The Müntz–Szász theorem is a basic result of

approximation theory In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...

, proved by Herman Müntz in 1914 and Otto Szász in 1916. Roughly speaking, the theorem shows to what extent the Weierstrass theorem on polynomial approximation can have holes dug into it, by restricting certain coefficients in the polynomials to be zero. The form of the result had been conjectured by

Sergei Bernstein Sergei Natanovich Bernstein (, sometimes Romanized as ; 5 March 1880 – 26 October 1968) was a Ukrainian and Soviet mathematician of Jewish origin known for contributions to partial differential equations, differential geometry, probability theo ...

before it was proved. The theorem, in a special case, states that a necessary and sufficient condition for the

monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with n ...

s :

x^n,\quad n\in S\subset\mathbb N

to span a

dense subset In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...

of the

Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...

''C'' 'a'',''b''of all

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

s with

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

values on the

closed interval In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...

'a'',''b''with ''a'' > 0, with the

uniform norm In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when t ...

, is that the sum :

\sum_\frac\

of the reciprocals, taken over ''S'', should diverge, i.e. ''S'' is a large set. For an interval , ''b'' the

constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. Basic properties As a real-valued function of a real-valued argument, a constant function has the general form or just For example, ...

s are necessary: assuming therefore that 0 is in ''S'', the condition on the other exponents is as before. More generally, one can take exponents from any

strictly increasing In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusiv ...

sequence of positive real numbers, and the same result holds. Szász showed that for complex number exponents, the same condition applied to the sequence of

real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s. There are also versions for the ''L''_''p'' spaces.

References

Scanned at University of Michigan

Scanned at digizeitschriften.de
* {{DEFAULTSORT:Muntz-Szasz Theorem Functional analysis Theorems in approximation theory

See also

References