Hardy's Theorem Research Articles

Regularly Oscillating Mappings Between Metric Spaces and a Theorem of Hardy and Littlewood

Regularly Oscillating Mappings Between Metric Spaces and a Theorem of Hardy and Littlewood

An elementary proof of a theorem of Hardy and Ramanujan

An elementary proof of a theorem of Hardy and Ramanujan

An analog of Hardy’s theorem for the second Hankel-Clifford transformation

In this paper, we generalize theorem of Hardy for the second Hankel-Clifford transform .

A note on parabolic frequency and a theorem of Hardy–Pólya–Szegö

In this note, we are concerned with the monotonicity of parabolic frequency for weighted [Formula: see text]-Laplacian heat equation on Riemannian manifolds and a theorem of Hardy–Pólya–Szegö on Kähler manifolds under the Kähler-Ricci flow.

Multiplicative property of series used in the Nevanlinna-Pick problem

In the paper we obtained substantially new sufficient condition for negativity of coefficients of power series inverse to series with positive ones. It has been proved that element-wise product of power series retains this property. In particular, it gives rise to generalization of the classical Hardy theorem about power series. These results are generalized for cases of series with multiple variables. Such results are useful in Nevanlinna – Pick theory. For example, if function k(x, y) can be represented as power series Pn≥0 an(x¯y)n, an > 0, and reciprocal function 1/k(x, y) can be represented as power series Pn≥0 bn(x¯y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna – Pick property. The reproducing kernel 1/(1 − x¯y) of the classical Hardy space H2(D) is a prime example for our theorems.

Power series of several variables with condition of logarithmical convexity

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence {as} = as1,s2,...,sn, ||s|| ≥ K satisfies condition of logarithmically convexity and the first coefficient a0 is sufficiently large then reciprocal power series has only negative coefficients {bs} = bs1,s2,...,sn, except b0,0,...,0 for any K. The classical Hardy theorem corresponds to the case K = 0, n = 1. Such results are useful in Nevanlinna - Pick theory. For example, if function k(x, y) can be represented as power series Σn≥0 an(x-y)n, an > 0, and reciprocal function 1/k(x,y) can be represented as power series Σn≥0 bn(x-y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna-Pick property. The reproducing kernel 1/1-x-y of the classical Hardy space H2(D) is a prime example for our theorems.

Research on Quantum Non-local Correlation Testing of Orbital Angular Momentum Mixed State Based on Hardy Theorem

Research on Quantum Non-local Correlation Testing of Orbital Angular Momentum Mixed State Based on Hardy Theorem

Power Series of Several Variables with the Condition of the Logarithmical Convexity of the Coefficients

In the paper, the generalization of the Hardy theorem for power series of several variables inverse to the power series with positive coefficients is considered. It is proved that if the sequence of coefficients {as} = $${{a}_{{{{s}_{1}},{{s}_{2}}, \ldots ,{{s}_{n}}}}}$$ of a power series satisfies the condition similar to the condition of the logarithmical convexity of the coefficients and the first coefficient a0 is sufficiently large starting from a certain place ||s|| ≥ K, then all the coefficients of the inverse power series are negative, except the first coefficient. The classical Hardy theorem corresponds to the case K = 0 and n = 1. These results are applicable in the Nevanlinna–Pick theory.

Harmonic analysis associated to the canonical Fourier Bessel transform

ABSTRACT The aim of this paper is to develop a new harmonic analysis related to a Bessel type operator on the real line: We define the canonical Fourier Bessel transform and study some of its important properties. We prove a Riemann–Lebesgue lemma, inversion formula and operational formulas for this transformation. We derive Plancherel theorem and Babenko inequality for In the present paper, several uncertainty inequalities and theorems for the canonical Fourier Bessel transform are given, including the Heisenberg inequality, Hardy theorem, Nash-type inequality, Carlson-type inequality, global uncertainty principle, local uncertainty principle, logarithmic uncertainty principle in terms of entropy and Miyachi uncertainty principle.

Hardy's theorem and rotation for Dunkl transform

We extend Hardy's theorem for rotation due to Lakey and Hogan for Dunkl transform associated with the reflection group in by using Dunkl-Hermite semigroup. In fact, we characterize by considering the decay of f and it's Dunkl transform along some lines in the complex plane, and we give a d-dimensional analogue for the same corresponding to the reflection group Moreover, we prove a version of Hardy's theorem, assuming the decay of Dunkl-Hermite coefficients.

