Difference between revisions of "Weierstrass theorem"
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+ | There are a number of theorems named after Karl Theodor Wilhelm Weierstrass (1815-1897). | ||
+ | |||
+ | ===Infinite product theorem=== | ||
+ | |||
Weierstrass' infinite product theorem [[#References|[1]]]: For any given sequence of points in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w0975101.png" />, | Weierstrass' infinite product theorem [[#References|[1]]]: For any given sequence of points in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w0975101.png" />, | ||
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Remmert, "Funktionentheorie" , '''II''' , Springer (1991) {{MR|1150243}} {{ZBL|0748.30002}} </TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Remmert, "Funktionentheorie" , '''II''' , Springer (1991) {{MR|1150243}} {{ZBL|0748.30002}} </TD></TR></table> | ||
+ | ===Approximation of functions=== | ||
Weierstrass' theorem on the approximation of functions: For any continuous real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751035.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751036.png" /> there exists a sequence of algebraic polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751037.png" /> which converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751038.png" /> to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751039.png" />; established by K. Weierstrass . | Weierstrass' theorem on the approximation of functions: For any continuous real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751035.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751036.png" /> there exists a sequence of algebraic polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751037.png" /> which converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751038.png" /> to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751039.png" />; established by K. Weierstrass . | ||
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Gaier, "Vorlesungen über Approximation im Komplexen" , Birkhäuser (1980) {{MR|0604011}} {{ZBL|0442.30038}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Wermer, "Banach algebras and several complex variables" , Springer (1976) {{MR|0394218}} {{ZBL|0336.46055}} </TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Gaier, "Vorlesungen über Approximation im Komplexen" , Birkhäuser (1980) {{MR|0604011}} {{ZBL|0442.30038}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Wermer, "Banach algebras and several complex variables" , Springer (1976) {{MR|0394218}} {{ZBL|0336.46055}} </TD></TR></table> | ||
+ | ===Uniformly convergent series of analytic functions=== | ||
Weierstrass' theorem on uniformly convergent series of analytic functions : If the terms of a series | Weierstrass' theorem on uniformly convergent series of analytic functions : If the terms of a series | ||
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Remmert, "Theory of complex functions" , '''1''' , Springer (1990) (Translated from German) {{MR|1084167}} {{ZBL|0780.30001}} </TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Remmert, "Theory of complex functions" , '''1''' , Springer (1990) (Translated from German) {{MR|1084167}} {{ZBL|0780.30001}} </TD></TR></table> | ||
− | Weierstrass' preparation theorem. A theorem obtained and originally formulated by K. Weierstrass | + | ===Preparation theorem=== |
− | + | Weierstrass' preparation theorem. A theorem obtained and originally formulated by K. Weierstrass in 1860 as a preparation lemma, used in the proofs of the existence and analytic nature of the implicit function of a complex variable defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751069.png" /> whose left-hand side is a holomorphic function of two complex variables. This theorem generalizes the following important property of holomorphic functions of one complex variable to functions of several complex variables: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751070.png" /> is a holomorphic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751071.png" /> in a neighbourhood of the coordinate origin with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751073.png" />, then it may be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751074.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751075.png" /> is the multiplicity of vanishing of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751076.png" /> at the coordinate origin, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751077.png" />, while the holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751078.png" /> is non-zero in a certain neighbourhood of the origin. | |
− | in 1860 as a preparation lemma, used in the proofs of the existence and analytic nature of the implicit function of a complex variable defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751069.png" /> whose left-hand side is a holomorphic function of two complex variables. This theorem generalizes the following important property of holomorphic functions of one complex variable to functions of several complex variables: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751070.png" /> is a holomorphic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751071.png" /> in a neighbourhood of the coordinate origin with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751073.png" />, then it may be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751074.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751075.png" /> is the multiplicity of vanishing of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751076.png" /> at the coordinate origin, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751077.png" />, while the holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751078.png" /> is non-zero in a certain neighbourhood of the origin. | ||
The formulation of the Weierstrass preparation theorem for functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751079.png" /> complex variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751080.png" />. Let | The formulation of the Weierstrass preparation theorem for functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751079.png" /> complex variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097510/w09751080.png" />. Let |
Revision as of 18:16, 7 December 2015
There are a number of theorems named after Karl Theodor Wilhelm Weierstrass (1815-1897).
