Quasianalytic function

Quasianalytic functions in mathematical analysis are a class of functions that, loosely speaking, can be completely reconstructed from their values in a small area (for example, on the boundary of a region). This property greatly facilitates the solution of differential equations and the study of other problems of analysis. Since this property holds for analytic functions (see Complex analysis ), the class of quasianalytic functions contains the class of ordinary analytic functions and can be considered as its extension ^[1] .

Content

Definitions

Single Variable Functions

One of the many defining features of an analytic function : let a function $f(x)$ ${\ displaystyle f (x)}$ infinitely differentiable at all points of the segment $[a,b]$ ${\ displaystyle [a, b]}$ and let there be a number $A$ ${\ displaystyle A}$ (depending on the function) such that for all points $x\in [a,b]$ ${\ displaystyle x \ in [a, b]}$ the inequality holds:

|f^{(n)}(x)|\leqslant A^{n}n!

{\ displaystyle | f ^ {(n)} (x) | \ leqslant A ^ {n} n!}

(one)

Then the function $f(x)$ ${\ displaystyle f (x)}$ analytic (the converse theorem is also true) ^[2] .

Jacques Hadamard in 1912 proposed to generalize the above inequality, replacing the sequence $n!$ ${\ displaystyle n!}$ general sequence $M=\{M_{k}\}_{k=0}^{\infty }$ ${\ displaystyle M = \ {M_ {k} \} _ {k = 0} ^ {\ infty}}$ positive real numbers . He defined on the interval [ a , b ] the class of functions C ^M ([ a , b ]) as follows:

Every function $f$ ${\ displaystyle f}$ from the class is infinitely differentiable ( f ∈ C ^∞ ([ a , b ])), and at all points x ∈ [ a , b ] and for all $k\geqslant 0$ ${\ displaystyle k \ geqslant 0}$ the condition is satisfied:

\left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leqslant A^{k}k!M_{k}

{\ displaystyle \ left | {\ frac {d ^ {k} f} {dx ^ {k}}} (x) \ right | \ leqslant A ^ {k} k! M_ {k}}

(2)

where A is some constant (depending on the function).

If we take the sequence M _k = 1, then, according to what was said at the beginning of the section, we obtain exactly the class of ordinary real analytic functions on the interval [ a , b ].

A class C ^M ([ a , b ]) is called quasianalytic if for any function f ∈ C ^M ([ a , b ]) the uniqueness condition holds: if ${\frac {d^{k}f}{dx^{k}}}(x)=0$ ${\ displaystyle {\ frac {d ^ {k} f} {dx ^ {k}}} (x) = 0}$ at some point x ∈ [ a , b ] for all k , then f is identically equal to zero.

Elements of a quasianalytic class are called quasianalytic functions . The above condition means that two functions that coincide at some point along with all their derivatives coincide everywhere. In other words, the values of the function on an arbitrarily small section completely determine all its values.

Functions of Several Variables

For function $f:\mathbb {R} ^{n}\to \mathbb {R}$ ${\ displaystyle f: \ mathbb {R} ^ {n} \ to \ mathbb {R}}$ and for a set of indices $j=(j_{1},j_{2},\ldots ,j_{n})\in \mathbb {N} ^{n}$ ${\ displaystyle j = (j_ {1}, j_ {2}, \ ldots, j_ {n}) \ in \ mathbb {N} ^ {n}}$ we denote:

|j|=j_{1}+j_{2}+\ldots +j_{n}

{\ displaystyle | j | = j_ {1} + j_ {2} + \ ldots + j_ {n}}

D^{j}={\frac {\partial ^{j}}{\partial x_{1}^{j_{1}}\partial x_{2}^{j_{2}}\ldots \partial x_{n}^{j_{n}}}}

{\ displaystyle D ^ {j} = {\ frac {\ partial ^ {j}} {\ partial x_ {1} ^ {j_ {1}} \ partial x_ {2} ^ {j_ {2}} \ ldots \ partial x_ {n} ^ {j_ {n}}}}}

j!=j_{1}!j_{2}!\ldots j_{n}!

{\ displaystyle j! = j_ {1}! j_ {2}! \ ldots j_ {n}!}

x^{j}=x_{1}^{j_{1}}x_{2}^{j_{2}}\ldots x_{n}^{j_{n}}.