Power series of one variable with condition of logarithmical convexity

We obtain a new version of the Hardy theorem about power series reciprocal to the power series with positive coefficients in this work. We prove that, if the sequence {an}, n ≥ K is logarithmically convex, then the reciprocal power series has only negative coefficients bn, n > 0 for any K if the first coefficient a0 is sufficiently large. The classical Hardy theorem corresponds to the case K = 0. These results are useful in the Nevanlinna–Pick theory. For instance, if the function k(x, y) can be represented as the power series $$\sum\nolimits_{n \geqslant 0} {{{a}_{n}}{{{(x\bar {y})}}^{n}}} $$, an > 0, and the reciprocal function $$\frac{1}{{k(x,y)}}$$ can be represented as the power series $$\sum\nolimits_{n \geqslant 0} {{{b}_{n}}{{{(x\bar {y})}}^{n}}} $$ such that bn 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc $$\mathbb{D}$$ with the Nevanlinna–Pick property. The reproducing kernel $$\frac{1}{{1 - x\bar {y}}}$$ of the classical Hardy space H2($$\mathbb{D}$$) is a prime example for our theorems. We also obtained new estimates on the growth of analytic functions reciprocal to analytic functions with positive Taylor coefficients.

Power Series of One Variable with Condition of Logarithmical Convexity

We obtain a new version of the Hardy theorem about power series reciprocal to the power series with positive coefficients in this work. We prove that, if the sequence {an}, n ≥ K is logarithmically convex, then the reciprocal power series has only negative coefficients bn, n > 0 for any K if the first coefficient a0 is sufficiently large. The classical Hardy theorem corresponds to the case K = 0. These results are useful in the Nevanlinna–Pick theory. For instance, if the function k(x, y) can be represented as the power series $$\sum\nolimits_{n \geqslant 0} {{{a}_{n}}{{{(x\bar {y})}}^{n}}} $$, an > 0, and the reciprocal function $$\frac{1}{{k(x,y)}}$$ can be represented as the power series $$\sum\nolimits_{n \geqslant 0} {{{b}_{n}}{{{(x\bar {y})}}^{n}}} $$ such that bn 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc $$\mathbb{D}$$ with the Nevanlinna–Pick property. The reproducing kernel $$\frac{1}{{1 - x\bar {y}}}$$ of the classical Hardy space H2($$\mathbb{D}$$) is a prime example for our theorems. We also obtained new estimates on the growth of analytic functions reciprocal to analytic functions with positive Taylor coefficients.

Zeros of combinations of the Riemann Ξ-function and the confluent hypergeometric function on bounded vertical shifts

In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation involving the Riemann Ξ-function and the confluent hypergeometric function linked to the general theta transformation. Using this result, we show that a series consisting of bounded vertical shifts of a product of the Riemann Ξ-function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line, thereby generalizing a previous result due to the first and the last authors along with Roy and Robles. The latter itself is a generalization of Hardy's theorem.

The Uncertainty Principle for the Two-Sided Quaternion Fourier Transform

In this paper, we provide the Heisenberg's inequality and the Hardy's theorem for the two-sided quaternion Fourier transform.

A note on Hardy's theorem

Hardy's theorem for the Riemann zeta-function ζ(s) says that it admits infinitely many complex zeros on the line (s) = 1 2. In this note, we give a simple proof of this statement which, to the best of our knowledge, is new.

Multidimensional Tauberian theorems for generalized functions

This is a brief survey of multidimensional Tauberian theorems for generalized functions. Included are theorems of Hardy–Littlewood type, Tauberian and Abelian comparison theorems of Keldysh type, theorems of Wiener type, and Tauberian theorems for generalized functions with values in Banach spaces. Bibliography: 58 titles.

Beurling–Hörmander's theorem related to Bessel–Struve transform

ABSTRACTIn this paper, we consider the Bessel–Struve transform and we establish an analogous of Beurling-Hörmander's theorem for each . More precisely, we determine the form of nonzero functions satisfying weaken condition of Beurling–Hörmander's theorem which differ from α half integer or non-half integer. As applications, we obtain a Cowling–Price-type theorem and analogous of Hardy's theorem.

Some uncertainty principles for diamond Lie groups

AbstractSo far, the uncertainty principles for solvable non-exponential Lie groups have been treated only in few cases. The first author and Kaniuth produced an analogue of Hardy's theorem for a diamond Lie group, which is a semi-direct product of ℝ

Modified Bessel -integrals and -derivatives and their properties

We study the modified Bessel -integral, whose properties are similar to those of the Bessel potential, and the modified Bessel - derivative. These operators are inverse to each other. We prove analogues of the embedding theorems of Hardy, Littlewood, Stein, Zygmund and Lizorkin concerning the images of under the action of Bessel potentials. We give applications of the Bessel integral and derivative to the integrability of the -adic Fourier transform and to approximation theory (an embedding theorem of Ul'yanov type).

On summability of Fourier coefficients of functions from Lebesgue space

In this paper we state new theorems of Hardy–Littlewood type for functions with general monotone Fourier coefficients. Sharpness of stated results is discusses.