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Hadamard's gap theorem: If the indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h0460901.png" /> of all non-zero coefficients of the power series
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{{TEX|done}}
 +
==Hadamard's gap theorem==
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h0460902.png" /></td> </tr></table>
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If the indices $n_1,n_2,\ldots,$ of all non-zero coefficients of the power series
 +
 
 +
$$f(z)=\sum_{n=0}^\infty a_nz^n$$
  
 
satisfy the condition
 
satisfy the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h0460903.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
\begin{equation}\label{had}n_{k+1}>(1+\theta)n_k,\end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h0460904.png" />, then the boundary of the disc of convergence of this series is its natural boundary, i.e. the function has no [[Analytic continuation|analytic continuation]] across the boundary of this disc. Condition (*) is known as Hadamard's condition; the gaps which satisfy the Hadamard condition are called Hadamard gaps. See also [[Lacunary series|Lacunary series]]; [[Fabry theorem|Fabry theorem]].
+
where $\theta>0$, then the boundary of the disc of convergence of this series is its natural boundary, i.e. the function has no [[Analytic continuation|analytic continuation]] across the boundary of this disc. Condition \ref{had} is known as Hadamard's condition; the gaps which satisfy the Hadamard condition are called Hadamard gaps. See also [[Lacunary series|Lacunary series]]; [[Fabry theorem|Fabry theorem]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hadamard,  "Essai sur l'étude des fonctions données par leurs développement de Taylor"  ''J. Math. Pures Appl. (4)'' , '''8'''  (1892)  pp. 101–186</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Bieberbach,  "Analytische Fortsetzung" , Springer  (1955)  pp. Sect. 3</TD></TR></table>
 
 
 
 
====Comments====
 
  
 +
{|
 +
|-
 +
|valign="top"|{{Ref|Bi}}||valign="top"| L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3 {{MR|0068621}} {{ZBL|0064.06902}}
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| G. Bourion, "L'ultraconvergence dans les séries de Taylor" , Hermann (1937)
 +
|-
 +
|valign="top"|{{Ref|Di}}||valign="top"| P. Dienes, "The Taylor series" , Oxford Univ. Press (1931) {{MR|0089895}} {{MR|1522577}} {{ZBL|0003.15502}} {{ZBL|57.0339.10}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}||valign="top"| J. Hadamard, "Essai sur l'étude des fonctions données par leurs développement de Taylor" ''J. Math. Pures Appl. (4)'' , '''8''' (1892) pp. 101–186
 +
|-
 +
|}
  
====References====
+
==Hadamard's theorem on entire functions==
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Dienes,  "The Taylor series" , Oxford Univ. Press  (1931)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Bourion,  "L'ultraconvergence dans les séries de Taylor" , Hermann  (1937)</TD></TR></table>
 
  
Hadamard's theorem on entire functions: A theorem on the representation of an [[Entire function|entire function]] by means of its zeros; it makes more precise the [[Weierstrass theorem|Weierstrass theorem]] on infinite products in the case of an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h0460905.png" /> of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h0460906.png" />. If, for the sake of simplicity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h0460907.png" />, then
+
A theorem on the representation of an [[Entire function|entire function]] by means of its zeros; it makes more precise the [[Weierstrass theorem|Weierstrass theorem]] on infinite products in the case of an entire function $f(z)$ of finite order $\rho$. If, for the sake of simplicity, $f(0)\neq 0$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h0460908.png" /></td> </tr></table>
+
$$f(z)=e^{Q(z)}P(z),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h0460909.png" /> is a polynomial of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609010.png" /> and
+
where $Q(z)$ is a polynomial of degree not exceeding $\rho$ and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609011.png" /></td> </tr></table>
+
$$P(z)=\prod_{k=1}^\infty W\left(\frac{z}{a_k},q\right)=\prod_{k=1}^\infty\left(1-\frac{z}{a_k}\right)e^{P_k(z)}$$
  
