</a> z {\displaystyle z} was written as z = r e i θ {\displaystyle z=re^{i\theta }} to emphasize that the limit must hold in all directions θ {\displaystyle \theta } . Letting τ {\displaystyle \tau } stand for the <a href="/facts/Infimum/oXCKVopz">infimum</a> of all such τ {\displaystyle \tau } , one then says that the function f {\displaystyle f} is of <i>exponential type τ {\displaystyle \tau } </i>. </p><p>For example, let f ( z ) = sin ( π z ) {\displaystyle f(z)=\sin(\pi z)} . Then one says that sin ( π z ) {\displaystyle \sin(\pi z)} is of exponential type π {\displaystyle \pi } , since π {\displaystyle \pi } is the smallest number that bounds the growth of sin ( π z ) {\displaystyle \sin(\pi z)} along the imaginary axis. So, for this example, <a href="/facts/Carlson%2527s_theorem/p7524I9a">Carlson's theorem</a> cannot apply, as it requires functions of exponential type less than π {\displaystyle \pi } . Similarly, the <a href="/facts/Euler%25E2%2580%2593Maclaurin_formula/mN6uNCzN">Euler–Maclaurin formula</a> cannot be applied either, as it, too, expresses a theorem ultimately anchored in the theory of <a href="/facts/Finite_difference/aY1txFhj">finite differences</a>. </p> <h2 id="formal-definition">Formal definition</h2> <p>A <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a> F ( z ) {\displaystyle F(z)} is said to be of exponential type σ > 0 {\displaystyle \sigma >0} if for every ε > 0 {\displaystyle \varepsilon >0} there exists a real-valued constant A ε {\displaystyle A_{\varepsilon }} such that </p> | F ( z ) | ≤ A ε e ( σ + ε ) | z | {\displaystyle |F(z)|\leq A_{\varepsilon }e^{(\sigma +\varepsilon )|z|}} <p>for | z | → ∞ {\displaystyle |z|\to \infty } where z ∈ C {\displaystyle z\in \mathbb {C} } . We say F ( z ) {\displaystyle F(z)} is of exponential type if F ( z ) {\displaystyle F(z)} is of exponential type σ {\displaystyle \sigma } for some σ > 0 {\displaystyle \sigma >0} . The number </p> τ ( F ) = σ = lim sup | z | → ∞ | z | − 1 log | F ( z ) | {\displaystyle \tau (F)=\sigma =\displaystyle \limsup _{|z|\rightarrow \infty }|z|^{-1}\log |F(z)|} <p>is the exponential type of F ( z ) {\displaystyle F(z)} . The <a href="/facts/Limit_superior/QeAhlygs">limit superior</a> here means the limit of the <a href="/facts/Supremum/oXCKVopz">supremum</a> of the ratio outside a given radius as the radius goes to infinity. This is also the limit superior of the maximum of the ratio at a given radius as the radius goes to infinity. The limit superior may exist even if the maximum at radius r {\displaystyle r} does not have a limit as r {\displaystyle r} goes to infinity. For example, for the function </p> F ( z ) = ∑ n = 1 ∞ z 10 n ! ( 10 n ! ) ! {\displaystyle F(z)=\sum _{n=1}^{\infty }{\frac {z^{10^{n!}}}{(10^{n!})!}}} <p>the value of </p> ( max | z | = r log | F ( z ) | ) / r {\displaystyle (\max _{|z|=r}\log |F(z)|)/r} <p>at r = 10 n ! − 1 {\displaystyle r=10^{n!-1}} is dominated by the n − 1 st {\displaystyle n-1^{\text{st}}} term so we have the asymptotic expressions: </p> ( max | z | = 10 n ! − 1 log | F ( z ) | ) / 10 n ! − 1 ∼ ( log ( 10 n ! − 1 ) 10 ( n − 1 ) ! ( 10 ( n − 1 ) ! ) ! ) / 10 n ! − 1 ∼ ( log 10 ) [ ( n ! − 1 ) 10 ( n − 1 ) ! − 10 ( n − 1 ) ! ( n − 1 ) ! ] / 10 n ! − 1 ∼ ( log 10 ) ( n ! − 1 − ( n − 1 ) ! ) / 10 n ! − 1 − ( n − 1 ) ! {\displaystyle {\begin{aligned}\left(\max _{|z|=10^{n!-1}}\log |F(z)|\right)/10^{n!-1}&\sim \left(\log {\frac {(10^{n!-1})^{10^{(n-1)!}}}{(10^{(n-1)!})!}}\right)/10^{n!-1}\\&\sim (\log 10)\left[(n!-1)10^{(n-1)!}-10^{(n-1)!}(n-1)!\right]/10^{n!-1}\\&\sim (\log 10)(n!-1-(n-1)!)/10^{n!-1-(n-1)!}\\\end{aligned}}} <p>and this goes to zero as n {\displaystyle n} goes to infinity,<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> but F ( z ) {\displaystyle F(z)} is nevertheless of exponential type 1, as can be seen by looking at the points z = 10 n ! {\displaystyle z=10^{n!