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We show that if f f is in an anisotropic Hardy space H A p H_A^p , 0 > p ≤ 1 0 > p \leq 1 , with respect to a dilation matrix A A , then its Fourier transform f ^ \hat {f} satisfies the pointwise estimate \[ | f ^ ( ξ ) | ≤ C | | f | | H A p ρ ∗ ( ξ ) 1 p − 1 . |\hat f(\xi )| \le C ||f||_{H^p_A} \rho _*(\xi )^{\frac {1}{p}-1}. \] Here, ρ ∗ \rho _* is a quasi-norm associated with the transposed matrix A ∗ A^* . This leads to necessary conditions for functions m m to be multipliers on H A p H_A^p , as well as further pointwise characterizations on f ^ \hat {f} and a generalization of the Hardy-Littlewood inequality on the integrability of f ^ \hat {f} . This last result is strengthened through the use of rearrangement functions.

We prove two versions of Beurling's theorem for Riemannian symmetric spaces of arbitrary rank. One of them uses the group Fourier transform and the other uses the Helgason Fourier transform. This is the master theorem in the quantitative uncertainty principle.

Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. The best-known example of this type is the Hermite functions, i.e., the Hermite polynomials multiplied by exp(-x 2 /2), which are eigenfunctions of the Fourier transform. In this paper, we introduce two new examples of finite systems of this type and obtain their orthogonality relations. We also estimate a complicated integral and propose a conjecture for a further example of finite orthogonal sequences.

We consider the classical theorems of Mikhlin and Littlewood-Paley from Fourier analysis in the context of the distorted Fourier transform. The latter is defined as the analogue of the usual Fourier transform as that transformation which diagonalizes a Schrödinger operator − Δ + V -\Delta +V . We show that for such operators which display a zero energy resonance the full range 1 > p > ∞ 1>p> \infty in the Mikhlin theorem cannot be obtained: in the radial, three-dimensional case it shrinks to 3 2 > p > 3 \frac 32>p>3 .

Hardy's uncertainty principle states that it is impossible for a function and its Fourier transform to be simultaneously very rapidly decreasing. In this paper we prove L P versions of this principle for the Jacobi transform and for the Fourier transform on real hyperbolic spaces.

We prove that the Cesàro mean σ \sigma of a multiplier λ \lambda on L 1 ( R ) {L^1}({\mathbf {R}}) is also a multiplier on L 1 ( R ) {L^1}({\mathbf {R}}) . In the particular cases when (i) λ \lambda is odd, we prove that σ \sigma is the Fourier transform of an odd function in the Hardy space H 1 ( R ) {H^1}({\mathbf {R}}) , and (ii) λ \lambda is even, we give a necessary and sufficient condition in order that σ \sigma be a Fourier transform of an even function in L 1 ( R ) {L^1}({\mathbf {R}}) . As a corollary, we obtain a nontrivial condition for λ \lambda in order to be a multiplier on L 1 ( R ) {L^1}({\mathbf {R}}) ; namely, \[ ∫ 0 ∞ | 1 t ∫ 0 t { λ ( ξ ) − λ ( − ξ ) } d ξ | d t t > ∞ . \int _0^\infty {\left | {\frac {1}{t}\int _0^t {\{ \lambda (\xi ) - \lambda ( - \xi )\} \,d\xi } } \right |} \frac {{dt}}{t} > \infty . \] We also prove Hardy type inequalities for multipliers and Hilbert transforms.