is Weierstrass' [[Canonical product|canonical product]] of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609012.png" />, constructed from the zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609014.png" />. In other words, Hadamard's theorem postulates that the genus of an entire function does not exceed its order. This theorem was used by J. Hadamard in proving an asymptotic law for the distribution of prime numbers.
+
is Weierstrass' [[Canonical product|canonical product]] of genus $q\leq \rho$, constructed from the zeros $a_k$ of $f(z)$. In other words, Hadamard's theorem postulates that the genus of an entire function does not exceed its order. This theorem was used by J. Hadamard in proving an asymptotic law for the distribution of prime numbers.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hadamard,  "Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann"  ''J. Math. Pures Appl. (4)'' , '''9'''  (1893)  pp. 171–215</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.Ya. Levin,  "Distribution of zeros of entire functions" , Amer. Math. Soc.  (1964)  (Translated from Russian)</TD></TR></table>
 
  
 +
{|
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| R.P. Boas, "Entire functions" , Acad. Press (1954) {{MR|0068627}} {{ZBL|0058.30201}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}||valign="top"|J. Hadamard, "Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann" ''J. Math. Pures Appl. (4)'' , '''9''' (1893) pp. 171–215
 +
|-
 +
|valign="top"|{{Ref|Ma}}||valign="top"|A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}}
 +
|-
 +
|valign="top"|{{Ref|Le}}||valign="top"|B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1964) (Translated from Russian) {{MR|0156975}} {{ZBL|0152.06703}}
 +
|-
 +
|valign="top"|{{Ref|Ti}}||valign="top"| E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) {{MR|0593142}} {{MR|0197687}} {{MR|1523319}} {{ZBL|0477.30001}} {{ZBL|0336.30001}} {{ZBL|0005.21004}} {{ZBL|65.0302.01}} {{ZBL|58.0297.01}}
 +
|-
 +
|}
  
 +
==Hadamard's theorem on determinants==
  
====Comments====
+
Let $D$ be the determinant of the matrix $A$ with complex entries $a_{\mu\nu}$, $\mu,\nu=1,\ldots,n$. The following inequality is then valid:
  
 +
\begin{equation}\label{haddet}\lvert D\rvert^2\leq\prod_{\mu=1}^n\left(\sum_{\nu=1}^n\lvert a_{\mu\nu}\rvert^2\right).\end{equation}
  
====References====
+
This inequality becomes an equality if and only if
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.P. Boas,  "Entire functions" , Acad. Press  (1954)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of functions" , Oxford Univ. Press  (1979)</TD></TR></table>
 
  
Hadamard's theorem on determinants: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609015.png" /> be the determinant of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609016.png" /> with complex entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609018.png" />. The following inequality is then valid:
+
$$a_{\mu1}\overline{a}_{\nu 1}+\cdots +a_{\mu n}\overline{a}_{\nu n}=0$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
for each pair of different $\mu,\nu$, or if at least one of the factors on the right-hand side of \ref{haddet} is zero. The geometrical meaning of this theorem is that the volume of a parallelepipedon in an $n$-dimensional space is never larger than the product of the lengths of its sides issuing from one vertex, and is equal to this product if the sides are mutually perpendicular or if the length of one of the sides is zero.
  
This inequality becomes an equality if and only if
+
In the special case when all entries $a_{\mu\nu}$ of $A$ are real numbers with $\lvert a_{\mu\nu}\rvert\leq 1$, one obtains $D^2\neq n^2$, with equality if and only if all entries are either $+1$ or $-1$ and $A$ satisfies the condition $AA^T=nI$. Such a matrix is called a [[Hadamard matrix|Hadamard matrix]] of order $n$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609020.png" /></td> </tr></table>
+
For references see [[Hadamard matrix|Hadamard matrix]].
 
 
for each pair of different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609021.png" />, or if at least one of the factors on the right-hand side of (*) is zero. The geometrical meaning of this theorem is that the volume of a parallelepipedon in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609022.png" />-dimensional space is never larger than the product of the lengths of its sides issuing from one vertex, and is equal to this product if the sides are mutually perpendicular or if the length of one of the sides is zero.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hadamard,  "Résolution d'une question relative aux déterminants"  ''Bull. Sci. Math. (2)'' , '''17'''  (1893)  pp. 240–246</TD></TR></table>
 