Infinite product theorem
Weierstrass' infinite product theorem [1]: For any given sequence of points in the complex plane ,
![]() | (1) |
![]() |
there exists an entire function with zeros at the points of this sequence and only at these points. This function may be constructed as a canonical product:
![]() | (2) |
where is the multiplicity of zero in the sequence (1), and
![]() |
The multipliers
![]() |
are called Weierstrass prime multipliers or elementary factors. The exponents are chosen so as to ensure the convergence of the product (2); for instance, the choice
ensures the convergence of (2) for any sequence of the form (1).
It also follows from this theorem that any entire function with zeros (1) has the form
![]() |
where is the canonical product (2) and
is an entire function (see also Hadamard theorem on entire functions).
Weierstrass' infinite product theorem can be generalized to the case of an arbitrary domain : Whatever a sequence of points
without limit points in
, there exists a holomorphic function
in
with zeros at the points
and only at these points.
The part of the theorem concerning the existence of an entire function with arbitrarily specified zeros may be generalized to functions of several complex variables as follows: Let each point of the complex space
,
, be brought into correspondence with one of its neighbourhoods
and with a function
which is holomorphic in
. Moreover, suppose this is done in such a way that if the intersection
of the neighbourhoods of the points
is non-empty, then the fraction
is a holomorphic function in
. Under these conditions there exists an entire function
in
such that the fraction
is a holomorphic function at every point
. This theorem is known as Cousin's second theorem (see also Cousin problems).
References
[1] | K. Weierstrass, "Math. Werke" , 1–7 , Mayer & Müller (1894–1895) |
[2] | A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002 |
[3] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1985) (In Russian) Zbl 0578.32001 Zbl 0574.30001 |
Comments
References
[a1] | R. Remmert, "Funktionentheorie" , II , Springer (1991) MR1150243 Zbl 0748.30002 |
Approximation of functions
Weierstrass' theorem on the approximation of functions: For any continuous real-valued function on the interval
there exists a sequence of algebraic polynomials
which converges uniformly on
to the function
; established by K. Weierstrass .
Similar results are valid for all spaces . The Jackson theorem is a strengthening of this theorem.
The theorem is also valid for real-valued continuous -periodic functions and trigonometric polynomials, e.g. for real-valued functions which are continuous on a bounded closed domain in an
-dimensional space, or for polynomials in
variables. For generalizations, see Stone–Weierstrass theorem. For the approximation of functions of a complex variable by polynomials, see [3].
References
[1a] | K. Weierstrass, "Über die analytische Darstellbarkeit sogenannter willkülicher Funktionen reeller Argumente" Sitzungsber. Akad. Wiss. Berlin (1885) pp. 633–639; 789–805 |
[1b] | K. Weierstrass, "Über die analytische Darstellbarkeit sogenannter willkülicher Funktionen reeller Argumente" , Werke , 3 , Preuss. Akad. Wiss. (1903) |
[2] | N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian) |
[3] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1985) (In Russian) Zbl 0578.32001 Zbl 0574.30001 |
Yu.N. Subbotin
Comments
References
[a1] | D. Gaier, "Vorlesungen über Approximation im Komplexen" , Birkhäuser (1980) MR0604011 Zbl 0442.30038 |
[a2] | J. Wermer, "Banach algebras and several complex variables" , Springer (1976) MR0394218 Zbl 0336.46055 |
Uniformly convergent series of analytic functions
Weierstrass' theorem on uniformly convergent series of analytic functions : If the terms of a series
![]() | (*) |
which converges uniformly on compacta inside a domain of the complex plane
, are analytic functions in
, then the sum
is an analytic function in
. Moreover, the series
![]() |
obtained by successive term-by-term differentiations of the series (*), for any
, also converges uniformly on compacta inside
towards the derivative
of the sum of the series (*). This theorem has been generalized to series of analytic functions of several complex variables converging uniformly on compacta inside a domain
of the complex space
,
, and the series of partial derivatives of a fixed order of the terms of the series (*) converges uniformly to the respective partial derivative of the sum of the series:
![]() |
![]() |
Weierstrass' theorem on uniform convergence on the boundary of a domain : If the terms of a series
![]() |
are continuous in a closed bounded domain of the complex plane
and are analytic in
, then uniform convergence of this series on the boundary of the domain implies that it converges uniformly on the closed domain
.
