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Journal articles on the topic 'Harmonic functions'

Author: Grafiati

Published: 4 June 2021

Last updated: 30 July 2024

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1

Mustafa, M. T. "Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions." Abstract and Applied Analysis 2012 (2012): 1–8. http://dx.doi.org/10.1155/2012/315757.

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For Riemannian manifoldsMandN, admitting a submersionϕwith compact fibres, we introduce the projection of a function via its decomposition into horizontal and vertical components. By comparing the Laplacians onMandN, we determine conditions under which a harmonic function onU=ϕ−1(V)⊂Mprojects down, via its horizontal component, to a harmonic function onV⊂N.

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2

Wu, Shan-He, Imran Abbas Baloch, and İmdat İşcan. "On Harmonically(p,h,m)-Preinvex Functions." Journal of Function Spaces 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/2148529.

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We define a new generalized class of harmonically preinvex functions named harmonically(p,h,m)-preinvex functions, which includes harmonic(p,h)-preinvex functions, harmonicp-preinvex functions, harmonich-preinvex functions, andm-convex functions as special cases. We also investigate the properties and characterizations of harmonically(p,h,m)-preinvex functions. Finally, we establish some integral inequalities to show the applications of harmonically(p,h,m)-preinvex functions.

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3

Ba�uelos, R., I. Kleme?, and C. Moore. "harmonic functions." Duke Mathematical Journal 60, no. 3 (June 1990): 689–715. http://dx.doi.org/10.1215/s0012-7094-90-06028-4.

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4

Wu, Pengcheng. "Increasing functions, harmonic bloch and harmonic normal functions." Complex Variables, Theory and Application: An International Journal 40, no. 2 (December 1999): 133–37. http://dx.doi.org/10.1080/17476939908815212.

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5

Gillespie, Mark, Denise Yang, Mario Botsch, and Keenan Crane. "Ray Tracing Harmonic Functions." ACM Transactions on Graphics 43, no. 4 (July 19, 2024): 1–18. http://dx.doi.org/10.1145/3658201.

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Sphere tracing is a fast and high-quality method for visualizing surfaces encoded by signed distance functions (SDFs). We introduce a similar method for a completely different class of surfaces encoded by harmonic functions , opening up rich new possibilities for visual computing. Our starting point is similar in spirit to sphere tracing: using conservative Harnack bounds on the growth of harmonic functions, we develop a Harnack tracing algorithm for visualizing level sets of harmonic functions, including those that are angle-valued and exhibit singularities. The method takes much larger steps than naïve ray marching, avoids numerical issues common to generic root finding methods and, like sphere tracing, needs only perform pointwise evaluation of the function at each step. For many use cases, the method is fast enough to run real time in a shader program. We use it to visualize smooth surfaces directly from point clouds (via Poisson surface reconstruction) or polygon soup (via generalized winding numbers) without linear solves or mesh extraction. We also use it to visualize nonplanar polygons (possibly with holes), surfaces from architectural geometry, mesh "exoskeletons", and key mathematical objects including knots, links, spherical harmonics, and Riemann surfaces. Finally we show that, at least in theory, Harnack tracing provides an alternative mechanism for visualizing arbitrary implicit surfaces.

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6

Khoroshchak, Vasylyna Stepanivna. "Stationary harmonic functions on homogeneous spaces." Ufimskii Matematicheskii Zhurnal 7, no. 4 (2015): 155–59. http://dx.doi.org/10.13108/2015-7-4-149.

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7

Hayman, W. K., Tewodros Amdeberhan, and Irl C. Bivens. "Harmonic Products of Harmonic Functions: 10651." American Mathematical Monthly 106, no. 8 (October 1999): 782. http://dx.doi.org/10.2307/2589039.

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8

Bonilla, A. "Universal harmonic functions." Quaestiones Mathematicae 25, no. 4 (December 2002): 527–30. http://dx.doi.org/10.2989/16073600209486036.

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9

Hengartner, W., and G. Schober. "Univalent harmonic functions." Transactions of the American Mathematical Society 299, no. 1 (January 1, 1987): 1. http://dx.doi.org/10.1090/s0002-9947-1987-0869396-9.