  
''O.A. Ivanova''
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ha}}||valign="top"| J. Hadamard, "Résolution d'une question relative aux déterminants" ''Bull. Sci. Math. (2)'' , '''17''' (1893) pp. 240–246
 +
|-
 +
|}
  
====Comments====
+
==Hadamard's three-circle theorem==
In the special case when all entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609024.png" /> are real numbers with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609025.png" />, one obtains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609026.png" />, with equality if and only if all enties are either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609027.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609029.png" /> satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609030.png" />. Such a matrix is called a [[Hadamard matrix|Hadamard matrix]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609031.png" />.
 
  
For references see [[Hadamard matrix|Hadamard matrix]].
+
If $f(z)$ is a holomorphic function of a complex variable $z=re^{i\phi}$ in the annulus $0<r_1<r<r_2<\infty$, which is continuous in the closed annulus $r_1\leq r\leq r_2$, and if $M(r)=\max\lvert f(z)\rvert$ where $\lvert z\rvert=r$, then the following inequality is valid for $r_1\leq r\leq r_2$:
  
Hadamard's three-circle theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609032.png" /> is a holomorphic function of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609033.png" /> in the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609034.png" />, which is continuous in the closed annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609035.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609036.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609037.png" />, then the following inequality is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609038.png" />:
+
$$\log M(r)\leq\frac{\log\frac{r_2}{r}}{\log\frac{r_2}{r_1}}\log M(r_1)+\frac{\log\frac{r}{r_1}}{\log\frac{r_2}{r_1}}\log M(r_2).$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609039.png" /></td> </tr></table>
+
The meaning of this inequality is that $\log M(r)$ is a [[Convex function (of a real variable)|convex function (of a real variable)]] of $\log r$. This theorem of Hadamard is a special case of the [[Two-constants theorem|two-constants theorem]].
 
 
The meaning of this inequality is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609040.png" /> is a [[Convex function (of a real variable)|convex function (of a real variable)]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609041.png" />. This theorem of Hadamard is a special case of the [[Two-constants theorem|two-constants theorem]].
 
  
 
Hadamard's theorem can be generalized in various directions; in particular, there are generalizations for other metrics and for harmonic and subharmonic functions.
 
Hadamard's theorem can be generalized in various directions; in particular, there are generalizations for other metrics and for harmonic and subharmonic functions.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hadamard,  "Sur la distribution des zéros de la fonction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609042.png" /> et ses conséquences arithmétiques"  ''Bull. Soc. Math. France'' , '''24'''  (1896)  pp. 199–220</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. Privalov,  "Subharmonic functions" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.D. Solomentsev,  "A three-sphere theorem for harmonic functions"  ''Dokl. Akad. Nauk ArmSSR'' , '''42''' :  5  (1966)  pp. 274–278  (In Russian)</TD></TR></table>
 
  
 +
{|
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"|R.P. Boas, "Entire functions" , Acad. Press (1954) {{MR|0068627}} {{ZBL|0058.30201}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}||valign="top"| J. Hadamard, "Sur la distribution des zéros de la fonction $\zeta(s)$ et ses conséquences arithmétiques" ''Bull. Soc. Math. France'' , '''24''' (1896) pp. 199–220 {{MR|}} {{ZBL|27.0154.01}}
 +
|-
 +
|valign="top"|{{Ref|Ma}}||valign="top"| A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}}
 +
|-
 +
|valign="top"|{{Ref|Pr}}||valign="top"| I.I. Privalov, "Subharmonic functions" , Moscow-Leningrad (1937) (In Russian)
 +
|-
 +
|valign="top"|{{Ref|So}}||valign="top"| E.D. Solomentsev, "A three-sphere theorem for harmonic functions" ''Dokl. Akad. Nauk ArmSSR'' , '''42''' : 5 (1966) pp. 274–278 (In Russian)
 +
|-
 +
|valign="top"|{{Ref|Ti}}||valign="top"| E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) {{MR|0593142}} {{MR|0197687}} {{MR|1523319}} {{ZBL|0477.30001}} {{ZBL|0336.30001}} {{ZBL|0005.21004}} {{ZBL|65.0302.01}} {{ZBL|58.0297.01}}
 +
|-
 +
|}
  