An analytic function F F on the disc belongs to B B if | | F | | B = ∫ 0 1 ∫ 0 2 π | F ′ ( r e i θ ) | d θ d r > ∞ ||F|{|_B} = \int _0^1 {\int _0^{2\pi } {|{F’}(r{e^{i\theta }})|d\theta \,dr > \infty } } . Notice that B ⫋ H 1 ⫋ L 1 B \varsubsetneqq {H^1} \varsubsetneqq {L^1} , where H 1 {H^1} is the Hardy space of all analytic functions F F so that \[ | | F | | H 1 = sup 0 > r > 1 ∫ 0 2 π | F ( r e i θ ) | d θ > ∞ , ||F|{|_{{H^1}}} = \sup \limits _{0 > r > 1} \int _0^{2\pi } {|F(r{e^{i\theta }})|\,d\theta > \infty ,} \] L 1 {L^1} is the Lebesgue space of integrable functions on [ 0 , 2 π ] [0,2\pi ] , and the inclusion H 1 ⫋ L 1 {H^1} \varsubsetneqq {L^1} is taken in the sense of boundary values, that is, F ∈ H 1 ⇒ lim r → 1 − ℜ F ( r e i θ ) ∈ L 1 F \in {H^1} \Rightarrow {\lim _{r \to 1}} - \Re F(r{e^{i\theta }}) \in {L^1} . Kolmogorov in 1923 showed that there exists an f f in L 1 {L^1} so that its Fourier series diverges almost everywhere. In 1953 Sunouchi showed that there exists an f f in H 1 {H^1} with an almost everywhere divergent Fourier series. The purpose of this note is to announce. Theorem 1. There exists an f ∈ B f \in B , whose Fourier series diverges a.e. This problem was mentioned to the first author by Professor Guido Weiss.

Let d u du be a smooth positive measure carried by a smooth compact hypersurface S S that is strictly convex and without boundary in R n ( n ⩾ 2 ) {R^n}(n \geqslant 2) . Assume that both S S and d u du are symmetric about the origin. If d ^ u \hat du denotes the Fourier transform of d u du then we show that the null set of d ^ u \hat du is a disjoint union of a compact set and countably many hypersurfaces that are all diffeomorphic to the unit sphere S n − 1 {S^{n - 1}} .

We study some problems related to convergence and divergence a.e. for Fourier series in systems { ϕ k } \{ {\phi _k}\} , where { ϕ k } \{ {\phi _k}\} is either a system of orthonormal polynomials with respect to a measure d μ d\mu on [ − 1 , 1 ] [-1,1] or a Bessel system on [ 0 , 1 ] [0,1] . We obtain boundedness in weighted L p {L^p} spaces for the maximal operators associated to Fourier-Jacobi and Fourier-Bessel series. On the other hand, we find general results about divergence a.e. of the Fourier series associated to Bessel systems and systems of orthonormal polynomials on [ − 1 , 1 ] [-1,1] .

In this note we give a short proof of a classical theorem in the theory of Fourier integrals. We will say that an even, continuous function f : R [0, o,) with f(O) = 1 is of Polya-type if f is convex on (0, ox) and limt+c f(t) = 0. Titchmarsh ([6], Theorem 124) has proved (using a different terminology) that every Polyatype function is the characteristic function of an absolutely continuous probability distribution.1 A proof of this result is essentially contained already in [4]. Esseen [1] and Polya [5] were the first to point out the usefulness of Polya-type functions in probability theory. Polya also noted that the corresponding density is continuous except perhaps at zero. In their proofs, Titchmarsh and Polya first show that p(x) -lim 1 (t)cosxtdt (x #8 0) exists and is nonnegative. Then, applying Fourier's inversion theorem, they prove that p is Lebesgue integrable and that f (t) = jp(x) cos xt dx (t E R). The aim of the present note is to point out that the result above has a short proof that does not use inversion theorems. Moreover, p can be given explicitly with a certain Lebesgue integral. Our proof is based on Fubini's theorem and the following simple fact (see e.g. Lukacs [3]): A function f is of Polya-type if and only if it admits a representation (1) f(t)= j b(t/y) dv(y), where v is a probability distribution on (0, ox) and +b(t) = max(0, 1 jtl). The "if part" of this statement is trivial, while the "only if part" follows from the equation j( t/y) d[1 f(y) + yf'(y)] =-tj + [(Y)] dy=f(t), t>0 Received by the editors March 4, 1996. 1991 Mathematics Subject Classification. Primary 42A38; Secondary 60E10. ISince this theorem seems to have most of its applications in probability theory we use the terminology of that field. In textbooks on probability theory the result is usually referred to as P61ya's theorem. (?)1998 American Mathematical Society 711 This content downloaded from 207.46.13.78 on Thu, 21 Jul 2016 05:26:29 UTC All use subject to http://about.jstor.org/terms