This property of series of analytic functions is also applicable to analytic and harmonic functions defined, respectively, in a domain of the complex space ,
, or in the Euclidean space
,
. As a general rule it remains valid in all situations in which the maximum-modulus principle is applicable.
References
[1a] | K. Weierstrass, "Abhandlungen aus der Funktionenlehre" , Springer (1866) |
[1b] | K. Weierstrass, "Math. Werke" , 1–7 , Mayer & Müller (1894–1895) |
[2] | E.T. Whittaker, G.N. Watson, "A course of modern analysis" , 1 , Cambridge Univ. Press (1952) pp. Chapt. 3 MR1424469 MR0595076 MR0178117 MR1519757 Zbl 0951.30002 Zbl 0108.26903 Zbl 0105.26901 Zbl 53.0180.04 Zbl 47.0190.17 Zbl 45.0433.02 Zbl 33.0390.01 |
[3] | A.I. Markushevich, "Theory of functions of a complex variable" , 1–2 , Chelsea (1977) pp. Chapts. 3; 7 (Translated from Russian) MR0444912 Zbl 0357.30002 |
Comments
References
[a1] | R. Remmert, "Theory of complex functions" , 1 , Springer (1990) (Translated from German) MR1084167 Zbl 0780.30001 |
Preparation theorem
Weierstrass' preparation theorem. A theorem obtained and originally formulated by K. Weierstrass in 1860 as a preparation lemma, used in the proofs of the existence and analytic nature of the implicit function of a complex variable defined by an equation whose left-hand side is a holomorphic function of two complex variables. This theorem generalizes the following important property of holomorphic functions of one complex variable to functions of several complex variables: If
is a holomorphic function of
in a neighbourhood of the coordinate origin with
,
, then it may be represented in the form
, where
is the multiplicity of vanishing of
at the coordinate origin,
, while the holomorphic function
is non-zero in a certain neighbourhood of the origin.
The formulation of the Weierstrass preparation theorem for functions of complex variables,
. Let
![]() |
be a holomorphic function of in the polydisc
![]() |
and let
![]() |
Then, in some polydisc
![]() |
the function can be represented in the form
![]() |
![]() |
where is the multiplicity of vanishing of the function
![]() |
at the coordinate origin, ; the functions
are holomorphic in the polydisc
![]() |
![]() |
the function is holomorphic and does not vanish in
. The functions
,
, and
are uniquely determined by the conditions of the theorem.
If the formulation is suitably modified, the coordinate origin may be replaced by any point of the complex space
. It follows from the Weierstrass preparation theorem that for
, as distinct from the case of one complex variable, every neighbourhood of a zero of a holomorphic function contains an infinite set of other zeros of this function.
Weierstrass' preparation theorem is purely algebraic, and may be formulated for formal power series. Let be the ring of formal power series in the variables
with coefficients in the field of complex numbers
; let
be a series of this ring whose terms have lowest possible degree
, and assume that a term of the form
,
, exists. The series
can then be represented as
![]() |
where are series in
whose constant terms are zero, and
is a series in
with non-zero constant term. The formal power series
and
are uniquely determined by
.