 +
==Hadamard's multiplication theorem==
  
====Comments====
+
If the power series
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.P. Boas,  "Entire functions" , Acad. Press  (1954)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of functions" , Oxford Univ. Press  (1979)</TD></TR></table>
 
 
 
Hadamard's multiplication theorem (Hadamard's theorem on the multiplication of singularities): If the power series
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\begin{equation}\label{mul1}f(z)=\sum_{n=0}^\infty a_nz^n,\quad g(z)=\sum_{n=0}^\infty b_nz^n\end{equation}
  
have convergence radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609045.png" />, respectively, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609047.png" /> are the Mittag-Leffler stars (cf. [[Star of a function element|Star of a function element]]) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609049.png" />, respectively, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609050.png" /> is the set of singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609051.png" /> on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609052.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609053.png" /> is the set of singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609054.png" /> on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609055.png" />, then the power series
+
have convergence radii $r_1>0$ and $r_2>0$, respectively, if $S_1$ and $S_2$ are the Mittag-Leffler stars (cf. [[Star of a function element|Star of a function element]]) for $f(z)$ and $g(z)$, respectively, if $\alpha$ is the set of singular points of $f(z)$ on the boundary of $S_1$, and if $\beta$ is the set of singular points of $g(z)$ on the boundary of $S_2$, then the power series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\begin{equation}\label{mul2} P(z)=\sum_{n=0}^\infty a_nb_nz^n\end{equation}
  
has radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609057.png" />, and its Mittag-Leffler star <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609058.png" /> contains the star product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609059.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609060.png" /> is the complement of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609062.png" /> is the set of all products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609063.png" /> of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609065.png" />. Moreover, among the corners and readily accessible points of the boundary of the star product, only the points of the product set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609066.png" /> can be singular points of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609067.png" />. The original statements of the theorem [[#References|[1]]], [[#References|[2]]] were somewhat different from the ones given above, and needed precization [[#References|[2]]].
+
has radius of convergence $r>r_1r_2$, and its Mittag-Leffler star $S$ contains the star product $C(CS_1\times CS_2)$, where $CA$ is the complement of the set $A$ and $A\times B$ is the set of all products $pq$ of the numbers $p\in A$, $q\in B$. Moreover, among the corners and readily accessible points of the boundary of the star product, only the points of the product set $\alpha\times \beta$ can be singular points of the function $P(z)$. The original statements of the theorem {{Cite|Ha}}, {{Cite|Ha2}} were somewhat different from the ones given above, and needed precization {{Cite|Ha2}}.
  
The power series (2) is known as the Hadamard product or Hadamard composition of the power series (1). The properties of the Hadamard product revealed by this theorem (and also in subsequent studies [[#References|[3]]]) made it possible to use it in problems of analytic continuation of power series, the coefficients of the series (2) yielding some indication of the singularities of the analytic function they represent.
+
The power series \ref{mul2} is known as the Hadamard product or Hadamard composition of the power series \ref{mul1}. The properties of the Hadamard product revealed by this theorem (and also in subsequent studies {{Cite|Bi}} made it possible to use it in problems of analytic continuation of power series, the coefficients of the series \ref{mul2} yielding some indication of the singularities of the analytic function they represent.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609068.png" /> is an arbitrary compact set inside the star product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609069.png" />, there exists a closed rectifiable contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609070.png" />, located inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609071.png" /> and including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609072.png" />, such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609073.png" /> the following integral representation of the Hadamard product:
+
If $K$ is an arbitrary compact set inside the star product $C(CS_1\times CS_2)$, there exists a closed rectifiable contour $L$, located inside $C(CS_1\times CS_2)$ and including $K$, such that for all $z\in K$ the following integral representation of the Hadamard product:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046090/h04609074.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\begin{equation}\label{mul3} P(z)=\frac{1}{2\pi i}\int_Lf(t)g\left(\frac{z}{t}\right)\frac{\mathrm{d}t}{t},\end{equation}
  
