# Extremal properties of functions

Properties of individual functions that distinguish them as solutions of some extremal problems. The majority of the special functions arising in mathematical analysis can be characterized by some extremal property. This applies, for example, to extremal properties of polynomials: the classical Laguerre polynomials, the Legendre polynomials, the Chebyshev polynomials, the Hermite polynomials, and the Jacobi polynomials can be characterized as polynomials that deviate least from zero in a weighted $L_2$-space. The classical polynomials are usually solutions of various extremal problems arising not infrequently in remote domains of analysis. For example, the Chebychev polynomials are extremal in the problem of an inequality for the derivatives of polynomials (see [1] and Markov inequality). The same can also be said of other special functions. Many of them are eigen functions of differential operators, that is, they are solutions of some isoperimetric problem. In this context the best known special functions are connected in some way with the existence of an invariant structure (see Harmonic analysis, abstract), when they are eigen functions of the Laplace–Beltrami equation, which is shift-invariant. This holds for the trigonometric polynomials, the spherical functions, the cylinder functions, etc. (see [2]). The majority of extremal properties of functions can be stated in the form of some exact inequality.

Connected with extremal problems of approximation theory are the Bernstein inequality, the Bohr–Favard inequality, etc. In particular, the Bohr–Favard inequality reflects the extremal property of the Bernoulli polynomials. Extremal properties of functions are studied in approximation theory (see [6] and [7]) and in the theory of numerical integration (see [8]).

Splines can be characterized by various extremal properties (see [9]). Many special splines have a number of extremal properties touching upon the approximation and interpolation of function classes (see [7] and [8]). Many extremal properties of functions are studied in complex analysis. In particular, the Koebe function is an extremal function for a number of problems in the theory of univalent functions. See also Isoperimetric inequality and Imbedding theorems.

#### References

 [1] S.N. Bernshtein, "Leçons sur les propriétés extrémales de la meilleure approximation des fonctions analytiques d'une variable réelle" , Chelsea, reprint (1970) (Translated from Russian) [2] N.Ya. Vilenkin, "Special functions and the theory of group representations" , Amer. Math. Soc. (1968) (Translated from Russian) [3] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1952) [4] E.F. Beckenbach, R. Bellman, "Inequalities" , Springer (1961) [5] D.S. Mitrinovič, "Analytic inequalities" , Springer (1970) ((Translated from the Servo-Croatian)) [6] N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian) [7] V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian) [8] S.M. Nikol'skii, "Quadrature formulas" , Moscow (1979) (In Russian) [9] J.H. Ahlberg, E.N. Nilson, J.F. Walsh, "Theory of splines and their applications" , Acad. Press (1967)