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We sharpen the bound n 2 k on the maximum modulus of the k th radial derivative of the Zernike circle polynomials (disk polynomials) of degree n to n 2 ( n 2 − 1 2 ) ⋅ . . . ⋅ ( n 2 − ( k − 1 ) 2 ) ∕ 2 k ( 1 ∕ 2 ) k . This bound is obtained from a result of Koornwinder on the non-negativity of connection coefficients of the radial parts of the circle polynomials when expanded into a series of Chebyshev polynomials of the first kind. The new bound is shown to be attained within a factor O ( 1 ∕ n ) for Zernike circle polynomials of degree n and azimuthal order m when m = O ( n ) by using an explicit expression for the connection coefficients in terms of squares of Jacobi polynomials evaluated at 0.

In this paper, we will give characterizations for weighted Bergman spaces with admissible weights. The first criterion is established by the invariant gradient on the unit ball of $${\mathbb {C}}^N$$. The other criterion is given by the radial derivative of holomorphic functions in the unit ball.

Let H ( B ) denote the space of holomorphic functions on the unit ball B of C n , n ∈ N , ψ , ϕ ∈ H ( B ) , ϕ ( B ) ⊂ B . Let C ϕ , M ψ and R denote the composition, multiplication and radial deriv...

We prove that the norm of a radial derivative Rf of an entire function f with respect to Gaussian type measure e−φ(z)dV is equivalent to the norm of f supplemented with |z|φ′(z). This means that |z...

In this paper, we provide uncertainty type principles for radial derivatives on an open bounded set \(\varOmega \subset \mathbb {R}^{n}\). We obtain several inequalities of uncertainty type principles of the form $$ \int _\varOmega |f(x)|^p \, \phi (|x|)\, dx \le p \, \left( \int _\varOmega \frac{\left| \mathscr {R}_{|x|}f(x)\right| ^p \, |\tilde{\phi }(|x|)|^p}{|x|^{n-p}}\, dx\right) ^\frac{1}{p} \left( \int _\varOmega \frac{|f(x)|^p}{|x|^n}\,dx\right) ^\frac{p-1}{p}, $$ for \(1

Let $$\mathbb{B}$$ be the unit ball of a complex Banach space X. In this paper, we generalize the Bloch-type spaces and the little Bloch-type spaces to the open unit ball $$\mathbb{B}$$ by using the radial derivative. Next, we define an extended Cesaro operator Tφ with holomorphic symbol φ and characterize those φ for which Tφ is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those φ for which Tφ is compact between the Bloch-type spaces and the little Bloch-type spaces under some additional assumption on the symbol φ. When $$\mathbb{B}$$ is the open unit ball of a finite dimensional complex Banach space X, this additional assumption is automatically satisfied.

While it has been relatively easy to determine solar-cycle related changes in solar dynamics, determining changes in structure in the deeper layers of the Sun has proved to be difficult. By using helioseismic data obtained over two solar cycles, and sacrificing resolution in favour of lower uncertainties, we show that there are significant changes in the solar convection zone, and perhaps even below it. Using MDI data, we find a relative squared sound-speed difference of $(2.56\pm 0.71)\times 10^{-5}$ at the convection-zone base between the maximum of solar Cycle~23 and the minimum between Cycles~23 and 24. The squared sound-speed difference for the maximum of Cycle~24 obtained with HMI data is $(1.95\pm 0.69)\times 10^{-5}$. GONG data support these results. We also find that the sound speed in the solar convection zone decreases compared to the sound speed below it as the Sun becomes more active. We find evidence of changes in the radial derivative of the sound-speed difference between the solar minimum and other epochs at the base of the convection zone implying possible small changes in the position of the convection-zone base, however, the results are too noisy to make any definitive estimates of the change., Comment: Accepted for publication in ApJ

The aim of this paper is to find weights W in the unit ball of C n for which characterization of the area integrals of Bergman spaces A p ( W ) holds. The area functions, related to those used to describe Hardy spaces, involve the radial derivative, the complex gradient and the invariant gradient. We extend to certain Bekolle weights the characterization by Z. Chen and W. Ouyang of the Bergman spaces with the classical weights W ( z ) = ( 1 − | z | ) α using area functions.