is valid. The representation (3) is also used in problems of analytic continuation.
+
is valid. The representation \ref{mul3} is also used in problems of analytic continuation.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hadamard,  "Théorème sur les series entières"  ''Acta Math.'' , '''22'''  (1899)  pp. 55–63</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Hadamard,  "La série de Taylor et son prolongement analytique"  ''Scientia Phys.-Math.'' :  12  (1901)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Bieberbach,  "Analytische Fortsetzung" , Springer  (1955)  pp. Sect. 3</TD></TR></table>
 
 
  
 
+
{|
====Comments====
+
|-
 
+
|valign="top"|{{Ref|Bi}}||valign="top"| L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3 {{MR|0068621}} {{ZBL|0064.06902}}
 
+
|-
====References====
+
|valign="top"|{{Ref|Bo}}||valign="top"| R.P. Boas, "Entire functions" , Acad. Press (1954) {{MR|0068627}} {{ZBL|0058.30201}}
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"R.P. Boas,   "Entire functions" , Acad. Press (1954)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"E.C. Titchmarsh,   "The theory of functions" , Oxford Univ. Press (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"C. Pommerenke,   "Univalent functions" , Vandenhoeck &amp; Ruprecht (1975)</TD></TR></table>
+
|-
 +
|valign="top"|{{Ref|Ha}}||valign="top"|J. Hadamard, "Théorème sur les series entières" ''Acta Math.'' , '''22''' (1899) pp. 55–63 {{MR|1554900}} {{ZBL|29.0210.02}} {{ZBL|28.0222.01}}
 +
|-
 +
|valign="top"|{{Ref|Ha2}}||valign="top"|J. Hadamard, "La série de Taylor et son prolongement analytique" ''Scientia Phys.-Math.'' : 12 (1901) {{MR|}} {{ZBL|32.0412.03}}
 +
|-
 +
|valign="top"|{{Ref|Ti}}||valign="top"| E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) {{MR|0593142}} {{MR|0197687}} {{MR|1523319}} {{ZBL|0477.30001}} {{ZBL|0336.30001}} {{ZBL|0005.21004}} {{ZBL|65.0302.01}} {{ZBL|58.0297.01}}
 +
|-
 +
|valign="top"|{{Ref|Po}}||valign="top"| C. Pommerenke, "Univalent functions" , Vandenhoeck &amp; Ruprecht (1975) {{MR|0507768}} {{ZBL|0298.30014}}
 +
|-
 +
|}

Latest revision as of 12:44, 21 April 2012

Hadamard's gap theorem

If the indices $n_1,n_2,\ldots,$ of all non-zero coefficients of the power series

$$f(z)=\sum_{n=0}^\infty a_nz^n$$

satisfy the condition

\begin{equation}\label{had}n_{k+1}>(1+\theta)n_k,\end{equation}

where $\theta>0$, then the boundary of the disc of convergence of this series is its natural boundary, i.e. the function has no analytic continuation across the boundary of this disc. Condition \ref{had} is known as Hadamard's condition; the gaps which satisfy the Hadamard condition are called Hadamard gaps. See also Lacunary series; Fabry theorem.