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10

Gindikin, S. "Holomorphic harmonic functions." Russian Journal of Mathematical Physics 15, no. 2 (June 2008): 243–45. http://dx.doi.org/10.1134/s1061920808020088.

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11

Al-Khal, R. A., and H. A. Al-Kharsani. "Harmonic hypergeometric functions." Tamkang Journal of Mathematics 37, no. 3 (September 30, 2006): 273–83. http://dx.doi.org/10.5556/j.tkjm.37.2006.172.

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In this paper we try to uncover some of the inequalities associating hypergeometric functions with planer harmonic mappings. Sharp coefficient relations, distortion theorems and neighborhood are given for these functions. Furthermore, convolution products are considered.

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12

Matsuoka, Osamu. "Molecular integrals over Laguerre Gaussian-type functions of real spherical harmonics." Canadian Journal of Chemistry 70, no. 2 (February 1, 1992): 388–92. http://dx.doi.org/10.1139/v92-055.

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Molecular integrals are formulated over the Laguerre Gaussian-type functions (LGTF) of real spherical harmonics. They include the overlap integrals and the energy integrals of kinetic, nuclear attraction, and electron repulsion. For the nuclear-attraction integrals the formulations based on the point as well as the Gaussian nuclear charge distribution models are presented. Integral formulas over the LGTFs of real spherical harmonics are found a little more complicated than those of the LGTFs of complex spherical harmonics due to the summations over magnetic quantum numbers. Keywords: molecular integral, Gaussian-type function, spherical harmonic, solid harmonic, Sonine polynomial.

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13

Juhas, Anamarija, and Ladislav A. Novak. "Maximally Flat Waveforms with Finite Number of Harmonics in Class-FPower Amplifiers." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/169590.

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In this paper general solution to the problem of finding maximally flat waveforms with finite number of harmonics (maximally flat trigonometric polynomials) is provided. Waveform coefficients are expressed in closed form as functions of harmonic orders. Two special cases of maximally flat waveforms (so-called maximally flat even harmonic and maximally flat odd harmonic waveforms), which proved to play an important role in class-Fand inverse class-Fpower amplifier (PA) operations, are also considered. For these two special types of waveforms, coefficients are expressed as functions of two parameters only. Closed form expressions for efficiency and power output capability of class-Fand inverse class-FPA operations with maximally flat waveforms are also provided as explicit functions of number of a harmonics.

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14

Kim, Hong Oh. "M-harmonic functions with M-harmonic square." Bulletin of the Australian Mathematical Society 53, no. 1 (February 1996): 123–29. http://dx.doi.org/10.1017/s0004972700016786.

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ℳ-harmonic functions with ℳ-harmonic square are proved to be either holomorphic or antiholomorphic in the unit ball of complex n-space under certain additional conditions. For example, if u and u2 are ℳ-harmonic in the unit ball of ℂ2 and if u is continuously differentiable up to the boundary then u is either holomorphic or antiholomorphic.

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15

Gardiner, Stephen J. "Coincidence of Harmonic and Finely Harmonic Functions." Potential Analysis 34, no. 1 (April 27, 2010): 81–88. http://dx.doi.org/10.1007/s11118-010-9182-0.

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16

Foskolos, Georgios. "Measurement-Based Current-Harmonics Modeling of Aggregated Electric-Vehicle Loads Using Power-Exponential Functions." World Electric Vehicle Journal 11, no. 3 (July 28, 2020): 51. http://dx.doi.org/10.3390/wevj11030051.

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This paper presents an aggregate current-harmonic load model using power exponential functions and built from actual measurement data during the individual charging of four different fully electric vehicles. The model is based on individual emitted current harmonics as a function of state of charge (SOC), and was used to deterministically simulate the simultaneous charging of six vehicles fed from the same bus. The aggregation of current harmonics up to the 11th was simulated in order to find the circumstances when maximal current-harmonic magnitude occurs, and the phase-angle location. The number of possible identical vehicles was set to four, while battery SOC, the start of charging, and the kind of vehicle were randomized. The results are presented in tables, graphs, and polar plots. Even though simulations did not consider the surrounding harmonics, supply-voltage variation, or network impedance, this paper presents an innovative modeling approach that gives valuable information on the individual current-harmonic contribution of aggregated electric-vehicle loads. With the future implementation of vehicle-to-grid technology, this way of modeling presents new opportunities to predict the harmonic outcome of multiple electric vehicles charging.

