Lecture
In this section we prove three technical lemmas used
in the study of singularities of various «continuous» objects.
LEMMA 5.1 (Borel–Whitney). For any formal series (5.1)
there exists a smooth function Е(т1,..., тр): ВР > Е, whose Taylor series at the point 0 coincides with (5.1). For any formal
series (5.2) and any open bounded domain @ С В" there exists a smooth function Е(т,у1,..., ут) : Ех О - В, whose Taylor series
in the variable x at the point x = 0 coincides with (5.2).
Proof. So as not to obscure the main idea with technical details, let us first consider the simplest case: let us show
that for any formal series
со
ат", тЕВ, (5.5)
=0
there exists a smooth function ЁР(5) : В - В, whose Taylor series at the
point () coincides with (5.5). We shall seek the function РЁ in the form of the series
Е(х) = У` авт" (=) р (5.6)
и=0
in which the coefficients а„ coincide with the coefficients of the given
formal series (5.5), and the function 4 is constructed in Problem 5.2. We shall try to choose the numbers,
т’ > 0 in such a way that the series on the right-hand
side of (5.6) converges absolutely and uniformly, and that all series obtained by its k-fold differentiation also converge absolutely and
uniformly. To satisfy this condition we set
1
Ги тает: (5.7)
It is easy to verify that the following estimate holds:
р р +)
т. п Ее 1" [ат | 1 — [ат ф(т/т»)| < [апт [п — (=) (1+ а» |)" ы > п < ©.
= = п=1 м=1
Here we first used the fact that |5" (т/т.)| < |5" for |1| < тп
and |110 (т/т»)| = 0 for || > т» (both statements follow from
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the definition of the function 4). Then we used the inequality
(1+ а, |)" > |ап|, which holds for all п > 1 (this is why
we began the summation from one rather than from zero, but discarding
a finite number of terms of the series does not affect its convergence).
Similarly, for the series obtained by differentiating (5.6) once, we obtain the following estimate:
оо оо
[авт (т / тп) )' | —
=: 2
= 1 и хЩ |+ М)) 2 (п + М!) кои + М = = 2 нае" | < 2 т 9%
[аи (пар (т ть) + 2” /тыч' (х/ть,))| <
=. 2
=
where the number М: = ах |1)" (т)|.
Using similar arguments, it is easy to obtain a similar estimate (implying absolute and uniform convergence)
for the series obtained by k-fold differentiation of the given series,
for any k. Hence, the series on the right-hand side of formula (5.6) converges to the function ЁР(5т), defined and smooth on the whole
space В. To complete the proof it remains only to verify that the Taylor series of the resulting function Ё(т) at the point 0 coincides
with (5.5), i.e., Е (0) = па». The verification of this statement is trivial and
is left to the reader.
The reader is also left to carry out on their own the proof of the first statement of the lemma in the case р > 1. It is entirely
analogous to the special case р = 1 treated above.
Let us give the proof of the second statement of the lemma, restricting ourselves for simplicity to the case т = 1 (in the general case the proof is
the same, but the formulas are more cumbersome). Let us show that for any
formal series
=.
>. ав (у) т", туЕВ, (5.8)
п=0
with smooth coefficients а„(у) and for any open bounded domain ® С Е there exists a smooth function Р(т, у): Ех 0 >В,
whose Taylor series in the variable at т = 0 coincides with (5.8).
We shall seek the function Р(т, у) in the form of the series
вый = УЗ ( =). (5.5)
in which а„(у) coincide with the coefficients of the given formal
series (5.8). The numbers г, > 0 are defined by a formula analogous to (5.7),
but with the obvious modification arising from the fact that the coefficients ап
are not numbers but functions:
1
та = ‚ where а» = шах шах |а(® (5). (5.10) п (1 ов) 0<1<п уЕб
Here @ denotes the closure of the domain О.
Arguing as before, for any integer 1 > 0 we
obtain the following estimate:
У` [а (у) ‘у(т/т»)| у быГи =
п=#+1 п=#+1
ыы 1" [6 < 1
. .) т — У. 1 ь ве" (1 +а») вал
which shows that the series obtained by 1-fold differentiation with respect to the variable у of the series on the right-hand side of formula (5.9), converges absolutely and uniformly on В х 9. It follows from this,
that the series (5.9) defines a function Р(т,у) having, on Вх @, continuous partial derivatives 0'Е/ду' of all orders # > 0.
Similarly, for the series obtained from (5.9) by differentiating once with respect to the variable x and 1-fold differentiation with respect to the
variable у, we obtain the following estimate:
р
У [а (у (аа, <
1-2
о
< У` ап|пх” (т ть) + 2” /тоф (с/т) | < у отт" (п + М,) =
и—4-2 п=4+2
> оп(п + М п+ М
>. ИИ а = у и т! у 5 < ИЕ и=е-2
Using similar arguments, it is easy to obtain a similar
estimate (implying absolute and uniform convergence) for the series obtained from (5.9) by k-fold differentiation with respect to the variable т and r-fold differentiation with respect to the variable у.