Let G be a homogeneous group with the graded Lie algebra or a noncompact semisimple Lie group with finite center. We define the Fourier transform f ^ \hat f of f as a family of operators f ^ ( π ) = ∫ G f ( x ) π ( x ) d x ( π ∈ G ^ ) \hat f(\pi ) = {\smallint _G}f(x)\pi (x)dx(\pi \in \hat G) , and we say that f ^ \hat f is positive if all f ^ ( π ) \hat f(\pi ) are positive. Then, we construct an integrable function f on G with positive f ^ \hat f and the restriction of f to any ball centered at the origin of G is square-integrable, however, f is not square-integrable on G.

f f is the Fourier transform of an infinitely differentiable function of compact support on R {\mathbf {R}} if, and only if, f f is entire and of exponential type with | f ( x ) | = O ( | x | − N ) \left | {f\left ( x \right )} \right | = O\left ( {|x{|^{ - N}}} \right ) for each N > 0 N > 0 as | x | → ∞ |x| \to \infty for real x x . In some sense, such an f f has its zeros close to the real axis and has positive density of zeros F F with n ( r ) = D r + o ( r ) n\left ( r \right ) = Dr + o\left ( r \right ) . It is shown here that if the zeros of f f are in a strip parallel to the real axis and if n ( r ) = D r + O ( 1 ) n\left ( r \right ) = Dr + O\left ( 1 \right ) , then f f is the product of two such transforms with zero densities D / 2 D/2 and indicators one-half of the indicator of f f . There is a factorable f f in D ^ ( R ) \widehat {\mathcal {D}}\left ( {\mathbf {R}} \right ) with zeros on a line and not satisfying the stricter density condition. Analogous results hold for transforms of distributions of compact support on R {\mathbf {R}} . The study was motivated by the outstanding problem of Ehrenpreis that asks if D ( R ) ∗ D ( R ) = D ( R ) \mathcal {D}\left ( {\mathbf {R}} \right ) * \mathcal {D}\left ( {\mathbf {R}} \right ) = \mathcal {D}\left ( {\mathbf {R}} \right ) .

Recent papers have given two different conditions on pairs of nonnegative weight functions that insure that a Fourier transform norm inequality holds in R n {{\mathbf {R}}^n} . With additional assumptions these conditions were also shown to be implied by the norm inequality. A direct proof is given here that these conditions are equivalent; this can be used to simplify some of the proofs in those papers.

In this paper degree of approximation of Lebesgue integrable functions by means of Fourier series is examined.

Let Λ \Lambda be a closed and discrete or compact subset of a second countable LCA {\text {LCA}} group G G and E E a subset of the dual group. Balayage is said to be possible for ( Λ , E ) (\Lambda ,\;E) if for every finite measure μ \mu on G G there is some measure ν \nu on Λ \Lambda whose Fourier transform, ν ^ \hat \nu , agrees on E E with μ ^ \hat \mu . If balayage is assumed possible just when μ \mu is a point measure (with the norms of all the measures ν \nu bounded by some constant), then there is a bounded linear projection, B Λ {B_\Lambda } , from the measures on G G onto those on Λ \Lambda with ( B Λ μ ) ∧ = μ ^ {({B_\Lambda }\mu )^ \wedge } = \hat \mu on E E . An application is made to balayage in product groups.