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17

Murugusundaramoorthy, G., and S. Porwal. "О гармонических функций типа Яновского, связанных с гипергеометрическими функциями Райта." Владикавказский математический журнал 25, no. 4 (December 22, 2023): 91–102. http://dx.doi.org/10.46698/b2503-7977-9793-e.

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In our present study we consider Janowski type harmonic functions class introduced and studied by Dziok, whose members are given by $h(z) = z + \sum_{n=2}^{\infty} h_n z^n$ and $g(z) = \sum_{n=1}^{\infty} g_n z^n$, such that $\mathcal{ST}_{H}(F,G)=\big\{ f = h + \bar{g} \in {H}:\frac{\mathfrak{D}_H f(z)}{f(z)}\prec\frac{1+Fz}{1+G z};\, (-G \leq F < G \leq 1, \text{ with } g_1=0)\big\},$ where $\mathfrak{D}_H f(z) = zh'(z)-\overline{zg'(z)}\,$ and $z\in \mathbb{U}=\{z:z\in \mathbb{C} \text{ and }|z| < 1 \}.$ We investigate an~association between these subclasses of harmonic univalent functions by applying certain convolution operator concerning Wright's generalized hypergeometric functions and several special cases are given as a corollary. Moreover we pointed out certain connections between Janowski-type harmonic functions class involving the generalized Mittag–Leffler functions. Relevant connections of the results presented herewith various well-known results are briefly indicated.

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18

Singh, Saransh, Donald E. Boyce, Joel V. Bernier, and Nathan R. Barton. "Discrete spherical harmonic functions for texture representation and analysis." Journal of Applied Crystallography 53, no. 5 (September 23, 2020): 1299–309. http://dx.doi.org/10.1107/s1600576720011097.

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A basis of discrete harmonic functions for efficient representation and analysis of crystallographic texture is presented. Discrete harmonics are a numerical representation of the harmonics on the sphere. A finite element formulation is utilized to calculate these orthonormal basis functions, which provides several advantageous features for quantitative texture analysis. These include high-precision numerical integration, a simple implementation of the non-negativity constraint and computational efficiency. Simple examples of pole figure and texture interpolation and of Fourier filtering using these basis sets are presented.

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19

Sabirin, Nor Hidalina Mohd, Zainuddin Mat Isa, Mohd Hafiz Arshad, Baharuddin Ismail, Md Hairul Nizam Talib, and Ernie Che Mid. "Harmonics Elimination in 7-Level Multilevel Inverter Using Animal Migration Optimization Algorithm with Different Objective Functions." International Journal of Emerging Technology and Advanced Engineering 13, no. 1 (January 3, 2023): 18–27. http://dx.doi.org/10.46338/ijetae0123_03.

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Harmonics can degrade the power quality of a multilevel inverter by causing the voltage to be distorted and vary from sinusoidal waveforms. Harmonics can be reduced by increasing the number of voltage levels or by employing suitable modulation techniques. In this paper, The Selective Harmonic Elimination Pulse Width Modulation (SHEPWM) modulation method is employed to obtain the optimal switching angles that able to reduce the specific individual harmonic and the Total Harmonic Distortion (THD) in singlephase 7-level Cascaded H-Bridge multilevel inverter. The Animal Migration Optimization (AMO) is proposed to acquire these angles using two difference objective functions. The performance is examined and evaluated. Both objective functions able to determine the optimal switching angles starting from modulation index of 0.34. However, the comparative study demonstratethat objective function number 2 has better performance in term of lowering selective individual harmonics as well as THD.

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20

Dorff, Michael, Maria Nowak, and Wojciech Szapiel. "Typically real harmonic functions." Rocky Mountain Journal of Mathematics 42, no. 2 (April 2012): 567–81. http://dx.doi.org/10.1216/rmj-2012-42-2-567.

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21

Poletsky, Evgeny A. "Approximation by harmonic functions." Transactions of the American Mathematical Society 349, no. 11 (1997): 4415–27. http://dx.doi.org/10.1090/s0002-9947-97-02041-2.