{\ displaystyle x ^ {j} = x_ {1} ^ {j_ {1}} x_ {2} ^ {j_ {2}} \ ldots x_ {n} ^ {j_ {n}}.}

Then $f$ ${\ displaystyle f}$ called quasianalytic in the open area $U\subset \mathbb {R} ^{n},$ ${\ displaystyle U \ subset \ mathbb {R} ^ {n},}$ if for each compact $K\subset U$ ${\ displaystyle K \ subset U}$ there is a constant $A$ ${\ displaystyle A}$ such that:

\left|D^{j}f(x)\right|\leqslant A^{|j|+1}j!M_{|j|}

{\ displaystyle \ left | D ^ {j} f (x) \ right | \ leqslant A ^ {| j | +1} j! M_ {| j |}}

for all indices from the set $j\in \mathbb {N} ^{n}$ ${\ displaystyle j \ in \ mathbb {N} ^ {n}}$ and at all points $x\in K$ ${\ displaystyle x \ in K}$ .

The class of quasianalytic functions of $n$ ${\ displaystyle n}$ variables with respect to the sequence $M$ ${\ displaystyle M}$ on the set $U$ ${\ displaystyle U}$ can be designated $C_{n}^{M}(U)$ ${\ displaystyle C_ {n} ^ {M} (U)}$ , although other notations are also found in the sources.

Quasianalytic classes for log-convex sequences

Assume in the above definition $M_{1}=1$ ${\ displaystyle M_ {1} = 1}$ and sequence $M_{k}$ ${\ displaystyle M_ {k}}$ non-decreasing. This sequence is called logarithmically convex if the condition is satisfied:

Sequence

{M_{k+1} \over M_{k}}

{\ displaystyle {M_ {k + 1} \ over M_ {k}}}

increasing.

If the sequence $M_{k}$ ${\ displaystyle M_ {k}}$ logarithmically convex, then:

(M_{k})^{1/k}

{\ displaystyle (M_ {k}) ^ {1 / k}}

also increasing.

M_{r}M_{s}\leqslant M_{r+s}

{\ displaystyle M_ {r} M_ {s} \ leqslant M_ {r + s}}

for all

(r,s)\in \mathbb {N} ^{2}

{\ displaystyle (r, s) \ in \ mathbb {N} ^ {2}}

.

For logarithmically convex $M$ ${\ displaystyle M}$ quasianalytic class $C_{n}^{M}$ ${\ displaystyle C_ {n} ^ {M}}$ is a ring . In particular, it is closed with respect to multiplication and composition . The latter means:

If a

f=(f_{1},f_{2},\ldots f_{p})\in (C_{n}^{M})^{p}

{\ displaystyle f = (f_ {1}, f_ {2}, \ ldots f_ {p}) \ in (C_ {n} ^ {M}) ^ {p}}

and

g\in C_{p}^{M}

{\ displaystyle g \ in C_ {p} ^ {M}}

then

g\circ f\in C_{n}^{M}

{\ displaystyle g \ circ f \ in C_ {n} ^ {M}}

.

Denjoy - Carleman theorem

The Denjoy-Carleman theorem was formulated and partially solved by Arno Denjoy ( Denjoy (1921 )) and fully proved in the work of Torsten Carleman ( Carleman (1926 )). This theorem provides a criterion for solving the question under which sequences M the functions C ^M ([ a , b ]) form a quasianalytic class.

According to the theorem, the following statements are equivalent:

C ^M ([ a , b ]) is a quasianalytic class.
$\sum 1/L_{j}=\infty ,$ ${\ displaystyle \ sum 1 / L_ {j} = \ infty,}$ Where $L_{j}=\inf _{k\geq j}(k\cdot M_{k}^{1/k})$ ${\ displaystyle L_ {j} = \ inf _ {k \ geq j} (k \ cdot M_ {k} ^ {1 / k})}$ .
$\sum _{j}(jM_{j}^{*})^{-1/j}=\infty ,$ ${\ displaystyle \ sum _ {j} (jM_ {j} ^ {*}) ^ {- 1 / j} = \ infty,}$ where M _j ^* is the largest logarithmically convex sequence bounded above by M _j .
$\sum _{j}{\frac {M_{j-1}^{*}}{(j+1)M_{j}^{*}}}=\infty .$ ${\ displaystyle \ sum _ {j} {\ frac {M_ {j-1} ^ {*}} {(j + 1) M_ {j} ^ {*}}} = \ infty.}$

To prove that statements 3, 4 are equivalent to the 2nd, Carleman's inequality is used.