Let f f be a continuous function defined on the interval ( 0 , 1 ) (0,1) . For n = 1 , 2 , … n = 1,2, \ldots and 0 > s > t > 1 0 > s > t > 1 , denote by a n ( f ; s , t ) , b n ( f ; s , t ) {a_n}(f;s,t),{b_n}(f;s,t) the n n th Fourier coefficients of f | ( s , t ) f|(s,t) . It is shown that the following statements are equivalent: (i) f f is strictly convex on ( 0 , 1 ) (0,1) . (ii) b n ( f ; s , t ) > ( 2 / n π ) [ f ( s ) − f ( ( s + t ) ) / 2 ] {b_n}(f;s,t) > (2/n\pi )[f(s) - f((s + t))/2] for all n = 1 , 2 , … n = 1,2, \ldots and whenever 0 > s > t > 1 0 > s > t > 1 . (iii) b n ( f ; s , t ) > ( 2 / n π ) [ f ( ( s + t ) / 2 ) − f ( t ) ] {b_n}(f;s,t) > (2/n\pi )[f((s + t)/2) - f(t)] for all n = 1 , 2 , … n = 1,2, \ldots and whenever 0 > s > t > 1 0 > s > t > 1 . If, in addition, f f is twice differentiable, then (i) and the following statement are also equivalent: (iv) a n ( f ; s , t ) > 0 {a_n}(f;s,t) > 0 for all n = 1 , 2 , … n = 1,2, \ldots and whenever 0 > s > t > 1 0 > s > t > 1 .

Denoting by S ∗ {S^\ast } the maximal partial sum operator of Fourier series, we prove that S ∗ ( f 1 , f 2 , … , f k , … ) = ( S ∗ f 1 , S ∗ f 2 , … , S ∗ f k , … ) {S^\ast }({f_1},{f_2}, \ldots ,{f_k}, \ldots ) = ({S^\ast }{f_1},{S^\ast }{f_2}, \ldots ,{S^\ast }{f_k}, \ldots ) is a bounded operator from L p ( l r ) {L^p}({l^r}) to itself, 1 > p , r > ∞ 1 > p,r > \infty . Thus, we extend the theorem of Carleson and Hunt on pointwise convergence of Fourier series to the case of vector valued functions. We give also an application to the rectangular convergence of double Fourier series.

We apply Hörmander’s technique to prove a weak-type ( 1 , 1 ) (1,1) estimate for multiplier operators with respect to the Fourier-Bessel transform. This improves a result in [4, 5].

There exists a continuous function f f on the real line, vanishing at infinity, such that, for every measure-preserving transformation h h , the composition f ∘ h f \circ h fails to be a Fourier transform. This fact is a consequence of a theorem about measurable functions which is obtained from the theory of idempotents.

We consider operators of the form∫−∞∞Ftϕ(t)dt\int _{-\infty }^\infty {{F_t}} \phi \left ( t \right )dtwhereFt{F_t}is a11-parameter family of Fourier integral operators andϕ(t)dt\phi \left ( t \right )dta tempered distribution on the real line. We extend the result given in [1].

Let G denote a compact abelian group, and A p {A^p} the space of functions continuous on G and having p-summable Fourier transforms. The idempotent multipliers from A p {A^p} to A q {A^q} are characterised for p , q ∈ [ 1 , 2 ] p,q \in [1,2] .

Let T T be a bounded linear operator from L 1 ( G , λ ) {L_1}(G,\lambda ) into L 1 ( Ω , F , P ) {L_1}(\Omega ,\mathcal {F},P) , where ( G , λ ) (G,\lambda ) is a compact abelian metric group with its Haar measure, and ( Ω , F , P ) (\Omega ,\mathcal {F},P) is a probability space. Let ( μ ω ) ({\mu _\omega }) be the random measure on G G associated to T T ; that is, T f ( ω ) = ∫ G f ( t ) d μ ω ( t ) Tf(\omega ) = \int _G {f(t)d{\mu _\omega }(t)} for each f f in L 1 ( G ) {L_1}(G) . We show that, unlike the ideals of representable and Kalton operators, there is no subideal B B of M ( G ) \mathcal {M}(G) such that T T is completely continuous if and only if μ ω ∈ B {\mu _\omega } \in B for almost ω \omega in Ω \Omega . We actually exhibit a noncompletely continuous operator T T such that μ ^ ω ∈ l 2 + ε ( G ^ ) {\hat \mu _\omega } \in {l_{2 + \varepsilon }}(\hat G) for each ε > 0 \varepsilon > 0 .

