1. Let us recall the original definition of g-juxtapolynomials /see C h □,C 7 □ / :
De finition 5. y 6 c£» is a g-juxtapolynomial of f on Q if there is no y^G^5 such that
(1.1) f(x)-f(x)
0 if f (x) = y?(x)
*
L < |f(x)-^(x)| if f(x) y y>(x)
hold for all x € Q. □ We can prove the following
Assertion 1. The Definition 2 is equivalent to the Definition 5.
a
Proof: Assume that there is 1 e such that
oii
X for xGQ : f (x ) = y?(x) ,
U . 2 ) ^(x) > 0 for xGQ : f ( x) > y7 ( x) , j(x) < 0 for xGQ : f ( x ) < fix) .
/I.e. y5 is not a g-juxtapolynomial according to De f. 21
Then for any 0 < £ < m i n i f : c(x) f 0} we have
f(x) J
for all x G Q:
, =
0 (1.3) If(x)-^(x) - £ £(x) I •f(x) - fix)
if f (x) = y? (x)
if f(x) f f (x).
according to Def. 5. for algebraic juxtapolynomials, see Section 1.
They introduced a following concept:
We say that a sequence of real numbers , ...,0^, m ^ n + 1 , n weak sign changes/the term "weakly interpolating g-polynomial"/.
it can be proved that our and their conditions are equivalent, namely the following assertion holds.
Assertion 2. The sequence of real numbers c*, , . c* ,
97
regarding the combinatorial properties of finite sequences of signs are also proved. /e.g. strict g-juxtapolynomials/ and the results proved seem to be new o n e s . that spline functions constitute such a system.
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m : L. Fejér, Über die Lage der Nullstellen von Polynomen die aus Minimumfolgerungen gewisser Art entspringen, Math. Ann., 85, 42-48 /1922/.
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31, 125-138 /1922/.
C 3 3 M. Fekete, On the structure of polynomials of least devi
ation, Bull. Res. Council Israle, 5A, 11-19 /19 5 5 /,
[1+3 T.S. Motzkin, J.L. Walsh, Polynomials of best approxima
tion on a real finite point set I, Trans. Amer. Math. Soc, 9_1, No.2, 231-245 /19 59 / .
C53 0. Shisha, Best approximation on some finite sets, J. of Math. Anal. Appl., 21, 347-355 /19 6 8 /.
l6i I. Marusciac, On linear juxtaoperators, in: Z. Cieselski, J. Musielak, e d s . , "Approximation Theory", PWN, Warszawa,
• 1975.
C73 J. Rice, "Approximation of Functions. Vol II: Non-linear and Multivariate Theory", Addison-Wesley, Reading, Mass.,
1969.
[83 B. TaHTMaxep, M . r . KpefiH, "OcuHJiaiiHOHHue MaTpHasi h Majibie KOJieöaHHH MexaHH'iecKHx CHCTeM", TTTH, MocKBa, 1950.
191 M . r . KpefiH, A.A. HyflejiBMaH, "npobneMa MOMeHTOB MapnoBa
h BKCTpeMaJiBHtie 3ana^iH", Hayna, MocKBa, 1973.
[103 S. Karlin. W.J. Studden, "Tschebysheff-Systems: With Applications to Anal, and Stat.", Interscience, N.Y.,1966 [113 S. Karlin, "Total Positivity", Stanford Univ. Press, 1968
99
E121 B. Uhrin, "Theorem of Helly and the Existence of G e n eral
ized Polynomials with Prescribed Values and Signs", Semi
nar Notes, Mathematics No.3, Hungarian Committee for Systems Analysis, Budapest, 1975.
1131 A. Tihanyi, B. Uhrin, About the generalized polynomials with prescribed signs on discrete sets, Ann. Univ. Sei.
Budapest, Sec. Math., _19, 165-169 / 19 7 6 / .
El^H B. Uhrin, A characterization of finite Chebyshev sequen
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E15H B. Uhrin, A characterization of generalized juxtapolyno- mials on finite point sets, Coll. Math. Soc. J. Bolyai, to appear.