Motivated and inspired by the improved Hardy inequalities studied in their well-known works by Brezis and Vazquez (Rev Mat Univ Complut Madrid 10:443–469, 1997) and Brezis and Marcus (Ann Scuola Norm Sup Pisa Cl Sci 25(1–2):217–237, 1997), we establish in this paper several identities that imply many sharpened forms of the Hardy type inequalities on upper half spaces $$\left\{ x_{N}>0\right\} $$. We set up these results for the distance to the origin, the distance to the boundary of any strip $$ \mathbb {R} ^{N-1}\times \left( 0,R\right) $$ and the distance to the hyperplane $$\left\{ x_{N}=0\right\} $$, using both the usual full gradient and radial derivative (in the case of distance to the origin) or only the partial derivative $$\frac{\partial u}{\partial x_{N}}$$ (in the case of distance to the boundary of the strip or hyperplane). One of the applications of our main results is that when $$\Omega $$ is the strip $$\mathbb {R}^{N-1}\times \left( 0,2R\right) $$, the bound $$\lambda \left( \Omega \right) $$ given by Brezis and Marcus in Brezis and Marcus (1997) can be improved to $$\frac{z_{0}^{2}}{R^{2}}$$, where $$z_{0} =2.4048 \ldots $$ is the first zero of the Bessel function $$J_{0}\left( z\right) $$. Our approach makes use of the notion of Bessel pairs introduced by Ghoussoub and Moradifam (Math Ann 349(1):1–57, 2011) and (Functional inequalities: new perspectives and new applications. Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2013) and the method of factorizations of differential operators. In particular, our identities and inequalities offer sharpened and more precise estimates of the second remainder term in the existing Hardy type inequalities on upper half spaces in the literature, including the Hardy-Sobolev-Maz’ya type inequalities.

Basic physics of drift-wave turbulence and zonal flows has long been studied within the framework of wave-kinetic theory. Recently, this framework has been re-examined from first principles, which has led to more accurate yet still tractable "improved" wave-kinetic equations. In particular, these equations reveal an important effect of the zonal-flow "curvature" (the second radial derivative of the flow velocity) on dynamics and stability of drift waves and zonal flows. We overview these recent findings and present a consolidated high-level picture of (mostly quasilinear) zonal-flow physics within reduced models of drift-wave turbulence., Paper BI2 6, Bull. Am. Phys. Soc. 65 (2020)

Possibilities for defining the radial derivative of the delta distribution $\delta(\underline{x})$ in the setting of spherical coordinates are explored. This leads to the introduction of a new class of continuous linear functionals similar to but different from the standard distributions. The radial derivative of $\delta(\underline{x})$ then belongs to that new class of so-called signumdistributions. It is shown that these signumdistributions obey easy-to-handle calculus rules which are in accordance with those for the standard distributions in $\mathbb{R}^m$., Comment: 18 pages

For \(s\in \mathbb {R}\) the weighted Besov space on the unit ball \(\mathbb {B}_d\) of \(\mathbb {C}^d\) is defined by $$\displaystyle B^s_\omega =\{f\in \operatorname {Hol}(\mathbb {B}_d): \int _{\mathbb {B}_d}|R^sf|{ }^2 \omega dV

When expressing a distribution in Euclidean space in spherical co-ordinates, derivation with respect to the radial and angular co-ordinates is far from trivial. Exploring the possibilities of defining a radial derivative of the delta distribution \(\delta ( \underline {x})\) (the angular derivatives of \(\delta ( \underline {x})\) being zero since the delta distribution is itself radial) led to the introduction of a new kind of distributions, the so-called signumdistributions, as continuous linear functionals on a space of test functions showing a singularity at the origin. In this paper we search for a definition of the radial and angular derivatives of a general standard distribution and again, as expected, we are inevitably led to consider signumdistributions. Although these signumdistributions provide an adequate framework for the actions on distributions aimed at, it turns out that the derivation with respect to the radial distance of a general (signum)distribution is still not yet unambiguous.

