scholarly journals Multipliers on spaces of functions on compact groups with p-summable Fourier transforms

1993 ◽  
Vol 47 (3) ◽  
pp. 435-442 ◽  
Author(s):  
Sanjiv Kumar Gupta ◽  
Shobha Madan ◽  
U.B. Tewari
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Compact Groups ◽  
Circle Group ◽  
Integrable Functions ◽  
Space Of Integrable Functions

Let G be a compact abelian group with dual group Γ. For 1 ≤ p < ∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to lp(Γ). We investigate several problems related to multipliers from Ap(G) to Aq(G). In particular, we prove that (Ap, Ap) ⊊ (Aq, Aq). For the circle group, we characterise permutation invariant multipliers from Ap to Ar for 1 ≤ r ≤ 2.

scholarly journals Multipliers between some function spaces on groups

1978 ◽  
Vol 18 (1) ◽  
pp. 1-11 ◽  
Author(s):  
A.K. Gupta ◽  
U.B. Tewari
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Nonabelian Group ◽  
Integrable Functions ◽  
Abelian Case ◽  
Space Of Integrable Functions

Let G be a nondiscrete locally compact abelian group with dual group Γ. For 1 ≤ p ≤ ∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to Lp(Γ). We investigate multipliers from Ap(G) to Aq(G). If G is compact and 2 < p1, p2 < ∞, we show that multipliers of and multipliers of are different, provided Pl ≠ P2. For compact G, we also exhibit a relationship between lr (Γ) and the multipliers from Ap(G) to Aq(G). If G is a compact nonabelian group we observe that the spaces Ap(G) behave in the same way as in the abelian case as far as the multiplier problems are concerned.

scholarly journals The algebra of functions with Fourier transforms in a given function space

1973 ◽  
Vol 9 (1) ◽  
pp. 73-82 ◽  
Author(s):  
U.B. Tewari ◽  
A.K. Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Function Space ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Algebra Of Functions

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

scholarly journals The conjugation operator onAq(G)

2003 ◽  
Vol 2003 (37) ◽  
pp. 2345-2347
Author(s):  
Sanjiv Kumar Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Bounded Operator ◽  
Conjugation Operator ◽  
Integrable Functions ◽  
The Fourier Transform ◽  
Space Of Integrable Functions

Letq>2. We prove that the conjugation operatorHdoes not extend to a bounded operator on the space of integrable functions defined on any compact abelian group with the Fourier transform inlq.

scholarly journals Wiener Tauberian theorems for vector-valued functions

1994 ◽  
Vol 17 (3) ◽  
pp. 475-478 ◽  
Author(s):  
K. Parthasarathy ◽  
Sujatha Varma
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Weak Form ◽  
Locally Compact Abelian Group ◽  
Tauberian Theorems ◽  
Locally Compact ◽  
Integrable Functions ◽  
Vector Valued Functions ◽  
Vector Valued

Different versions of Wiener's Tauberian theorem are discussed for the generalized group algebraL1(G,A)(of integrable functions on a locally compact abelian groupGtaking values in a commutative semisimple regular Banach algebraA) usingA-valued Fourier transforms. A weak form of Wiener's Tauberian property is introduced and it is proved thatL1(G,A)is weakly Tauberian if and only ifAis. The vector analogue of Wiener'sL2-span of translates theorem is examined.

scholarly journals The class L(log L)α and some lacunary sets

1995 ◽  
Vol 58 (3) ◽  
pp. 387-403
Author(s):  
Sanjiv Kumar Gupta ◽  
Shobha Madan ◽  
U. B. Tewari
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Circle Group ◽  
New Class ◽  
Special Cases ◽  
L Log L

AbstractA well-known result of Zygmund states that if f ∈ L (log+L) ½ on the circle group T and E is a Hadamard set of integers, then . In this paper we investigate similar results for the classes on an arbitrary infinite compact abelian group G and Sidon subsets E of the dual Γ. These results are obtained as special cases of more general results concerning a new class of lacunary sets Sαβ, 0 < α ≤ β, where a subset E of Γ is an Sα β set if . We also prove partial results on the distinctness of the Sαβ sets in the index β.

scholarly journals Multipliers for Amalgams and the Algebra S0(G)

1987 ◽  
Vol 39 (1) ◽  
pp. 123-148 ◽  
Author(s):  
Maria L. Torres De Squire
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
The Difference ◽  
Image Position

Throughout the whole paper G will be a locally compact abelian group with Haar measure m and dual group Ĝ. The difference of two sets A and B will be denoted by A ∼ B, i.e.,For a function f on G and s ∊ G, the functions f′ and fs will be defined by

On Multipliers with Unconditionally Converging Fourier Series

1972 ◽  
Vol 24 (3) ◽  
pp. 477-484 ◽  
Author(s):  
Gregory F. Bachelis ◽  
Louis Pigno
Keyword(s):  
Banach Space ◽  
Fourier Series ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Complex Valued

Let G be a compact abelian group with dual group Γ. For 1 ≦ p < ∞, 1 ≦ q < ∞, let denote the Banach space of complex-valued functions on Γ which are multipliers of type (p, q) and the subspace of compact multipliers.Grothendieck [10; 11] has proven that a function in LP(G), 1 ≦ p < 2, has an unconditionally converging Fourier series in LP(G) if and only if it is in L2(G), and Helgason [12] has proven that the derived algebra of LP(G), 1 ≦ p < 2, is L2(G). Using these results we show in § 2 that a multiplier of type (p, g), 1 ≦ p ≦ 2, 1 ≦ q ≦ 2, has an unconditionally converging Fourier series in if and only if it is in (Theorem 2.1), and that, for 1 ≦ p ≦ q ≦ 2, the derived algebra of is (Theorem 2.2). Statements equivalent to the above are also given.

Gap Series on Groups and Spheres

1966 ◽  
Vol 18 ◽  
pp. 389-398 ◽  
Author(s):  
Daniel Rider
Keyword(s):  
Abelian Group ◽  
Trigonometric Polynomial ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Gap Series ◽  
Image Position

Let G be a compact abelian group and E a subset of its dual group Γ. A function ƒ ∈ L1(G) is called an E-function if for all γ ∉ E wheredx is the Haar measure on G. A trigonometric polynomial that is also an E-function is called an E-polynomial.

On multipliers of Fourier transforms

1972 ◽  
Vol 71 (1) ◽  
pp. 63-66 ◽  
Author(s):  
Louis Pigno
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Lebesgue Space ◽  
Fourier Transforms ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Continuous Complex ◽  
Complex Valued

In this paper G is a locally compact Abelian group, φ a complex-valued function defined on the dual Γ, Lp(G) (1 ≤ p ≤ ∞) the usual Lebesgue space of index p formed with respect to Haar measure, C(G) the set of all bounded continuous complex-valued functions on G, and C0(G) the set of all f ∈ C(G) which vanish at infinity.

scholarly journals Union results for thin sets

1990 ◽  
Vol 32 (2) ◽  
pp. 241-254
Author(s):  
Kathryn E. Hare
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Group Operation ◽  
Borel Measures ◽  
Thin Sets

Let G be a compact abelian group and let Γ be its (discrete) dual group. Denote by M(G) the space of complex regular Borel measures on G.Let E be a subset of Γ. Then:(i) E is called a Rajchman set if, for all μ ∈M(G) implies (ii) E is called a set of continuity if given ε > 0 there exists δ > 0 such that if and(iii) E is called a parallelepiped of dimension N if |E| = 2N and there are two-element sets . (The multiplication indicated here is the group operation.)

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