A meaning which is sometimes given to the theorem is the following division theorem: Let the series
![]() |
satisfy the conditions just specified, and let be an arbitrary series in
. Then there exists a series
![]() |
and series
![]() |
![]() |
which satisfy the following equation:
![]() |
Weierstrass' preparation theorem also applies to rings of formally bounded series. It provides a method of inductive transition, e.g. from to
. It is possible to establish certain properties of the rings
and
in this way, such as being Noetherian and having the unique factorization property. There exists a generalization of this theorem to differentiable functions [6].
References
[1a] | K. Weierstrass, "Abhandlungen aus der Funktionenlehre" , Springer (1866) |
[1b] | K. Weierstrass, "Math. Werke" , 1–7 , Mayer & Müller (1894–1895) |
[2] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1985) (In Russian) Zbl 0578.32001 Zbl 0574.30001 |
[3] | S. Bochner, W.T. Martin, "Several complex variables" , Princeton Univ. Press (1948) MR0027863 Zbl 0041.05205 |
[4] | R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) MR0180696 Zbl 0141.08601 |
[5] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
[6] | B. Malgrange, "Ideals of differentiable functions" , Tata Inst. (1966) MR2065138 MR0212575 Zbl 0177.17902 |
Comments
The polynomial
![]() |
![]() |
which occurs in the Weierstrass preparation theorem, is called a Weierstrass polynomial of degree in
.
The analogue of the Weierstrass preparation theorem for differentiable functions is variously known as the differentiable preparation theorem, the Malgrange preparation theorem or the Malgrange–Mather preparation theorem. Let be a smooth real-valued function on some neighbourhood of
in
and let
with
and
smooth near
in
. Then the Malgrange preparation theorem says that there exists a smooth function
near zero such that
for suitable smooth
, and the Mather division theorem says that for any smooth
near
in
there exist smooth functions
and
on
near
such that
with
. For more sophisticated versions of the differentiable preparation and division theorems, cf. [a2]–[a4].
An important application is the differentiable symmetric function theorem (differentiable Newton theorem), which says that a germ of a symmetric differentiable function of
in
can be written as a germ of a differentiable function in the elementary symmetric functions
,
, [a7], [a8].
There exist also -adic analogues of the preparation and division theorems. Let
be a complete non-Archimedean normed field (cf. Norm on a field).
is the algebra of power series
,
,
,
, such that
as
,
. The norm on
is defined by
. The subring
consists of all
with
and
is the ideal of all
with
. Let
be the residue ring
, and let
be the quotient mapping. Then
, where
is the residue field of
. An element
with
is called regular in
of degree
if
is of the form
with
and
. Note that
is naturally a subalgebra of
. The
-adic Weierstrass preparation and division theorem now says: i) (division) Let
be regular of degree
in
and let
. Then there exist unique elements
and
,
, such that
and, moreover,
, where
; ii) (preparation) Let
be of norm
, then there exists a
-automorphism of
such that
is regular in
.
References
[a1] | L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4 MR0344507 Zbl 0271.32001 |
[a2] | M. Golubitsky, "Stable mappings and their singularities" , Springer (1973) pp. Chapt. IV MR0341518 Zbl 0294.58004 |
[a3] | J.C. Tougeron, "Ideaux de fonction différentiables" , Springer (1972) pp. Chapt. IX MR0440598 |
[a4] | B. Malgrange, "Ideals of differentiable functions" , Oxford Univ. Press (1966) pp. Chapt. V MR2065138 MR0212575 Zbl 0177.17902 |
[a5] | J. Fresnel, M. van der Put, "Géométrie analytique rigide et applications" , Birkhäuser (1981) pp. §II.2 MR0644799 Zbl 0479.14015 |
[a6] | N. Koblitz, "![]() ![]() |
[a7] | G. Glaeser, "Fonctions composés différentiables" Ann. of Math. , 77 (1963) pp. 193–209 |
[a8] | S. Łojasiewicz, "Whitney fields and the Malgrange–Mather preparation theorem" C.T.C. Wall (ed.) , Proc. Liverpool Singularities Symposium I , Lect. notes in math. , 192 , Springer (1971) pp. 106–115 Zbl 0224.58003 |
Weierstrass theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_theorem&oldid=24010