In this paper, we obtain some characterizations of the boundedness and compactness of the products of the radial derivative and multiplication operator RMu between mixed norm spaces H(p, q, φ) and Zygmund-type spaces on the unit ball.

In this note we obtain a remainder estimate for improved critical Hardy inequalities on a 2-dimensional quasi-ball on homogeneous Lie groups. These results are new even in the Abelian case of ℝ2 in terms of choosing any choice of homogeneous quasi-norm as well as replacing the full gradient by the radial derivative.

In this paper, we obtain complete characterizations of the boundedness and compactness of the products of the multiplication and the radial derivative operator MuR from mixed norm spaces \({H(p, q, \phi)}\) to Zygmund-type spaces on the unit ball.

Let B n be the unit ball in C n . If f is a bounded holomorphic function, we say that f is inner provided that lim r → 1 − | f ( r ζ ) | = 1 σ -a.e. , ζ ∈ S n where S n is the unit sphere and σ is normalized surface measure on S n . If β > − 1 and p > 0 then A β p denotes the weighted Bergman space of all holomorphic functions weighted by ( 1 − | z | 2 ) β . For 0 q 1 , set B q : = A n q − n − 1 1 and if p > 0 let H p denote the usual Hardy space of holomorphic functions on the ball. In this paper, we consider derivatives of inner functions in several spaces of holomorphic functions. If f is an inner function, membership of the radial derivative, R f = ∑ j = 1 n z j ∂ f ∂ z j , will be considered in the B p spaces for p > n n + 1 and will be related to membership in weighted Dirichlet spaces, weighted Bergman spaces A α 2 for 0 α 1 , and to the A p spaces for 1 p 2 . Moreover, it will be shown that if f is an inner function, n > 1 , and either R f ∈ B 2 n 2 n + 1 , R f ∈ A 3 / 2 , or R f ∈ H 1 / 2 then f must be constant.

We prove some Hardy-Sobolev inequalities for radial derivative and obtain the corresponding sharp constant. Mathematics subject classification (2010): Primary 26D10, 46E35.

The 4D gravity or time lapse gravity has been used many reseracher to identify fluid injection in oil reservoir. The objective of this study is to find the better way in interpreting 4D gravity anomaly due to fluid injection around the reservoir. Radial Derivatives are derivative values of gravity anomalies against horizontal distances in the radial direction. Radial inversion is a two-dimensional inversion of lines with radial directions resulting in a 3-dimension model. Time lapse microgravity research have been performed in “X Oil Field” with amount of 604 data point covering area of 4000 m x 5000 m. This Radial derivative and Radial inversion have been aplied at an injection well of the X oil field. The yield show that 4D gravity anomaly value due to injection is 0.02 mGal to 0.36 mGal. Radial derivative value in the area is 0 micro Gal/cm to 0,012 mGal/meter. Radial inversion shows radius of fluid front movement is 304 meters to 1120 meters. Radial derivative and Radial inversion have been proven fairly good to identify injected fluid movement in the reservoir.

Let H ( B ) denote the space of all holomorphic functions on the unit ball B ⊂ C n . The boundedness and compactness of the following integral-type operators T g ( f ) ( z ) = ∫ 0 1 f ( tz ) R g ( tz ) dt t and L g ( f ) ( z ) = ∫ 0 1 R f ( tz ) g ( tz ) dt t , z ∈ B , where g ∈ H ( B ) and R h ( z ) is the radial derivative of h, between α-Bloch spaces and Besov spaces on the unit ball are characterized. The results regarding the case when these operators map α-Bloch spaces to Besov spaces are nontrivial generalizations of recent one-dimensional results by Li and the present author.

Let B denote the unit ball in C^n and H(B) the space of all holomorphic functions on B. We study the boundedness and compactness of the following integral-type operatorsI"@f^gf(z)[email protected]!"0^1Rf(@f(tz))g(tz)dtt,[email protected]?B,where [email protected]?H(B),g(0)=0,@f is a holomorphic self-map of B and Rf is the radial derivative of f, between weighted-type spaces on the unit ball.