View source for Ditkin set

A closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d1301801.png" /> of a [[Locally compact space|locally compact space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d1301802.png" /> is called a Ditkin set (with respect to a regular function algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d1301803.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d1301804.png" />; cf. [[Algebra of functions|Algebra of functions]]) if each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d1301805.png" /> vanishing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d1301806.png" /> can be approximated, arbitrarily closely, by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d1301807.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d1301808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d1301809.png" /> vanishing  "near"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018010.png" /> (i.e. on a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018011.png" />). The notion of a Ditkin set is closely related to, but more restrictive than, that of a set of spectral synthesis (cf. [[Spectral synthesis|Spectral synthesis]]): for such a set the requirement is that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018012.png" /> vanishing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018013.png" /> can be approximated by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018014.png" /> vanishing near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018015.png" />.

The closed ideal of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018016.png" /> vanishing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018017.png" /> is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018018.png" />. Denoting the ideal of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018019.png" /> vanishing near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018020.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018021.png" /> and its closure by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018022.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018023.png" />. Now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018024.png" /> is a set of spectral synthesis if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018025.png" />, whereas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018026.png" /> is a Ditkin set if each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018027.png" /> belongs to the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018028.png" /> (or, equivalently, to the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018029.png" />). 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 [[#References|[a1]]] it is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018032.png" />-set-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018033.png" />-set problem).

Ditkin sets were first studied for the [[Fourier-algebra(2)|Fourier algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018034.png" />, with the norm defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018035.png" />; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018036.png" /> is any locally compact Abelian group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018037.png" /> is its dual group, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018038.png" /> is the Fourier transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018039.png" /> (cf. also [[Harmonic analysis|Harmonic analysis]]; [[Fourier transform|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018041.png" />-sets; cf. [[#References|[a4]]] and [[#References|[a10]]], respectively. The name  "Ditkin set" , attributed in [[#References|[a6]]], p. 183, to C.S. Herz, refers to work of V.A. Ditkin (1910–1987) in his seminal paper [[#References|[a2]]]; results from this paper were later studied and generalized in [[#References|[a11]]]. In [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018042.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018044.png" /> are sets of spectral synthesis such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018045.png" /> is a Ditkin set, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018046.png" /> is a set of spectral synthesis.

It is also easy to prove that if the boundary of a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018047.png" /> is a Ditkin set, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018048.png" /> itself; cf. [[#References|[a10]]] (for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018049.png" />), [[#References|[a9]]] or [[#References|[a8]]].

Ditkin sets are of particular interest if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018050.png" /> satisfies Ditkin's condition, i.e. if single points are Ditkin sets for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018051.png" />. This notion is older than that of a Ditkin set; cf., e.g., [[#References|[a7]]], p. 86. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018052.png" /> has approximate units (i.e. if each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018053.png" /> can be approximated by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018054.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018055.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018056.png" /> satisfies Ditkin's condition if and only if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018057.png" /> and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018058.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018059.png" /> (i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018060.png" /> belongs to the maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018061.png" />), the zero function can be approximated, arbitrarily closely, by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018062.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018064.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018065.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018066.png" />. It follows that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018067.png" /> satisfies Ditkin's condition, then closed scattered sets (cf. [[Scattered space|Scattered space]]) are Ditkin sets.

The following results can be found in [[#References|[a8]]], Sec. 7.4. First, closed subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018068.png" /> are Ditkin sets for the Fourier algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018069.png" />, and the same result still holds for certain Beurling algebras. Secondly, the following injection theorem for Ditkin sets holds: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018070.png" /> is a closed subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018071.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018072.png" /> is a closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018073.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018074.png" /> is a Ditkin set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018075.png" /> if and only if it is one for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018076.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018077.png" />, the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018078.png" /> orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018079.png" /> (cf. also [[Orthogonality|Orthogonality]]).

In the literature a more restrictive class of sets is also considered, especially in the case of the Fourier algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018080.png" />. A closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018081.png" /> is called a strong Ditkin set if there exists a net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018082.png" /> (cf. also [[Net (directed set)|Net (directed set)]]), bounded in the [[operator norm]] (i.e. the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018083.png" /> is bounded), such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018084.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018085.png" /> is metrizable (cf. [[Metrizable space|Metrizable space]]), then one can require, equivalently, the existence of a [[Sequence|sequence]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018086.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018087.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018088.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018089.png" />, the boundedness in operator norm then being automatically satisfied, by the uniform boundedness theorem (cf. [[Uniform boundedness|Uniform boundedness]])

Strong Ditkin sets were first considered by I. Wik [[#References|[a12]]]. Subsequently it was proved that a closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018091.png" /> without interior is a strong Ditkin set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018092.png" /> if and only if E belongs to the coset ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018094.png" /> (cf., e.g., [[#References|[a4]]], [[#References|[a3]]], [[#References|[a8]]] for details).

A closed interval in the circle group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018095.png" /> is a strong Ditkin set; cf. [[#References|[a12]]]. Therefore, it is essential, for the criterion above, to consider closed sets with empty interior. Also, a line segment in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018096.png" /> is not a strong Ditkin set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018097.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018098.png" /> is not a set of spectral synthesis, then only functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d13018099.png" /> have a chance of being approximable in the Ditkin sense. This motivates the following definition, given in [[#References|[a8]]]. A closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d130180100.png" /> is called a Ditkin set in the wide sense if each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d130180101.png" /> can be approximated by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d130180102.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130180/d130180103.png" />. 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|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.J. Benedetto,   "Spectral synthesis" , Teubner  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.A. Ditkin,   "On the structure of ideals in certain normed rings"  ''Uchen. Zap. Mosk. Gos. Univ. Mat.'' , '''30'''  (1939)  pp. 83–120</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.C. Graham,   O.C. McGehee,   "Essays in commutative harmonic analysis" , Springer  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''2''' , Springer  (1970)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  C.S. Herz,   "The sprectral theory of bounded functions"  ''Trans. Amer. Math. Soc.'' , '''94'''  (1960)  pp. 181–232</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.-P. Kahane,   R. Salem,   "Ensembles parfaits et séries trigonométriques" , Hermann  (1963)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L.H. Loomis,   "An introduction to abstract harmonic analysis" , Van Nostrand  (1953)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  H. Reiter,   J.D. Stegeman,   "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press  (2000)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  H. Reiter,   "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press  (1968)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  W. Rudin,   "Fourier analysis on groups" , Interscience  (1962)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  G.E. Shilov,   "On regular normed rings"  ''Trav. Inst. Math. Steklov'' , '''21'''  (1947)  (In Russian)  (English summary)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  I. Wik,   "A strong form of spectral synthesis"  ''Ark. Mat.'' , '''6'''  (1965)  pp. 55–64</TD></TR></table>