Let Λ \Lambda be a discrete subset of a LCA group and E a compact subset of the dual group Γ \Gamma . Balayage is said to be possible for ( Λ \Lambda , E) if the Fourier transform of each measure on G is equal on E to the Fourier transform of some measure supported by Λ \Lambda . For a class of infinite compact metrizable Γ \Gamma , including all such torsion groups, we show how to construct E ⊂ Γ E \subset \Gamma such that there are arbitrarily sparse sets Λ \Lambda with balayage possible for ( Λ \Lambda , E). E is, moreover, large enough that the set of products E ⋅ E ⋅ E = Γ E \cdot E \cdot E = \Gamma .

A new necessary and sufficient condition for real entire functions, represented by Fourier transforms, to have only real zeros is proved. An application of this result to the Riemann ξ \xi -function is also given.

The theory of nonharmonic Fourier series in L 2 ( − π , π ) {L^2}( - \pi ,\pi ) is concerned with the completeness and expansion properties of sets of complex exponentials { e i λ n t } \{ {e^{i{\lambda _n}t}}\} . It is well known, for example, that the completeness of the set { e i λ n t } \{ {e^{i{\lambda _n}t}}\} ensures that of { e i μ n t } \{ {e^{i{\mu _n}t}}\} whenever ∑ | λ n − μ n | > ∞ \sum {|{\lambda _n} - {\mu _n}| > \infty } . In this note we establish two results which guarantees that if { e i λ n t } \{ {e^{i{\lambda _n}t}}\} is a Schauder basis for L 2 ( − π , π ) {L^2}( - \pi ,\pi ) , then { e i μ n t } \{ {e^{i{\mu _n}t}}\} is also a Schauder basis whenever { μ n } \{ {\mu _n}\} is “sufficiently close” to { λ n } \{ {\lambda _n}\} .

Necessary and sufficient conditions are given on α \alpha , β \beta and t t for there to exist a constant K K such that \[ ( ∑ n ∈ E | f ^ ( n ) | 2 ) 1 / 2 ⩽ K | E | α ‖ f | x | β ‖ t {\left ( {{{\sum \limits _{n \in E} {\left | {\hat f(n)} \right |} }^2}} \right )^{1/2}} \leqslant K{\left | E \right |^\alpha }{\left \| {f{{\left | x \right |}^\beta }} \right \|_t} \] for all f ∈ L 1 ( T d ) f \in {L^1}({T^d}) and finite E ⊂ Z d E \subset {Z^d} .

C. Fefferman [1, 2, 3] has shown that the multiple Fourier series of an f ∈ L p , p > 2 f \in {L^p},p > 2 , may diverge a.e. when summed over expanding spheres, but converges a.e. when summed over expanding polyhedral surfaces. We show this dichotomy does not prevail for multiple Walsh-Fourier series.

It is shown that the Fomin class F p ( 1 > p ⩽ 2 ) {\mathcal {F}_p}(1 > p \leqslant 2) is a subclass of C ∩ B V \mathcal {C} \cap \mathcal {B}\mathcal {V} , where C \mathcal {C} is the Garrett-Stanojević class and B V \mathcal {B}\mathcal {V} the class of sequences of bounded variation. Wider classes of Fourier and Fourier-Stieltjes series are found for which a n lg n = o ( 1 ) , n → ∞ {a_n}\;{\text {lg}}\;n = o(1),n \to \infty , is a necessary and sufficient condition for L 1 {L^1} -convergence. For cosine series with coefficients in B V \mathcal {B}\mathcal {V} and n Δ a n = O ( 1 ) n\Delta {a_n} = O(1) , n → ∞ n \to \infty , necessary and sufficient integrability conditions are obtained.

