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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.

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.

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.

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)

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

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.

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.

Let H(B) denote the space of all holomorphic functions on the unit ball [email protected]?C^n. This paper investigates the following integral-type operator with symbol [email protected]?H(B)T"g(f)(z)[email protected]!"0^1f(tz)Rg(tz)dtt,[email protected]?H(B),[email protected]?B,where Rg(z)[email protected]?"j"="1^nz"[email protected][email protected]?z"j(z) is the radial derivative of g. The boundedness and compactness of the operator T"g from Bloch-type spaces to Zygmund-type spaces are studied.

Let and τ n denote the invariant gradient and invariant measure on the unit ball B n of ℂ n , respectively. We show that a holomorphic function ϕ : B n → B 1 is in the hyperbolic Nevanlinna class if and only if We obtain similar descriptions in terms of the holomorphic gradient and radial derivative.

Least-squares collocation may be used for the estimation of spherical harmonic coefficients and their error and error correlations from GOCE data. Due to the extremely large number of data, this requires the use of the so-called method of Fast Spherical Collocation (FSC) which requires that data is gridded equidistantly on each parallel and have the same uncorrelated noise on the parallel. A consequence of this is that error-covariances will be zero except between coefficients of the same signed order (i.e., the same order and the same coefficient type C–C or S–S). If the data distribution and the characteristics of the data noise are symmetric with respect to the equator, then, within a given order and coefficient type, the error-covariances amongst coefficients whose degrees are of different parity also vanish. The deviation from this “ideal” pattern has been studied using data-sets of second order radial derivatives of the anomalous potential. A total number of points below 17,000 were used having an equi-angular or an equal area distribution or being associated with points on a realistic GOCE orbit but close to the nodes of a grid. Also the data were considered having a correlated or an uncorrelated noise and three different signal covariance functions. Grids including data or not including data in the polar areas were used. Using the functionals associated with the data, error estimates of coefficients and error-correlations between coefficients were calculated up to a maximal degree and order equal to 90. As expected, for the data-distributions with no data in the polar areas the error-estimates were found to be larger than when the polar areas contained data. In all cases it was found that only the error-correlations between coefficients of the same order were significantly different from zero (up to 88%). Error-correlations were significantly larger when data had been regarded as having non-zero error-correlations. Also the error-correlations were largest when the covariance function with the largest signal covariance distance was used. The main finding of this study was that the correlated noise has more pronounced impact on gridded data than on data distributed on a realistic GOCE orbit. This is useful information for methods using gridded data, such as FSC.