Hadamard theorem

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{mul1} 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