Hardy's Theorem Research Articles

On a Theorem of Hardy and Littlewood

In this paper, we give an extension of a classical theorem of Hardy and Littlewood on power series. Let $\varphi$ be a strictly positive function defined on some interval $\left ( {\delta ,1} \right )$, satisfying a certain condition of limit. We prove that if $f\left ( x \right )$ is the sum of a convergent power series for $0 < x < 1$ with nonnegative coefficients ${a_n}$ and $f\left ( x \right ) \sim \varphi \left ( x \right )\;\left ( {x \to 1} \right )$, then ${S_n} \sim \alpha \cdot \varphi \left ( {x_0^{1/n}} \right )\left ( {n \to \infty } \right )$, where ${S_n} = {a_0} + {a_1} + \cdots + {a_{n,\;}}{x_0} \in \left ( {0,1} \right )$ and $\alpha$ depends only upon $\varphi$.

Bounded Sequence-to-Function Hausdorff Transformations

Let \[ \left ( {Ta} \right )\left ( y \right ) = \sum \limits _{n = 0}^\infty {{{\left ( { - y} \right )}^n}} \frac {{{g^{\left ( n \right )}}\left ( y \right )}}{{n!}}{a_{n,\quad }}\quad y \geq 0\] be the sequence-to-function Hausdorff transformation generated by the completely monotone function $g$ or, what is equivalent, the Laplace transform of a finite positive measure $\sigma$ on $[0,\infty )$. It is shown that for $1 \leq p \leq \infty$, $T$ is a bounded transformation of ${l^p}$ with weight $\Gamma \left ( {n + s + 1} \right ) / n!$ into ${L^p}[0,\infty )$ with weight ${y^s},s > - 1$, whose norm $\left \| T \right \| = \int _0^\infty {{t^{ - \left ( {1 + s} \right ) / p}}} d\sigma \left ( t \right ) = C\left ( {p,s} \right )$ if and only if $C\left ( {p,s} \right ) < \infty$, and that for $1 < p < \infty ,{\left \| {Ta} \right \|_{p,s}} < C\left ( {p,s} \right ){\left \| a \right \|_{p,s}}$ unless ${a_n}$ is a null sequence. Furthermore, if $1 < p < r < \infty , \;0 < \lambda < 1$ and $\sigma$ is absolutely continuous with derivatives $\psi$ such that the function ${\psi _r}\left ( t \right ) = {t^{ - 1 / r}}\psi \left ( t \right )$ belongs to ${L^{1 / \lambda }}[0,\infty )$, then the transformation $\left ( {{T_\lambda }a} \right )\left ( y \right ) = {y^{1 - \lambda }}\left ( {Ta} \right )\left ( y \right )$ is bounded from ${l^p}$ to ${L^r}[0,\infty )$ and has norm $\left \| {{T_\lambda }} \right \| \leq {\left \| {{\psi _r}} \right \|_{1 / \lambda }}$. The transformation $T$ includes in particular the Borel transform and that of generalized Abel means. These results constitute an improved analogue of a theorem of Hardy concerning the discrete Hausdorff transformation on ${l^p}$ which corresponds to a totally monotone sequence, and lead to improved forms of some inequalities of Hardy and Littlewood for power series and moment sequences.

On an F-Algebra of Holomorphic Functions

The complex maximal theorem of Hardy and Little-wood states:(Mp). For 0 &lt; p &lt; ∞, there exists a positive constant Cp such that if f is holomorphic in the unit disc U of the complex plane thenwhereThe corresponding statement to the limiting case p = 0 can be stated as follows:(M0) There exists a positive constant C0 such that if f is holomorphic in Uwhere log+t = max(log t, 0).The statement (M0) is false as the following example shows.

On a theorem of Hardy and Littlewood

In this paper, we give an extension of a classical theorem of Hardy and Littlewood on power series. Let φ \varphi be a strictly positive function defined on some interval ( δ , 1 ) \left ( {\delta ,1} \right ) , satisfying a certain condition of limit. We prove that if f ( x ) f\left ( x \right ) is the sum of a convergent power series for 0 &gt; x &gt; 1 0 &gt; x &gt; 1 with nonnegative coefficients a n {a_n} and f ( x ) ∼ φ ( x ) ( x → 1 ) f\left ( x \right ) \sim \varphi \left ( x \right )\;\left ( {x \to 1} \right ) , then S n ∼ α ⋅ φ ( x 0 1 / n ) ( n → ∞ ) {S_n} \sim \alpha \cdot \varphi \left ( {x_0^{1/n}} \right )\left ( {n \to \infty } \right ) , where S n = a 0 + a 1 + ⋯ + a n , x 0 ∈ ( 0 , 1 ) {S_n} = {a_0} + {a_1} + \cdots + {a_{n,\;}}{x_0} \in \left ( {0,1} \right ) and α \alpha depends only upon φ \varphi .

