Hellmuth Kneser

Hellmuth Kneser, 1930

Hellmuth Kneser, 1958

Hellmuth Kneser (1898.04.16, Tartu (Dortpart), Estonia - 1973.08.23, Tubingen) is Estonian and German mathematician.

Biography
Since 1916, education at Wroclaw University

Since 1921, work with Dufit Hilbert on foundation of quantum mechanics ("Untersuchungen zur Quantentheorie")

Since 1944, work at the Mathematical Research Institute of Oberwolfach

at 1954, president of the German Mathematical Society.

Kneser-Suss inequality
In 1932, Kneser and Wilhelm Suss suggest the Kneser-Suss unequality :

In the convex geometry setting, let \( K, L ∈ \mathfrak{K}^n\) be convex bodies, and let \(M\) be the convex body satisfying

\( S(M^{n−1};·) = S(K^{n−1};·) + S(L^{n−1};·) \)

Then

\( \mathrm{vol}(M)^{n-1/n} \ge \mathrm{vol}(K)^{n-1/n} + \mathrm{vol}(L)^{n-1/n} \)

Tetration
\(y={\rm tet}_b(x)\) versus \(x\) for various \(b\)

Complex map \( p\!+\!\mathrm{i} q=\varphi(x\!+\!\mathrm{i} y) \)

In 1950, Kneser publishes paper entitled

Reelle analytische Lösungen der Gleichung \( \varphi(\varphi(x))=\mathrm e^x \) und verwandter Funktionalgleichungen.

There, the need to construct superfunction of exponential, id est, tetration, is formulated.

Tetration to base \( b\) appears as holomorphic solution \( F= \mathrm{tet}_b \)

of the Transfer equation \( F(z+1) = b^{F(z)} \)

with additional condition \( F(0)=1 \)

and certain behaviors at infinity (approaching the fixed point(s) of \( \exp_b \) at \( \pm \mathrm i\infty \) ). These conditions are necessary for the uniqueness of the solution.

Then, during a half-century, various approaches to evaluation of tetration had been reported.

Since 2009, the efficient algorithms for evaluation of tetration to various values of base (and the corresponding complex maps) are available.

With tetration \( F=\mathrm{tet} \)

and the inverse function \( G=F^{-1} \), id est, arctetration \( G=\mathrm{ate} \) ,

the function \( \varphi \), mentioned in the original title by Kneser, can be expressed as follows:

\( \varphi(x)= F(1/2+G(x)) \)

Complex map of function \( \varphi \) (metioned in the title of paper is shown in figure at right. This map is right hand side of figure 4 (or figure 5) of publication

Relatives
Adolf Kneser (1862.03.19, Grüssow, Mecklenburg, Germany - 1930.01.24, Breslau, Germany (Wrocław, Poland)), father

Laura Kneser (aka Laura Booth) (1869.07.18 - 1944.07.18), mother

Lorenz Friedrich Kneser (1896.oo.oo - 1918.oo.oo), brother

Hans Otto Kneser (1901.oo.oo - 1985.oo.oo), brother

Dorothee Beer (aka Dorothee Kneser) (1905.oo.oo - 1968.oo.oo), sister

Hertha Kneser (aka Hertha Scheuerlen) (1900.oo.oo - 1980.oo.oo), wife

Martin Kneser (1928.01.21 - 2004.02.16), son

Keywords
Kneser-Suss inequality, Square root of exponential, Superfunction, Tetration