Fourier transforms of distributions of finite order with left-sided bounded support are characterized. Furthermore, a product of these Fourier transforms is defined which corresponds to the convolution of the original distributions.

In a recent Tauberian theorem of Stanojević [3] for the L 1 {L^1} -convergence of Fourier series, the notion of asymptotically even sequences is introduced. These conditions are satisfied if the Fourier coefficients { f ^ ( n ) } \{ \hat f(n)\} are even ( f ^ ( − n ) = f ^ ( n ) ) (\hat f( - n) = \hat f(n)) , a case formally equivalent to cosine Fourier series. This paper applies the Tauberian method of Stanojević [3] separately to cosine and sine Fourier series and shows that the notion of asymptotic evenness can be circumvented in each case.

The class of even, nonnegative, finite measures ρ \rho on the real line such that for any b > 0 b > 0 the Fourier transform of exp ⁡ ( − b t 2 ) d ρ ( t ) \exp ( - b{t^2})d\rho (t) has only real zeros is completely determined. This result is then applied to the Riemann hypothesis.

Given the Fourier coefficients of an even continuous function, we find a necessary and sufficient condition such that their arithmetic means are the Fourier coefficients of an odd continuous function. A similar result is shown for those Lipschitz classes whose elements are automatically equivalent to continuous functions. Introduction. Let I' la"cos nx and ' lb,, sin nx be the Fourier series of fc and fS, respectively. Hardy showed that X= J(a, + * + a")/n]cos nx and . Xs [(b1 + + bn)/n]sin nx [2, p. 51] will also be Fourier series-let THJC and THfJ, respectively, represent these two series. It was shown by M. Kinukawa and S. Igari [5, Theorem 2] that fc E LI? = THfc E L??. In this paper we show that if fc E C, then fc(O) = O X= THfC E C. A. Konyushkov [6, Theorem 15] showed that for 0 < a < 1/p < l,fc E AP X THfc E AP. In this paper we consider the case 0 < I/p < a < 1, and show that iffc Ef AP, then I' Ian = 0= THfc E AP. All the notations, unless otherwise mentioned, are taken from [9]. Every function is supposed to be defined a.e., periodic with period 21r and integrable on [T, ir]. We shall not distinguish between equivalent functions. Given a function f, by f(t) we would mean limn,o an (t; f)-whenever it exists. For convenience we shall write the Lipschitz classes AP and Aa as A(a, p) and A(a), respectively. Definition. For f and E E L1, we define Tsf as the function representing the Fourier series 21'1[Sn(0; f)/n]sin nt. We can easily check the following: (i) TSfC = THJC. (ii) For 0 < oa < 1 < p < oo and ap # 1, Tsfc E A(a, p) X= THfc E A(a, p). (iii) For 0 < Il/p < a < 1, by [4, Theorem 5 (ii)]

For the class of Fourier series with quasi-monotone coefficients, it is proved that ‖ s n − σ n ‖ = o ( 1 ) , n → ∞ \left \| {{s_n} - {\sigma _n}} \right \| = o(1),n \to \infty , if and only if a n lg ⁡ n = o ( 1 ) , n → ∞ {a_n}\lg n = o(1),n \to \infty . This generalizes a theorem for monotone coefficients and provides a new proof for a result due to Telyakovskii and Fomin.