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22

Galperin, A., E. Ivanov, V. Ogievetsky, and E. Sokatchev. "Harmonic supergraphs: Green functions." Classical and Quantum Gravity 2, no. 5 (September 1, 1985): 601–16. http://dx.doi.org/10.1088/0264-9381/2/5/004.

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23

Sung, Chiung-Jue, Luen-Fai Tam, and Jiaping Wang. "Spaces of Harmonic Functions." Journal of the London Mathematical Society 61, no. 3 (June 2000): 789–806. http://dx.doi.org/10.1112/s0024610700008759.

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24

Kim, Yong Chan, Jay M. Jahangiri, and Jae Ho Choi. "Certain convex harmonic functions." International Journal of Mathematics and Mathematical Sciences 29, no. 8 (2002): 459–65. http://dx.doi.org/10.1155/s0161171202007585.

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We define and investigate a family of complex-valued harmonic convex univalent functions related to uniformly convex analytic functions. We obtain coefficient bounds, extreme points, distortion theorems, convolution and convex combinations for this family.

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25

Ferrada-Salas, Álvaro, and María J. Martín. "Generalized harmonic Koebe functions." Journal of Mathematical Analysis and Applications 435, no. 1 (March 2016): 860–73. http://dx.doi.org/10.1016/j.jmaa.2015.10.066.

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26

Abu-Muhanna, Yusuf. "On harmonic univalent functions." Complex Variables, Theory and Application: An International Journal 39, no. 4 (August 1999): 341–48. http://dx.doi.org/10.1080/17476939908815201.

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27

Colding, Tobias H., and William P. Minicozzi II. "Harmonic Functions on Manifolds." Annals of Mathematics 146, no. 3 (November 1997): 725. http://dx.doi.org/10.2307/2952459.

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28

Khrebtukov, D. B., and J. H. Macek. "Harmonic oscillator Green functions." Journal of Physics A: Mathematical and General 31, no. 12 (March 27, 1998): 2853–68. http://dx.doi.org/10.1088/0305-4470/31/12/010.

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29

Amini, Massoud, and Cho-Ho Chu. "Harmonic functions on hypergroups." Journal of Functional Analysis 261, no. 7 (October 2011): 1835–64. http://dx.doi.org/10.1016/j.jfa.2011.05.020.

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30

Knieper, Gerhard, and Norbert Peyerimhoff. "Harmonic Functions on Rank One Asymptotically Harmonic Manifolds." Journal of Geometric Analysis 26, no. 2 (February 3, 2015): 750–81. http://dx.doi.org/10.1007/s12220-015-9570-1.

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31

Noor, Muhammad, Khalida Noor, and Sabah Iftikhar. "Harmonic beta-convex functions involving hypergeometric functions." Publications de l'Institut Math?matique (Belgrade) 104, no. 118 (2018): 241–49. http://dx.doi.org/10.2298/pim1818241n.

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32

Li, Peter, and Luen-Fai Tam. "Green's functions, harmonic functions, and volume comparison." Journal of Differential Geometry 41, no. 2 (1995): 277–318. http://dx.doi.org/10.4310/jdg/1214456219.

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33

Choi, Gundon, and Gabjin Yun. "Harmonic morphisms and subharmonic functions." International Journal of Mathematics and Mathematical Sciences 2005, no. 3 (2005): 383–91. http://dx.doi.org/10.1155/ijmms.2005.383.

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LetMbe a complete Riemannian manifold andNa complete noncompact Riemannian manifold. Letϕ:M→Nbe a surjective harmonic morphism. We prove that ifNadmits a subharmonic function with finite Dirichlet integral which is not harmonic, andϕhas finite energy, thenϕis a constant map. Similarly, iffis a subharmonic function onNwhich is not harmonic and such that|df|is bounded, and if∫M|dϕ|<∞, thenϕis a constant map. We also show that ifNm(m≥3)has at least two ends of infinite volume satisfying the Sobolev inequality or positivity of the first eigenvalue of the Laplacian, then there are no nonconstant surjective harmonic morphisms with finite energy. Forp-harmonic morphisms, similar results hold.