References

[Bi] L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3 MR0068621 Zbl 0064.06902
[Bo] G. Bourion, "L'ultraconvergence dans les séries de Taylor" , Hermann (1937)
[Di] P. Dienes, "The Taylor series" , Oxford Univ. Press (1931) MR0089895 MR1522577 Zbl 0003.15502 Zbl 57.0339.10
[Ha] J. Hadamard, "Essai sur l'étude des fonctions données par leurs développement de Taylor" J. Math. Pures Appl. (4) , 8 (1892) pp. 101–186

Hadamard's theorem on entire functions

A theorem on the representation of an entire function by means of its zeros; it makes more precise the Weierstrass theorem on infinite products in the case of an entire function $f(z)$ of finite order $\rho$. If, for the sake of simplicity, $f(0)\neq 0$, then

$$f(z)=e^{Q(z)}P(z),$$

where $Q(z)$ is a polynomial of degree not exceeding $\rho$ and

$$P(z)=\prod_{k=1}^\infty W\left(\frac{z}{a_k},q\right)=\prod_{k=1}^\infty\left(1-\frac{z}{a_k}\right)e^{P_k(z)}$$

is Weierstrass' canonical product of genus $q\leq \rho$, constructed from the zeros $a_k$ of $f(z)$. In other words, Hadamard's theorem postulates that the genus of an entire function does not exceed its order. This theorem was used by J. Hadamard in proving an asymptotic law for the distribution of prime numbers.

References

[Bo] R.P. Boas, "Entire functions" , Acad. Press (1954) MR0068627 Zbl 0058.30201
[Ha] J. Hadamard, "Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann" J. Math. Pures Appl. (4) , 9 (1893) pp. 171–215
[Ma] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[Le] B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1964) (Translated from Russian) MR0156975 Zbl 0152.06703
[Ti] E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) MR0593142 MR0197687 MR1523319 Zbl 0477.30001 Zbl 0336.30001 Zbl 0005.21004 Zbl 65.0302.01 Zbl 58.0297.01

Hadamard's theorem on determinants

Let $D$ be the determinant of the matrix $A$ with complex entries $a_{\mu\nu}$, $\mu,\nu=1,\ldots,n$. The following inequality is then valid:

\begin{equation}\label{haddet}\lvert D\rvert^2\leq\prod_{\mu=1}^n\left(\sum_{\nu=1}^n\lvert a_{\mu\nu}\rvert^2\right).\end{equation}

This inequality becomes an equality if and only if

$$a_{\mu1}\overline{a}_{\nu 1}+\cdots +a_{\mu n}\overline{a}_{\nu n}=0$$

for each pair of different $\mu,\nu$, or if at least one of the factors on the right-hand side of \ref{haddet} is zero. The geometrical meaning of this theorem is that the volume of a parallelepipedon in an $n$-dimensional space is never larger than the product of the lengths of its sides issuing from one vertex, and is equal to this product if the sides are mutually perpendicular or if the length of one of the sides is zero.

In the special case when all entries $a_{\mu\nu}$ of $A$ are real numbers with $\lvert a_{\mu\nu}\rvert\leq 1$, one obtains $D^2\neq n^2$, with equality if and only if all entries are either $+1$ or $-1$ and $A$ satisfies the condition $AA^T=nI$. Such a matrix is called a Hadamard matrix of order $n$.

For references see Hadamard matrix.

References

[Ha] J. Hadamard, "Résolution d'une question relative aux déterminants" Bull. Sci. Math. (2) , 17 (1893) pp. 240–246

Hadamard's three-circle theorem

If $f(z)$ is a holomorphic function of a complex variable $z=re^{i\phi}$ in the annulus $0<r_1<r<r_2<\infty$, which is continuous in the closed annulus $r_1\leq r\leq r_2$, and if $M(r)=\max\lvert f(z)\rvert$ where $\lvert z\rvert=r$, then the following inequality is valid for $r_1\leq r\leq r_2$:

$$\log M(r)\leq\frac{\log\frac{r_2}{r}}{\log\frac{r_2}{r_1}}\log M(r_1)+\frac{\log\frac{r}{r_1}}{\log\frac{r_2}{r_1}}\log M(r_2).$$

The meaning of this inequality is that $\log M(r)$ is a convex function (of a real variable) of $\log r$. This theorem of Hadamard is a special case of the two-constants theorem.

Hadamard's theorem can be generalized in various directions; in particular, there are generalizations for other metrics and for harmonic and subharmonic functions.