The purpose of this note is to show that while I holds, when appropriately interpreted, for Fourier series on arbitrary compact groups, II does not hold in general if the group is noncommutative. The fact that a generalization of I holds for arbitrary groups follows easily from Billard's original proof and some results of Kahane on random series in Banach spaces [4, Theoreme 2] and [5, Chapter II, Theorem 1]. With the help of another result of Kahane [4, Theoreme 3] and [5, Chapter II, Theorem 5] one can easily show that II holds for Fourier series defined on an arbitrary commutative compact group. Our example in ?3 shows that II can fail at least for some noncommutative groups. One should notice that II is a stronger result than I, in fact an application of Fubini's theorem yields that if II is true, the conclusion also holds for random Fourier series of the type

not have a singular component. The present note contributes further evidence in this direction by showing that in many groups (including the real line and the integers) there are relatively small sets on which the Fourier transform of any absolutely continuous measure can be so modified that it becomes the transform of a singular measure. Let r be the dual of a locally compact abelian group G; L1(G) and M(G) denote the spaces of all Haar-integrable functions on G and of all complex Borel measures on G, respectively, and we identify L'(G) with the absolutely continuous members of M(G). The Fourier transform of ,u E M(G) is defined to be

In Theorem 1 it is shown that a function obtained by repeatedly integrating Z(s) has either a real or an imaginary part which is of constant sign when s is purely imaginary. Using this result, inequalities on the Fourier transforms of Z(it) are developed. The conclusions of Theorems 3 and 4 are generalizations of inequalities given in a previous paper [1, Theorems 1, 2, and 3]. These in turn are somewhat similar to an inequality obtained by Boas and Kac [2].

The Fourier transform of an unbounded spectral distribution is studied: An explicit integral representation is obtained; connections are drawn to the associated generalized scalar operator. It is proved that every generalized pseudo-hermitian operator is the infinitesimal generator of a temperate CO group. Introduction. In [6] the author introduced a theory of unbounded spectral distributions in Banach spaces, and a corresponding theory of the generalized scalar operators which they represent. Properties of these objects were studied, culminating in a spectral mapping theorem [6, Theorem 4]. In this paper we study the Fourier transform of an unbounded spectral distribution, deriving an explicit integral representation, as well as growth estimates at infinity. The case of real support is considered in some detail, leading to the proof that every generalized pseudo-hermitian operator (i.e., generalized scalar with real spectrum) is the infinitesimal generator of a temperate group. This result generalizes the corresponding result for bounded operators proved in [5]. In this paper, all definitions are as in [6]. Integral representation of the Jourier transform. THEOREM I (EXTENSION OF SPECTRAL DISTRIBUTIONS). Let T be a spectral distribution [6, Definition 2] in a Banach space X. Then (a) for each fuinction ffromn the space (R2) of C' functions wt ith bounded lerivatives, and for each x E X, limit,, Tf pnx exists, where {c p} is a sequence of test flnctions as in [6, Definition 2(c)]. (b) If we define Tfx=Jimitn Tfp,,x for all x E X, then Tf is in the space Y(X) of all bounded linear operators on X, and the correspondence f--*Tf is a continuous linear mapping of X(R2) into f(X). (c) 7 T= Tf T, for al/f, g E .6(R2). (d) The operator Tf is independent of the sequence {p7J PROOF. (a) Define a double sequence of functions m-f(%-pm). Then {y form i X(R 2)-bounded subset of ?/ (R2) (cf. [3, p. 91 ] for the Received by the editors October 20, 1971. AAIS 1970 sub/jecr ch ssifications. Primary 47A65, 47A60, 47B40; Secondary 46F99. ? American Mathematical Society 1972

There are some gaps between the conditions for the sequences of coefficients {a,} in these two theorems. In ?2, we shall prove that the conditions for {an } in Theorem B can be replaced by those in Theorem A. On the other hand Reves and Szasz [2 ] have proved the analogues of Cantor-Lebesgue's theorem and Denjoy-Lusin's theorem for the double trigonometric series. In ?3, we shall generalize these theorems: the former is on the line of Mazur-Orlicz' generalization for CantorLebesgue's theorem [1, Theorem 1 and its corollary] and the latter is in the direction of Salem's generalization for Denjoy-Lusin's theorem [3, Theorem X].