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34

Lee, Taeyoung. "Real Harmonic Analysis on the Special Orthogonal Group." International Journal of Analysis and Applications 20 (April 4, 2022): 21. http://dx.doi.org/10.28924/2291-8639-20-2022-21.

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This paper presents theoretical analysis and software implementation for real harmonics analysis on the special orthogonal group. Noncommutative harmonic analysis for complex-valued functions on the special orthogonal group has been studied extensively. However, it is customary to treat real harmonic analysis as a special case of complex harmonic analysis, and there have been limited results developed specifically for real-valued functions. Here, we develop a set of explicit formulas for real-valued irreducible unitary representations on the special orthogonal group, and provide several operational properties, such as derivatives, sampling, and Clebsch-Gordon coefficients. Furthermore, we implement both of complex and real harmonics analysis on the special orthogonal group into an open source software package that utilizes parallel processing through the OpenMP library. The efficacy of the presented results are illustrated by benchmark studies and an application to spherical shape matching.

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35

Jahangiri, Jay M., Herb Silverman, and Evelyn M. Silvia. "Construction of Planar Harmonic Functions." International Journal of Mathematics and Mathematical Sciences 2007 (2007): 1–11. http://dx.doi.org/10.1155/2007/70192.

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Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk can be written in the formf=h+g¯, wherehandgare analytic in the open unit disk. The functionshandgare called the analytic and coanalytic parts off, respectively. In this paper, we construct certain planar harmonic maps either by varying the coanalytic parts of harmonic functions that are known to be harmonic starlike or by adjoining analytic univalent functions with coanalytic parts that are related or derived from the analytic parts.

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36

NIKOLAEV, NIKOLAY Y., and HITOSHI IBA. "GENETIC PROGRAMMING OF POLYNOMIAL HARMONIC NETWORKS USING THE DISCRETE FOURIER TRANSFORM." International Journal of Neural Systems 12, no. 05 (October 2002): 399–410. http://dx.doi.org/10.1142/s0129065702001242.

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This paper presents a genetic programming system that evolves polynomial harmonic networks. These are multilayer feed-forward neural networks with polynomial activation functions. The novel hybrids assume that harmonics with non-multiple frequencies may enter as inputs the activation polynomials. The harmonics with non-multiple, irregular frequencies are derived analytically using the discrete Fourier transform. The polynomial harmonic networks have tree-structured topology which makes them especially suitable for evolutionary structural search. Empirical results show that this hybrid genetic programming system outperforms an evolutionary system manipulating polynomials, the traditional Koza-style genetic programming, and the harmonic GMDH network algorithm on processing time series.

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37

Medjber, Salim, Hacene Bekkar, Salah Menouar, and Jeong Ryeol Choi. "Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System." Advances in Mathematical Physics 2016 (2016): 1–6. http://dx.doi.org/10.1155/2016/3693572.

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The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.

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38

Graf, S. Yu. "The Schwarzian derivatives of harmonic functions and univalence conditions." Issues of Analysis 24, no. 2 (December 2017): 42–56. http://dx.doi.org/10.15393/j3.art.2017.4230.

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39

Jahangiri, Jay M., and Raj Kumar Garg. "Directional Convexity of Convolutions of Harmonic Functions." International Journal of Mathematics and Mathematical Sciences 2019 (April 1, 2019): 1–6. http://dx.doi.org/10.1155/2019/5731830.

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Harmonic functions can be constructed using two analytic functions acting as their analytic and coanalytic parts but the prediction of the behavior of convolution of harmonic functions, unlike the convolution of analytic functions, proved to be challenging. In this paper we use the shear construction of harmonic mappings and introduce dilatation conditions that guarantee the convolution of two harmonic functions to be harmonic and convex in the direction of imaginary axis.

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40

Bellan, Diego. "Three-Phase Distortion Analysis based on Space-Vector Locus Diagrams." WSEAS TRANSACTIONS ON POWER SYSTEMS 18 (December 31, 2023): 467–73. http://dx.doi.org/10.37394/232016.2023.18.46.