Bounded sequence-to-function Hausdorff transformations

Let \[ ( T a ) ( y ) = ∑ n = 0 ∞ ( − y ) n g ( n ) ( y ) n ! a n , y ≥ 0 \left ( {Ta} \right )\left ( y \right ) = \sum \limits _{n = 0}^\infty {{{\left ( { - y} \right )}^n}} \frac {{{g^{\left ( n \right )}}\left ( y \right )}}{{n!}}{a_{n,\quad }}\quad y \geq 0 \] be the sequence-to-function Hausdorff transformation generated by the completely monotone function g g or, what is equivalent, the Laplace transform of a finite positive measure σ \sigma on [ 0 , ∞ ) [0,\infty ) . It is shown that for 1 ≤ p ≤ ∞ 1 \leq p \leq \infty , T T is a bounded transformation of l p {l^p} with weight Γ ( n + s + 1 ) / n ! \Gamma \left ( {n + s + 1} \right ) / n! into L p [ 0 , ∞ ) {L^p}[0,\infty ) with weight y s , s &gt; − 1 {y^s},s &gt; - 1 , whose norm ‖ T ‖ = ∫ 0 ∞ t − ( 1 + s ) / p d σ ( t ) = C ( p , s ) \left \| T \right \| = \int _0^\infty {{t^{ - \left ( {1 + s} \right ) / p}}} d\sigma \left ( t \right ) = C\left ( {p,s} \right ) if and only if C ( p , s ) &gt; ∞ C\left ( {p,s} \right ) &gt; \infty , and that for 1 &gt; p &gt; ∞ , ‖ T a ‖ p , s &gt; C ( p , s ) ‖ a ‖ p , s 1 &gt; p &gt; \infty ,{\left \| {Ta} \right \|_{p,s}} &gt; C\left ( {p,s} \right ){\left \| a \right \|_{p,s}} unless a n {a_n} is a null sequence. Furthermore, if 1 &gt; p &gt; r &gt; ∞ , 0 &gt; λ &gt; 1 1 &gt; p &gt; r &gt; \infty ,\,\;0 &gt; \lambda &gt; 1 and σ \sigma is absolutely continuous with derivatives ψ \psi such that the function ψ r ( t ) = t − 1 / r ψ ( t ) {\psi _r}\left ( t \right ) = {t^{ - 1 / r}}\psi \left ( t \right ) belongs to L 1 / λ [ 0 , ∞ ) {L^{1 / \lambda }}[0,\infty ) , then the transformation ( T λ a ) ( y ) = y 1 − λ ( T a ) ( y ) \left ( {{T_\lambda }a} \right )\left ( y \right ) = {y^{1 - \lambda }}\left ( {Ta} \right )\left ( y \right ) is bounded from l p {l^p} to L r [ 0 , ∞ ) {L^r}[0,\infty ) and has norm ‖ T λ ‖ ≤ ‖ ψ r ‖ 1 / λ \left \| {{T_\lambda }} \right \| \leq {\left \| {{\psi _r}} \right \|_{1 / \lambda }} . The transformation T T includes in particular the Borel transform and that of generalized Abel means. These results constitute an improved analogue of a theorem of Hardy concerning the discrete Hausdorff transformation on l p {l^p} which corresponds to a totally monotone sequence, and lead to improved forms of some inequalities of Hardy and Littlewood for power series and moment sequences.

Pointwise Convergence of Trigonometric Series

We establish two results in the pointwise convergence problem of a trigonometric series for some nonnegative integer m. These results not only generalize Hardy's theorem, the Jordan test theorem and Fatou's theorem, but also complement the results on pointwise convergence of those Fourier series associated with known L1-convergence classes. A similar result is also established for the case that , where {ln} satisfies certain conditions.

On the Sieve of Eratosthenes

Let v(n) denote the number of distinct prime factors of a natural number n. A classical theorem of Hardy and Ramanujan states that the normal order of v(n) is log log n. That is, given any , the number of natural numbers not exceeding x which fail to satisfy the inequality1is o(x) as x → ∞. A very simple proof of this was subsequently given by Turán. He showed that2

Singular values of fractional integral operators: A unification of theorems of Hille, Tamarkin, and Chang

We obtain upper bounds on the singular values of fractional integral operators of the form L α · = ∝ 0 x (x − y) x − 1 Γ(α) · dy under the constraint α > 0. These bounds are employed to extend various results obtained over the last half century on the rate of decrease of eigenvalues and singular values of much more general integral operators. Apart from one relatively difficult theorem of Hardy and Littlewood ( Math. Z., 27 (1928), 565–606) the devices used are quite simple. They involve no complex variable arguments.

On a Theorem of Hardy and Littlewood on the Polydisc

We prove the polydisc version of the theorem of Hardy and Littlewood on the fractional integral: If $0 < \alpha < 1/p$ and if $f \in {H^p}$, then ${I^\alpha }f \in {H^q}$ with $q = p/(1 - \alpha p)$ where ${I^\alpha }f$ is the fractional integral of $f$ of order $\alpha$.