Hence, the series (5.9) converges to some function Р(т,у),
defined and smooth on Ех 0. The verification that the Taylor series of this
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function in the variable т at the point x = 0 coincides with (5.8), we again
leave to the reader. The reader is also left to carry out on their own the proof in the case т > 1. It is entirely analogous to the special case т = 1, and all the computations repeat almost
verbatim if 1 is regarded as a multi-index. ■
It follows from Lemma 5.1 that any smooth function in a neighborhood of
the point 0 is given by its formal Taylor series up to
a term that is infinitely flat at the point 0. This fact is often used to prove various statements about local properties of functions, mappings, differential equations, etc.
Later we shall consider several examples of this kind (the simplest ones).
Let В, = [0, +) and О С В" be an open bounded domain.
Suppose we are given a function
ЕГ(т,/,... т): В ХО >В,
smooth on its entire domain of definition. Let us ask the question: can
Р(т,у) be smoothly extended to the whole set Е х 9? This
means that we must construct a smooth function
Е(т,у,.... ут): Ех О-В,
such that Р(т,у) = Е(т, у) for all (1,9) Е В. х 9. Although the affirmative answer seems intuitively obvious, it requires justification, which can be obtained, for example, with the help of Lemma 5.1.
LEMMA 5.2. Any smooth function Е(т, у): В. хО > Е can
be smoothly extended to the set В х 0.
Proof. Let us write the Taylor series of the function Р(т, у) in the variable x at the point т = 0. This is the formal power series
— 1 0”Е 1 0"Е
т т", т = т 0, = в ] >. (у) х а» (у) И Эт ( у) +0 п! дат (2.5). (51)
Strictly speaking, the last phrase requires clarification. At all interior
points of the set Е Хх О, smoothness implies the existence of continuous
partial derivatives of all orders, understood in the usual sense. At the boundary
points of the set В, х О, i.e., points lying on the hyperplane т = 0, all derivatives (and limits) are understood as one-sided: «right-hand» with respect to the variable т.
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We need to prove the existence of a function
Е(т, у), 0, уЕОЙ,
С(т,у), 1<0, уЕЯ, Рей = |
where С(т, у): Вх О -+ Е has only one property: Ё(хт,у/) is smooth on the whole set В х <). It is clear that if we take as С’
a smooth function whose Taylor series in the variable x at 1 = 0 coincides with (5.11), this requirement will be satisfied. By Lemma 5.1,
such a function С’ exists. □
REMARK 5.1. The smooth extension of the function Р(х, у) constructed in Lemma 5.2
is not unique.
LEMMA 5.3. Let О С В" be an arbitrary open bounded domain. Any smooth function Е(т,у) : Ех О + Е can
be represented in the form
Е(т, у) = Е(12, у) +=Е5(12,у), у=(и,..- т), (5.12)
where ЕР\ 2(т,у): Ех Я > Е are some smooth functions.
Proof. Note that any function can be represented as the
sum of a function even and a function odd in т. Indeed,
Е(х, у) = Ееу(х, у) т Вох, у),
where
Е(т,у) + Е(-т,у) Е(т,у) — Е(-т, у) 2 Саи: — Еса(т, у) — р.
Applying representation (1.1) with с = 1, it is easy to see that any
smooth function Ра(т, у), odd in x, can be represented in the form
Еа(т, у) ГЕ 1 (т, у),
where /(т, у) is a smooth function even in т. Thus, to complete the proof of the lemma it suffices to show that for any
even function Р(х, у): Ех В” > Е the representation holds
Е(т,у) = Е(1°,у), у= (у... т). (5.13)
Since Р(т, у) = Е(]т|, у), equality (5.13) is equivalent to the equality Ё\ (&, у) = Р(\/&, у) holding for all & > 0. Thereby the function
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Е, (&, у) is uniquely defined for all values & > 0 and may take arbitrary values for & < 0:
Е(У$,у) for &>0, К 1 (5, у) — arbitrary for & < 0.
The only condition on the function Р1(6, у) that it must
satisfy is smoothness on the whole set В х ©.
For interior points of the set В. х О (i.e. for & > 0) the smoothness of
Е; obviously follows from the smoothness of Р. At points of the hyperplane & = 0
the function ($, у) = Е(\/&,у) has continuous partial derivatives (one-sided with respect to the variable &) of all orders. To prove this statement one can use the representation of the function
Е(т,у) in the form (1.5). In the case of a function even in x, monomials of odd degrees in 1 are absent from this representation, and we obtain
the expression
Е(т,у) = Лу) +2 Л(у) ++ у) +2 (у,
where ]:(у) and 9р(т,у) are smooth functions, with д»(т,у) even in т.
Substituting 1 = \/& into the resulting expression, we obtain
ВЕ) = УЕ.) + 29, УВ у), &>0.
From this it follows that the function Ё!\(&, у) has continuous partial
derivatives of all orders at all points of the set В, х О. Thus
we have the situation of Lemma 5.2, and the existence of the required extension of the function Ё1($5, у) to the region & < 0 follows from this
lemma. □
REMARK 5.2. For polynomials, formal series, and analytic functions the representation (5.12) is entirely obvious, and at
first glance it seems that it should be just as simple for smooth functions as well. However, in the smooth case additional justification is required,
which is precisely what Lemma 5.3 provides.
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