Example : Denjoy (1921 ) ^[3] indicated that if $M_{n}$ ${\ displaystyle M_ {n}}$ given by one of the sequences

1,\,{(\ln n)}^{n},\,{(\ln n)}^{n}\,{(\ln \ln n)}^{n},\,{(\ln n)}^{n}\,{(\ln \ln n)}^{n}\,{(\ln \ln \ln n)}^{n},\dots ,

{\ displaystyle 1, \, {(\ ln n)} ^ {n}, \, {(\ ln n)} ^ {n} \, {(\ ln \ ln n)} ^ {n}, \, {(\ ln n)} ^ {n} \, {(\ ln \ ln n)} ^ {n} \, {(\ ln \ ln \ ln n)} ^ {n}, \ dots,}

then the corresponding quasianalytic class. The first sequence (of units) gives the usual analytical functions.

Additional properties

For a logarithmically convex sequence $M$ ${\ displaystyle M}$ The following properties of the corresponding class of functions hold.

$C^{M}$ ${\ displaystyle C ^ {M}}$ coincides with the class of analytic functions if and only if $\sup _{j\geq 1}(M_{j})^{1/j}<\infty$ ${\ displaystyle \ sup _ {j \ geq 1} (M_ {j}) ^ {1 / j} <\ infty}$ .
If a $N$ ${\ displaystyle N}$ Is another logarithmically convex sequence in which $M_{j}\leqslant C^{j}N_{j}$ ${\ displaystyle M_ {j} \ leqslant C ^ {j} N_ {j}}$ (here $C$ ${\ displaystyle C}$ Is some constant), then $C^{M}\subset C^{N}$ ${\ displaystyle C ^ {M} \ subset C ^ {N}}$ .
$C^{M}$ ${\ displaystyle C ^ {M}}$ stable with respect to differentiation if and only if $\sup _{j\geq 1}(M_{j+1}/M_{j})^{1/j}<\infty$ ${\ displaystyle \ sup _ {j \ geq 1} (M_ {j + 1} / M_ {j}) ^ {1 / j} <\ infty}$ .
For any unboundedly differentiable function $f$ ${\ displaystyle f}$ quasianalytic rings can be found $C^{M}$ ${\ displaystyle C ^ {M}}$ and $C^{N}$ ${\ displaystyle C ^ {N}}$ and elements $g\in C^{M},h\in C^{N}$ ${\ displaystyle g \ in C ^ {M}, h \ in C ^ {N}}$ such that $f=g+h$ ${\ displaystyle f = g + h}$ .

Weierstrass Division

Definition Function $g:\mathbb {R} ^{n}\to \mathbb {R}$ ${\ displaystyle g: \ mathbb {R} ^ {n} \ to \ mathbb {R}}$ called regular order $d$ ${\ displaystyle d}$ towards $x_{n}$ ${\ displaystyle x_ {n}}$ , if a $g(0,x_{n})=h(x_{n})x_{n}^{d}$ ${\ displaystyle g (0, x_ {n}) = h (x_ {n}) x_ {n} ^ {d}}$ and $h(0)\neq 0$ ${\ displaystyle h (0) \ neq 0}$ .

Let be $g$ ${\ displaystyle g}$ - regular order function $d$ ${\ displaystyle d}$ towards $x_{n}$ ${\ displaystyle x_ {n}}$ . They say that the ring $A_{n}$ ${\ displaystyle A_ {n}}$ real or complex functions of $n$ ${\ displaystyle n}$ variables satisfies the Weierstrass division with respect to $g$ ${\ displaystyle g}$ if for each $f\in A_{n}$ ${\ displaystyle f \ in A_ {n}}$ exist $q\in A$ ${\ displaystyle q \ in A}$ and $h_{1},h_{2},\ldots ,h_{d-1}\in A_{n-1}$ ${\ displaystyle h_ {1}, h_ {2}, \ ldots, h_ {d-1} \ in A_ {n-1}}$ such that:

f=gq+h

{\ displaystyle f = gq + h}

where

h(x',x_{n})=\sum _{j=0}^{d-1}h_{j}(x')x_{n}^{j}

{\ displaystyle h (x ', x_ {n}) = \ sum _ {j = 0} ^ {d-1} h_ {j} (x') x_ {n} ^ {j}}

.