On a theorem of Hardy and Littlewood on the polydisc

We prove the polydisc version of the theorem of Hardy and Littlewood on the fractional integral: If 0 &gt; α &gt; 1 / p 0 &gt; \alpha &gt; 1/p and if f ∈ H p f \in {H^p} , then I α f ∈ H q {I^\alpha }f \in {H^q} with q = p / ( 1 − α p ) q = p/(1 - \alpha p) where I α f {I^\alpha }f is the fractional integral of f f of order α \alpha .

Application of a Tauberian theorem to finite model theory

An extension of a Tauberian theorem of Hardy and Littlewood is proved. It is used to show that, for classes of finite models satisfying certain combinatorial and growth properties, Cesàro probabilities (limits of average probabilities over second order sentences) exist. Examples of such classes include the class of unary functions and the class of partial unary functions. It is conjectured that the result holds for the usual notion of asymptotic probability as well as Cesàro probability. Evidence in support of the conjecture is presented.

A note on the attainability of states by equalizing processes

In this paper, we consider “regions of attainability” in certain state spaces as a function of the initial state under a well-defined and physically relevant class of processes. These processes are the continuous and “chaos-enhancing” processes (in the sense of Ruch and Uhlmann). It turns out that these sets have complicated geometrical structures: They are polyhedra which are generally non-convex. The proof — a rather geometrical one — is given. A theorem of Hardy, Littlewood, and Polya and properties of bistochastic matrices are used crucially.

On the asymptotics of distributions with support in cone

The asymptotic behaviors of tempered distributions with support in a convex, closed cone are classified by means of group theory. The notion of regularly varying distribution is introduced. An Abelian–Tauberian theorem for regularly varying tempered distributions, which generalizes the one-dimensional Abelian–Tauberian theorem of Hardy–Littlewood–Karamata and the many-dimensional extension due to Vladimirov, is proved. Applications to n-point functions are also presented.

A theorem of hardy and titchmarsh for generalized functions

A theorem of Hardy and Titchmarsh on unsymmetrical Fourier kernels is extended to a classof generalized functions interpreting convergence in the weak distributional sense. An inversion theorem for the distributional Hankel transformation is derived as a special case of the extended theorem

An Improvement on a Theorem of Hardy and Littlewood

An Improvement on a Theorem of Hardy and Littlewood

Mean growth and smoothness of analytic functions

Let G α {G_\alpha } denote the class of functions f ( z ) f(z) analytic in the unit disk such that \[ ∫ 0 1 ( 1 − r ) − α M ∞ ( f ′ , r ) d r &gt; ∞ , \int _0^1 {{{(1 - r)}^{ - \alpha }}{M_\infty }(f’,r)} dr &gt; \infty , \] for some α ( 0 &gt; α &gt; 1 ) \alpha (0 &gt; \alpha &gt; 1) . A characterization of G α {G_\alpha } is given in terms of moduli of continuity and an application is given to Riesz factorization of functions in G α {G_\alpha } .

On some extensions of a theorem of Hardy and Littlewood

On some extensions of a theorem of Hardy and Littlewood

Factorization of Lipschitz functions and absolute convergence of Vilenkin-Fourier series

LetG be a Vilenkin group (i.e., an infinite, compact, metrizable, zero-dimensional Abelian group). Our main result is a factorization theorem for functions in the Lipschitz spaces\(\mathfrak{L}\mathfrak{i}\mathfrak{p}\) (α,p; G). As colloraries of this theorem, we obtain (i) an extension of a factorization theorem ofY. Uno; (ii) a convolution formula which says that\(\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha , r; G) = \mathfrak{L}\mathfrak{i}\mathfrak{p} (\beta , l; G)*\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha - \beta , r; G)\) for 0<β<α<∞ and 1≤r≤∞; and (iii) an analogue, valid for allG, of a classical theorem ofHardy andLittlewood. We also present several results on absolute convergence of Fourier series defined onG, extending a theorem ofC. W. Onneweer and four results ofN. Ja. Vilenkin andA. I. Rubinshtein. The fourVilenkin-Rubinshtein results are analogues of classical theorems due, respectively, toO. Szasz, S. B. Bernshtein, A. Zygmund, andG. G. Lorentz.

Ordering of Distributions and Rearrangement of Functions

Some characterizations of semiorders defined on the set of all probability measures on $R^n$ by the set of Schur-convex functions and by some subsets of all convex functions are proved. A connection of these results to the theorem of Hardy, Littlewood and Polya on the rearrangement of functions is discussed. Furthermore, by means of the results on the ordering of probability measures a generalization of a theorem on doubly stochastic linear operators due to Ryff is proved.

A bibliographical note on a theorem of Hardy, Littlewood, and Polya

A well-known theorem of Hardy Littlewood and Polya together with its more recent generalizations is stated here. Some of these results were applied recently in the economic literature. The economic interpretation of the mathematical theorems is mentioned and references are included.