1. A theorem of Pal and Bohr, [l], [2], asserts that for every continuous/ of period 27r there is a homeomorphism g of [ — ir, ir] with itself such that the Fourier series of/og converges uniformly. Salem [3] has given a powerful test for the uniform convergence of a Fourier series. On the other hand, there is no criterion which gives necessary and sufficient conditions for the Fourier series of a continuous function to converge everywhere. In this note we show that the method of Salem may be used to determine a necessary and sufficient condition that a continuous function / be such that the Fourier series of / o g should converge everywhere for every homeomorphism g of [ — ir, ir] with itself. It is clear that this condition must be weaker than bounded variation since the continuous functions of bounded ^-variation with 4> = ^, p>l, have uniformly convergent Fourier series, and this class of functions is preserved by composition with homeomorphisms [3], [4].

1. Let G be a locally compact and abelian group, G' the dual group. Through this paper, 2c'y will denote the set of all bounded Radon measure ,uk on G; this set will be considered as a Banach algebra when provided with the customary norm as the dual of the space Co(G) defined below and with the ring product defined as convolution. If AEfl, its Fourier transform is by definition the bounded and continuous function on G given by the formula

Received by the editors September 2, 1958 and, in revised form, November 21, 1958. 1 This research was supported in part by the United States Air Force under Contract AF 18(600)-i 111 monitored by the Air Force Office of Scientific Research of the Air Research and Development Command, and in part by the National Science Foundation. The author expresses his hearty thanks to Dr. M. Noble for his kindness in giving many suggestions.

The radial behavior of its Fourier-Stieltjes transform in R 2 {R^2} is related to the modulus of continuity of a measure; in certain cases the Hausdorff dimension of an exceptional set of lines can be estimated. Converse results use the theory of Diophantine approximation established by Besicovitch and Jarník.

where I I| denotes the length of the interval I, AS denotes the length of the longest interval I in the subdivision S of (a, b) into nonoverlapping intervals, and the integral on the left is a Burkill integral [1]. Guided by this, we make the following definition: DEFINITION 1. A function u defined on subintervals of (0, oo) is said to belong to the class B2 if u is integrable on every finite range and the

(5) F(2) = 0(0+) This expression follows readily from a known theorem on Fourier transforms [1, Theorem 3, p. 13] if +(t) is defined in the interval (-x , 0) by the relation (-t) =c(t). In this paper an inequality on f(x) for monotonic decreasing +(t) and inequalities on g(x) and F(x) for non-negative +(t) are obtained when the bounds on these functions are known for x 2 1. The result on f(x) is similar to an inequality obtained by Boas and Kac [2]. In the more general case when 4(t) 2 0, they use the more restrictive condition, f(x) = 0 for x 2 1, to show that then

Let f ∈ L [ − π , π ] f \in L[ - \pi ,\pi ] be 2 π 2\pi -periodic. Noble [6] posed the following problem: if the fulfillment of some property of a function f on the whole interval [ − π , π ] [ - \pi ,\pi ] implies certain conclusions concerning the Fourier series σ ( f ) \sigma (f) of f, then what lacunae in σ ( f ) \sigma (f) guarantees the same conclusions when the property is fulfilled only locally? Applying the more powerful methods of approach to this kind of problems, originally developed by Paley and Wiener [7], the absolute convergence of a certain lacunary Fourier series is studied when the function f satisfies some hypothesis in terms of either the modulus of continuity or the modulus of smoothness of order l considered only at a fixed point of [ − π , π ] [ - \pi ,\pi ] . The results obtained here are a kind of generalization of the results due to Patadia [8].

Fourier transforms of distribution functions are an important tool in the theory of probability because they have certain properties. These properties are expressed in the uniqueness theorem, the convolution theorem, and the continuity theorem (cf. [1, pp. 93, 96, 188]). This note deals with the question of whether there are any other integral transforms of distribution functions which have these properties. We consider an integral transform