Example : the ring of analytic functions and the ring of formal power series both satisfy the Weierstrass division property. If, however, $M$ ${\ displaystyle M}$ logarithmically convex and $C^{M}$ ${\ displaystyle C ^ {M}}$ does not coincide with the class of analytic functions, then $C^{M}$ ${\ displaystyle C ^ {M}}$ does not satisfy the Weierstrass division property with respect to $g(x_{1},x_{2},\ldots ,x_{n})=x_{1}+x_{2}^{2}$ ${\ displaystyle g (x_ {1}, x_ {2}, \ ldots, x_ {n}) = x_ {1} + x_ {2} ^ {2}}$ .

History

The key question of this topic is the ability of an analytic function to uniquely restore its “global appearance” from the values of the function itself and its derivatives at an arbitrary regular point ^[4] . Emil Borel was the first to discover that this property holds not only for analytic functions.

In 1912, Jacques Hadamard formulated the question: what should be the sequence $M_{n},$ ${\ displaystyle M_ {n},}$ so that the above “ uniqueness condition ” is satisfied for any pair of functions from the corresponding class. Arno Denjoy in 1921 gave sufficient conditions for quasi-analyticity and a number of examples of quasi-analytic classes (see Denjoy (1921 )). A complete solution to the problem was given five years later by Torsten Carleman (see Carleman (1926 )), who established the necessary and sufficient conditions for quasianalyticity ^[1] .

Subsequently, S. N. Bernshtein and S. Mandelbright generalized the concept of quasianalyticity to classes of non-differentiable and even discontinuous functions. The simplest example is a set of solutions of a linear differential equation with continuous coefficients; the functions included in this solution, generally speaking, do not have an infinite number of derivatives ^[5] ..

Notes

↑ ¹ ² Mathematical Encyclopedia, 1979 , p. 798.
↑ Mandelbright, 1937 , p. 10-12.
↑ Leontiev, 2001 .
↑ Mandelbright, 1937 , p. 9-11.
↑ Mountain, 1938 , p. 171.

Literature

Gorny A. Quasi-analytic functions // Uspekhi Matematicheskikh Nauk . - M. , 1938. - No. 5 . - S. 171–186 .
Leontyev A.F. Quasianalytic class of functions // Mathematical Encyclopedia (in 5 volumes). - M .: Soviet Encyclopedia , 1979. - T. 2. - S. 798-800.
Mandelbright S. Quasianalytic classes of functions. - M. — L .: ONTI NKTP, 1937.
Mandelbrot S. Adjacent rows, regularization of sequences. Applications, per. from the French., M.: Foreign Literature, 1955.
Carleman, T. (1926), Les fonctions quasi-analytiques , Gauthier-Villars (fr.)
Denjoy, A. (1921), "Sur les fonctions quasi-analytiques de variable réelle", CR Acad. Sci. Paris T. 173: 1329–1331
Hörmander, Lars (1990), The Analysis of Linear Partial Differential Operators I , Springer-Verlag, ISBN 3-540-00662-1
Leont'ev, AF Quasi-analytic class // Hazewinkel, Michiel . Encyclopedia of Mathematics. - Springer Science + Business Media BV / Kluwer Academic Publishers, 2001. - ISBN 978-1-55608-010-4 .

Links

Cohen, Paul J. (1968), " A simple proof of the Denjoy-Carleman theorem ", The American Mathematical Monthly (Mathematical Association of America). - T. 75 (1): 26–31, ISSN 0002-9890 , DOI 10.2307 / 2315100

[ME798-1] ¹ ² Mathematical Encyclopedia, 1979 , p. 798.

[_2996fa783e56bfed-2] Mandelbright, 1937 , p. 10-12.

[_cf40e8715a13dece-3] Leontiev, 2001 .

[_e39c5dc30c579202-4] Mandelbright, 1937 , p. 9-11.

[GOR171-5] Mountain, 1938 , p. 171.