}} . </p> <h2 id="exponential-type-with-respect-to-a-symmetric-convex-body">Exponential type with respect to a symmetric convex body</h2> <p>Stein (1957) has given a generalization of exponential type for <a href="/facts/Entire_function/XNluKzu7">entire functions</a> of <a href="/facts/Several_complex_variables/axwNpV4a">several complex variables</a>. Suppose K {\displaystyle K} is a <a href="/facts/Convex_set/vdAuJRJl">convex</a>, <a href="/facts/Compact_element/1aLoXrfR">compact</a>, and <a href="/facts/Symmetric/hpu7DK8p">symmetric</a> subset of R n {\displaystyle \mathbb {R} ^{n}} . It is known that for every such K {\displaystyle K} there is an associated <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> ‖ ⋅ ‖ K {\displaystyle \|\cdot \|_{K}} with the property that </p> K = { x ∈ R n : ‖ x ‖ K ≤ 1 } . {\displaystyle K=\{x\in \mathbb {R} ^{n}:\|x\|_{K}\leq 1\}.} <p>In other words, K {\displaystyle K} is the unit ball in R n {\displaystyle \mathbb {R} ^{n}} with respect to ‖ ⋅ ‖ K {\displaystyle \|\cdot \|_{K}} . The set </p> K ∗ = { y ∈ R n : x ⋅ y ≤ 1 for all x ∈ K } {\displaystyle K^{*}=\{y\in \mathbb {R} ^{n}:x\cdot y\leq 1{\text{ for all }}x\in {K}\}} <p>is called the <a href="/facts/Polar_set/JY45pfbt">polar set</a> and is also a <a href="/facts/Convex_set/vdAuJRJl">convex</a>, <a href="/facts/Compact_element/1aLoXrfR">compact</a>, and <a href="/facts/Symmetric/hpu7DK8p">symmetric</a> subset of R n {\displaystyle \mathbb {R} ^{n}} . Furthermore, we can write </p> ‖ x ‖ K = sup y ∈ K ∗ | x ⋅ y | . {\displaystyle \|x\|_{K}=\displaystyle \sup _{y\in K^{*}}|x\cdot y|.} <p>We extend ‖ ⋅ ‖ K {\displaystyle \|\cdot \|_{K}} from R n {\displaystyle \mathbb {R} ^{n}} to C n {\displaystyle \mathbb {C} ^{n}} by </p> ‖ z ‖ K = sup y ∈ K ∗ | z ⋅ y | . {\displaystyle \|z\|_{K}=\displaystyle \sup _{y\in K^{*}}|z\cdot y|.} <p>An entire function F ( z ) {\displaystyle F(z)} of n {\displaystyle n} -complex variables is said to be of exponential type with respect to K {\displaystyle K} if for every ε > 0 {\displaystyle \varepsilon >0} there exists a real-valued constant A ε {\displaystyle A_{\varepsilon }} such that </p> | F ( z ) | < A ε e 2 π ( 1 + ε ) ‖ z ‖ K {\displaystyle |F(z)|<A_{\varepsilon }e^{2\pi (1+\varepsilon )\|z\|_{K}}} <p>for all z ∈ C n {\displaystyle z\in \mathbb {C} ^{n}} . </p> <h2 id="frchet-space">Fréchet space</h2> <p>Collections of functions of exponential type τ {\displaystyle \tau } can form a <a href="/facts/Complete_space/lsckmU8G">complete</a> <a href="/facts/Uniform_space/GKumHlHo">uniform space</a>, namely a <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet space</a>, by the <a href="/facts/Topological_space/v2ut7CbK">topology</a> induced by the countable family of <a href="/facts/Norm_(mathematics)/xIbR4uE1">norms</a> </p> ‖ f ‖ n = sup z ∈ C exp [ − ( τ + 1 n ) | z | ] | f ( z ) | . {\displaystyle \|f\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)|z|\right]|f(z)|.} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Paley%25E2%2580%2593Wiener_theorem/02pKjt9F">Paley–Wiener theorem</a></li> <li>Paley–Wiener space</li></ul> <ul><li><a href="/facts/Elias_M._Stein/9IRxdGWp">Stein, E.M.</a> (1957), "Functions of exponential type", <i>Ann. of Math.</i>, 2, 65 (3): 582–592, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1970066">10.2307/1970066</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1970066">1970066</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0085342">0085342</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>In fact, even ( max | z | = r log log | F ( z ) | ) / ( log r ) {\displaystyle (\max _{|z|=r}\log \log |F(z)|)/(\log r)} goes to zero at r = 10 n ! − 1 {\displaystyle r=10^{n!-1}} as n {\displaystyle n} goes to infinity. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>