Ditkin set
A closed subset of a locally compact space
is called a Ditkin set (with respect to a regular function algebra
defined on
; cf. Algebra of functions) if each
vanishing on
can be approximated, arbitrarily closely, by functions
with
and
vanishing "near"
(i.e. on a neighbourhood of
). The notion of a Ditkin set is closely related to, but more restrictive than, that of a set of spectral synthesis (cf. Spectral synthesis): for such a set the requirement is that each
vanishing on
can be approximated by functions
vanishing near
.
The closed ideal of all vanishing on
is usually denoted by
. Denoting the ideal of all
vanishing near
by
and its closure by
, one has
. Now
is a set of spectral synthesis if
, whereas
is a Ditkin set if each
belongs to the closure of
(or, equivalently, to the closure of
). It is a famous open problem (as of 2000) whether (in specific cases) each set of spectral synthesis is actually a Ditkin set (this problem may be called the synthesis-Ditkin problem; in [a1] it is called the
-set-
-set problem).
Ditkin sets were first studied for the Fourier algebra , with the norm defined by
; here,
is any locally compact Abelian group,
is its dual group, and
is the Fourier transform of
(cf. also Harmonic analysis; Fourier transform). A.P. Calderón (1956) studied this kind of set in an effort to obtain results about sets of spectral synthesis. Therefore, Ditkin sets are sometimes called Calderón sets or
-sets; cf. [a4] and [a10], respectively. The name "Ditkin set" , attributed in [a6], p. 183, to C.S. Herz, refers to work of V.A. Ditkin (1910–1987) in his seminal paper [a2]; results from this paper were later studied and generalized in [a11]. In [a9] the term Wiener–Ditkin set is used.
The union of two Ditkin sets is again a Ditkin set; this follows easily from a triangle inequality like . More generally, if a closed set is the union of countably many Ditkin sets, then it is again a Ditkin set. In contrast, for most function algebras it is unknown (as of 2000) whether the union of two sets of spectral synthesis is again of spectral synthesis: this is the famous union problem for this class of sets. Of course, the union problem becomes trivial if the synthesis-Ditkin problem gets a positive answer. Also, if
and
are sets of spectral synthesis such that
is a Ditkin set, then
is a set of spectral synthesis.
It is also easy to prove that if the boundary of a closed set is a Ditkin set, then so is
itself; cf. [a10] (for the case
), [a9] or [a8].
Ditkin sets are of particular interest if satisfies Ditkin's condition, i.e. if single points are Ditkin sets for
. This notion is older than that of a Ditkin set; cf., e.g., [a7], p. 86. If
has approximate units (i.e. if each
can be approximated by functions
with
), then
satisfies Ditkin's condition if and only if for each
and each
such that
(i.e. such that
belongs to the maximal ideal
), the zero function can be approximated, arbitrarily closely, by functions
with
and
equals
near
. It follows that if
satisfies Ditkin's condition, then closed scattered sets (cf. Scattered space) are Ditkin sets.
The following results can be found in [a8], Sec. 7.4. First, closed subgroups of are Ditkin sets for the Fourier algebra
, and the same result still holds for certain Beurling algebras. Secondly, the following injection theorem for Ditkin sets holds: If
is a closed subgroup of
, and
is a closed subset of
, then
is a Ditkin set for
if and only if it is one for
, where
, the subgroup of
orthogonal to
(cf. also Orthogonality).
In the literature a more restrictive class of sets is also considered, especially in the case of the Fourier algebra . A closed set
is called a strong Ditkin set if there exists a net
(cf. also Net (directed set)), bounded in the operator norm (i.e. the mapping
is bounded), such that
. If
is metrizable (cf. Metrizable space), then one can require, equivalently, the existence of a sequence
in
such that
for all
, the boundedness in operator norm then being automatically satisfied, by the uniform boundedness theorem (cf. Uniform boundedness)
Strong Ditkin sets were first considered by I. Wik [a12]. Subsequently it was proved that a closed subset of
without interior is a strong Ditkin set for
if and only if E belongs to the coset ring of
(cf., e.g., [a4], [a3], [a8] for details).
A closed interval in the circle group is a strong Ditkin set; cf. [a12]. Therefore, it is essential, for the criterion above, to consider closed sets with empty interior. Also, a line segment in
is not a strong Ditkin set for
, because it has empty interior but does not belong to the coset ring. Consequently, the above-mentioned injection theorem does not hold for strong Ditkin sets.
If is not a set of spectral synthesis, then only functions
have a chance of being approximable in the Ditkin sense. This motivates the following definition, given in [a8]. A closed set
is called a Ditkin set in the wide sense if each
can be approximated by functions
with
. This notion is, in a way, more natural than that of a Ditkin set; but in 1956 it was not yet known that sets not of spectral synthesis abound in the case of the Fourier algebra: Malliavin's result (cf. Spectral synthesis) dates from 1959. It is not known in general (for instance in the case of the Fourier algebra) whether all closed subsets are Ditkin sets in the wide sense. This problem is a natural generalization of the synthesis-Ditkin problem.
References
[a1] | J.J. Benedetto, "Spectral synthesis" , Teubner (1975) |
[a2] | V.A. Ditkin, "On the structure of ideals in certain normed rings" Uchen. Zap. Mosk. Gos. Univ. Mat. , 30 (1939) pp. 83–120 |
[a3] | C.C. Graham, O.C. McGehee, "Essays in commutative harmonic analysis" , Springer (1979) |
[a4] | E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 2 , Springer (1970) |
[a5] | C.S. Herz, "The sprectral theory of bounded functions" Trans. Amer. Math. Soc. , 94 (1960) pp. 181–232 |
[a6] | J.-P. Kahane, R. Salem, "Ensembles parfaits et séries trigonométriques" , Hermann (1963) |
[a7] | L.H. Loomis, "An introduction to abstract harmonic analysis" , Van Nostrand (1953) |
[a8] | H. Reiter, J.D. Stegeman, "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press (2000) |
[a9] | H. Reiter, "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press (1968) |
[a10] | W. Rudin, "Fourier analysis on groups" , Interscience (1962) |
[a11] | G.E. Shilov, "On regular normed rings" Trav. Inst. Math. Steklov , 21 (1947) (In Russian) (English summary) |
[a12] | I. Wik, "A strong form of spectral synthesis" Ark. Mat. , 6 (1965) pp. 55–64 |
Ditkin set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ditkin_set&oldid=18420