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This work deals with the use of the space vector concept to characterize the harmonic content of a three-phase voltage/current. It is shown that the shape of the trajectory of the space vector on the complex plane (i.e., the locus diagram) provides information about its harmonic content. In particular, it is shown that each harmonic contributes to the locus diagram with a number of lobes depending on the relative angular frequency between the harmonic and the fundamental component. To this aim, the different contributions of positive-sequence and negative-sequence harmonics is explained and put into evidence with specific examples. The expressions for the magnitude and phase of the space vector as functions of the harmonics are derived analytically. Numerical examples are provided to show how the locus diagram can represent a three-phase quantity with positive-sequence and negative-sequence harmonics.

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41

Chi, Dong-Pyo, Gun-Don Choi, and Jeong-Wook Chang. "ASYMPTOTIC BEHAVIOR OF HARMONIC MAPS AND EXPONENTIALLY HARMONIC FUNCTIONS." Journal of the Korean Mathematical Society 39, no. 5 (September 1, 2002): 731–43. http://dx.doi.org/10.4134/jkms.2002.39.5.731.

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42

BEJAN, Cornelia-Livia, and Simona-Luiza DRUTA-ROMANIUC. "Harmonic functions and quadratic harmonic morphisms on Walker spaces." TURKISH JOURNAL OF MATHEMATICS 40 (2016): 1004–19. http://dx.doi.org/10.3906/mat-1504-87.

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43

Zou, Wan Feng, and Lei Zhang. "An Active Power Filter Controller Based on Dual-DSPs in Low-Voltage Power System." Applied Mechanics and Materials 291-294 (February 2013): 2302–7. http://dx.doi.org/10.4028/www.scientific.net/amm.291-294.2302.

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To compensate harmonics from nonlinear load, the controller of Active Power Filter (APF) need to detect harmonic and generate compensation current in high speed. APF controller functions are divided into two parts: outside controller part and inside controller part. Inside controller part detect harmonic current and control the IGBT to generate grid harmonics current. Outside controller part detect common power parameters and generate signal of management in main circuit. An experiment circuit of APF Controller is established. Some experiments are presented here.

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44

Stojiljković, Vuk, Nicola Fabiano, and Vesna Šešum-Čavić. "Harmonic series with polylogarithmic functions." Vojnotehnicki glasnik 70, no. 1 (2022): 43–61. http://dx.doi.org/10.5937/vojtehg70-35148.

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Introduction/purpose: Some sums of the polylogarithmic function associated with harmonic numbers are established. Methods: The approach is based on using the summation methods. Results: This paper generalizes the results of the zeta function series associated with the harmonic numbers. Conclusions: Various interesting series as the consequence of the generalization are obtained.

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45

SALMAN JUMA, ABDULRAHMAN. "ON MULTIVALENT HARMONIC MEROMORPHIC FUNCTIONS INVOLVING HYPERGEOMETRIC FUNCTIONS." IRAQI JOURNAL OF STATISTICAL SCIENCES 11, no. 20 (December 28, 2011): 117–26. http://dx.doi.org/10.33899/iqjoss.2011.27883.

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46

Chung, Soon-Yeong, Dohan Kim, and Sang Moon Kim. "Isolated Singularities of Harmonic Functions and Analytic Functions∗." Integral Transforms and Special Functions 11, no. 3 (June 2001): 257–72. http://dx.doi.org/10.1080/10652460108819316.

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47

Rossi, Julio. "Functions of least gradient and 1-harmonic functions." Indiana University Mathematics Journal 63, no. 4 (2014): 1067–84. http://dx.doi.org/10.1512/iumj.2014.63.5327.

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48

Bishop, Christopher J. "Approximating continuous functions by holomorphic and harmonic functions." Transactions of the American Mathematical Society 311, no. 2 (February 1, 1989): 781. http://dx.doi.org/10.1090/s0002-9947-1989-0961619-2.

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49

Tolokonnikov, V. A. "Free interpolation of harmonic functions by analytic functions." Journal of Soviet Mathematics 36, no. 3 (February 1987): 426–28. http://dx.doi.org/10.1007/bf01839618.

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50

Stević, S. "Weighted integrals of harmonic functions." Studia Scientiarum Mathematicarum Hungarica 39, no. 1-2 (2002): 87–96. http://dx.doi.org/10.1556/sscmath.39.2002.1-2.5.

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