El
6l
B. Uhrin, Some combinatorial results concerning finite sequences of signs, in preparationEflHHbffi nOflXOfl K TEOPHH IIPHEJlHJKEHHHft HA KOHEHHLJX MH05KECTBAX B. yxpHH
Pe3iOMe
B cTaTbe BBOflHTca noHHTHe "obobmeHHoro mKCTa-MHoro^uieHa"
nax sjieMeHTa nonnpocTpaHCTBa $yHKUHíl, k o t o p h íI HBJineTCH "caMHM 6jih3k h m" k «aHHoß $yHKU.HH H flaeTCH h x xapaKTepHCTHKa b eny^ae gebbmieBCKHx noanpocTpaHCTB. HecKOJibKO pe3yjibTaTOB xacaiomHxcH cjiabbix HeöbniieBCKHX noanpocTpaHCTB Toxce noxa3aHO b CTaTbe.
MTA SZTAKI Tanulmányok 147/1983 101-109
The mathematical theory known generally as inductive in
ference (or algorithmic identification) is based on the b l a c k box principle. According to it the knowledge of some black- box-informations on a given system creates the possibility to derive certain statements, describing its strucutre. In the present paper we use the following modification of this prin-cimple: The task to synthesize a program for a certain general recursive function
f
is given to a "program-constructor". No description of / in a language, that he can understand, is available. The "program-constructor" works in steps and in any of them a program-hypothesis has to be built up, using onlysome inputs l,2,...,n and the corresponding outputs
f(l),f(2)a ...j f(n).
The increase in the number of steps must leed to a stabilization of the program-hypothesis on a real program off (f
is being identified in the limit). This problem was d i s cussed for the first time by GOLD in [5] .There are situations in which the "program-constructor"
is supplied with some extra knowledge about
f:
e.g. belongs to a certain class of functions, a program forf
is known to have a definite complexity etc. The whole process is then re-fered to as inductive inference with additional information.
The presence of additional information may substantially in
crease the power of the "program-constructor" or under certain conditions have no impact on it. Naturally there are many type of additional information. In the following we shall consider the one introduced by FREIWALD/WIEHAGEN in [4].
We can formalize the above described process of inductive inference with additional information as follows: the "program- constructor" is a partial recursive function, that is also cal
led identification strategy; a program for / is a number of f in a fixed Gödel numbering of all partial recursive function/see [8] /; any upper bound of the minimal number of f in this Gödel numbering is considered additional information. This is the usual formalization applied by BARZDIN [2] , BLUM/BLUM [l] and WIEHAGEN I 9 I .
An identification strategy is able to synthesize programs for a whole class of general recursive functions. Such strate
gies can in addition have some practically important properties:
consistency of the hypotheses to the function to be identified, finite number of steps leeding to the stabilization etc. In
[7], [8] and [ll] the power of identification strategies, cha
racterized by the volume of the corresponding families of iden
tified functional classes, is investigated. In [4] this is done in relation to additional information. The families of identi
fied functional classes are refered to as identification types.
These have been studied using means, provided by the theory of inductive inference itself. However it is important to consider the identification types from a different point of view with the purpose of obtaining deeper results. We can achieve this by means of an inductive inference-free characterization of the identification types. Characterization results were obtained by WIEHAGEN/LIEPE [ll] - using means from the theory of com_
plexity, JUNG [6] - using the apparatus of topology and WIEHAGEN/JUNG [ i o ] - exploiting only the possibilities and means, given by the classical recursion theory. Other charac
terization problems have been solved by BLUM/BLUM [l] and
FREIVALD [3]. All kinds of characterization consider the usual identification types. The following paper presents recursion theory characterizations of the identification types which require additional information. A similar topological or -theory of complexity-characterization of these remains an open problem.
103
Next we turn to the precise definitions and formulation of results.
Let Nz denote the set of all natural numbers, Pn and R n denote the class of all partial recursive respectively general recursive function of nGNz variables. For n=l we omit the upper