References

[Bo] R.P. Boas, "Entire functions" , Acad. Press (1954) MR0068627 Zbl 0058.30201
[Ha] J. Hadamard, "Sur la distribution des zéros de la fonction $\zeta(s)$ et ses conséquences arithmétiques" Bull. Soc. Math. France , 24 (1896) pp. 199–220 Zbl 27.0154.01
[Ma] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[Pr] I.I. Privalov, "Subharmonic functions" , Moscow-Leningrad (1937) (In Russian)
[So] E.D. Solomentsev, "A three-sphere theorem for harmonic functions" Dokl. Akad. Nauk ArmSSR , 42 : 5 (1966) pp. 274–278 (In Russian)
[Ti] E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) MR0593142 MR0197687 MR1523319 Zbl 0477.30001 Zbl 0336.30001 Zbl 0005.21004 Zbl 65.0302.01 Zbl 58.0297.01

Hadamard's multiplication theorem

If the power series

\begin{equation}\label{mul1}f(z)=\sum_{n=0}^\infty a_nz^n,\quad g(z)=\sum_{n=0}^\infty b_nz^n\end{equation}

have convergence radii $r_1>0$ and $r_2>0$, respectively, if $S_1$ and $S_2$ are the Mittag-Leffler stars (cf. Star of a function element) for $f(z)$ and $g(z)$, respectively, if $\alpha$ is the set of singular points of $f(z)$ on the boundary of $S_1$, and if $\beta$ is the set of singular points of $g(z)$ on the boundary of $S_2$, then the power series

\begin{equation}\label{mul2} P(z)=\sum_{n=0}^\infty a_nb_nz^n\end{equation}

has radius of convergence $r>r_1r_2$, and its Mittag-Leffler star $S$ contains the star product $C(CS_1\times CS_2)$, where $CA$ is the complement of the set $A$ and $A\times B$ is the set of all products $pq$ of the numbers $p\in A$, $q\in B$. Moreover, among the corners and readily accessible points of the boundary of the star product, only the points of the product set $\alpha\times \beta$ can be singular points of the function $P(z)$. The original statements of the theorem [Ha], [Ha2] were somewhat different from the ones given above, and needed precization [Ha2].

The power series \ref{mul2} is known as the Hadamard product or Hadamard composition of the power series \ref{mul1}. The properties of the Hadamard product revealed by this theorem (and also in subsequent studies [Bi] made it possible to use it in problems of analytic continuation of power series, the coefficients of the series \ref{mul2} yielding some indication of the singularities of the analytic function they represent.

If $K$ is an arbitrary compact set inside the star product $C(CS_1\times CS_2)$, there exists a closed rectifiable contour $L$, located inside $C(CS_1\times CS_2)$ and including $K$, such that for all $z\in K$ the following integral representation of the Hadamard product:

\begin{equation}\label{mul3} P(z)=\frac{1}{2\pi i}\int_Lf(t)g\left(\frac{z}{t}\right)\frac{\mathrm{d}t}{t},\end{equation}

is valid. The representation \ref{mul3} is also used in problems of analytic continuation.

References

[Bi] L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3 MR0068621 Zbl 0064.06902
[Bo] R.P. Boas, "Entire functions" , Acad. Press (1954) MR0068627 Zbl 0058.30201
[Ha] J. Hadamard, "Théorème sur les series entières" Acta Math. , 22 (1899) pp. 55–63 MR1554900 Zbl 29.0210.02 Zbl 28.0222.01
[Ha2] J. Hadamard, "La série de Taylor et son prolongement analytique" Scientia Phys.-Math. : 12 (1901) Zbl 32.0412.03
[Ti] E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) MR0593142 MR0197687 MR1523319 Zbl 0477.30001 Zbl 0336.30001 Zbl 0005.21004 Zbl 65.0302.01 Zbl 58.0297.01
[Po] C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975) MR0507768 Zbl 0298.30014
How to Cite This Entry:
Hadamard theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hadamard_theorem&oldid=15608
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article