This solves a problem proposed by Bank and Laine in In recent years the study of quaternionic linear spaces has been widely developed by mathematicians and has been widely used by physicists. At the same time it turns out that some basic and fundamental properties of those spaces are not treated properly and this requires to develop the corresponding theory. In this talk we will analyze certain peculiarities of the situation via the notion of quaternionic extension of real and complex linear spaces as well as using the notion of internal quaternionization.
Hecke, Ostrowski and Kesten characterized the intervals on the circle for which the ergodic sums of their indicator function, under an irrational rotation, stay at a bounded distance from their integral with respect to the Lebesgue measure on the circle. In this talk I will discuss this phenomenon in multi-dimensional setting. Based on joint work with Sigrid Grepstad. Topological methods based on the usage of degree theory have proved themselves to be an important tool for qualitative studying of solutions to nonlinear differential systems including such problems as existence, uniqueness, multiplicity, bifurcation, etc.
During the last twenty years the equivariant degree theory emerged in Non- linear Analysis. Golubitsky et al. In fact, the equivariant degree has different faces reflecting a diversity of symmetric equations related to applications. In the two talks, I will discuss three variants of the equivariant degree: i non-parameter equivariant degree, ii twisted equivariant degree with one parameter, and iii gradient equivariant degree.
Each of the three variants of equivariant degree will be illustrated by appropriate examples of applications: i boundary value problems for vector symmetric pendulum equation, ii Hopf bifurcation in symmetric neural networks simulation of legged locomotion , and iii bifurcation of relative equilibria in Lennard-Jones three-body problem. The following problems or a part of them will be discussed. Generalization of the Abel-Poisson summation method. Generalization of the Riemann-Lebesgue lemma. Generalization of the Euler-Maclaurin formula.
Absolute convergence of grouped Fourier series. Comparison of linear differential operators with constant coefficients. Positive definite functions and splines. Strong converse theorems in approximation theory. Bernstein-Stechkin polynomials. We study the influence the angular distribution of zeroes of the Taylor series with pseudo-random and random coefficients, and show that the distribution of zeroes is governed by certain autocorrelations of the coefficients.
Nodal sets are zero loci of Laplace eigenfunctions e. Study of nodal sets is important for understanding wave processes. The geometry of a single nodal set may be very complicated and hardly can be well understood. More realistic might be describing geometry of sets which are nodal for a large family of e. Indeed, it was proved that common nodal curves for large, in different senses, families of e. It was conjectured that in a Euclidean space of arbitrary dimension, common nodal hypersurfaces for large families of e. Relation to the injectivity problem for the spherical Radon transform will be explained.
In the talk, I will present a method of Beurling that gives a solution to both of the problems for some quasianalytic Carleman classes. If time permits, I will also discuss the image problem in some non-quasianalytic classes. Outline of the talk:. Two types of optimal estimates for derivatives of analytic functions with bounded real part are considered. An important property of exponential bases is their stability.
The result that I will present in my talk are part of joint projects with my students A. Kumar and S. This recent extension of a classical for Fourier series Hardy- Littlewood theorem gives rise to new thoughts and results. The aim of this talk is to present a simple two-sided estimate for the operator norm of a finite Hankel matrix in terms of its standard symbol.
We consider the wave flow on a surface of constant negative curvature.
Includes joint work with Roman Schubert. We construct a series of examples of the quadratic vector field to show the impact of their spectral properties into qualitative theory. We present new sufficient conditions for Fourier multipliers. These conditions are given in terms of simultaneous behavior of quasi- norms of a function in different Lebesgue and Besov spaces.
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We also provide some sufficient conditions for the representation of a function as an absolute convergence Fourier integrals in terms of belonging of a function simultaneously to several spaces of smooth functions. We discuss sharp continuity and regularity results for solutions of the polyharmonic equation in an arbitrary open set. Positive results have been available only when the domain is sufficiently smooth, Lipschitz or diffeomorphic to a polyhedron. First-order systems of partial differential equations appear in many areas of physics, from the Maxwell equations to the Dirac operator.
The beautiful and extensive Coifman-Meyer theory, developed in the 70s and 80s to study singular multi-linear operators does not apply to many naturally arising operators with positive kernels. We shall describe some elementary approaches to such operators and apply them to some problems in geometric measure theory and classical harmonic analysis. Bauschke and J. Borwein showed that in the space of all tuples of bounded, closed and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection.
We show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of closed and convex sets which are not necessarily bounded. This is joint work with A. We derive an extension of the standard time-dependent WKB theory, which can be applied to propagate coherent states and other strongly localized states for long times. It in particular allows us to give a uniform description of the transformation from a localized coherent state into a delocalized Lagrangian state, which takes place at the Ehrenfest time.
The main new ingredient is a metaplectic operator that is used to modify the initial state in a way that the standard time-dependent WKB theory can then be applied for the propagation. This is based on joint work with Raul Vallejos and Fabricio Toscano, but in this talk we will focus on the special case of propagation on a manifold of negative curvature.
The inequalities defining the continuity are sharp with respect to the order. It is well known that the geometric nature of semigroup trajectories essentially depends on the semigroup type. In this work, we concentrate on parabolic type semigroups of holomorphic self-mappings of the open unit disk and of the right half-plane, and study the structure of semigroup trajectories near the Denjoy--Wolff point. For these purposes, we suggest that two terms in the asymptotic power expansion of semigroup generators are known.
Inter alia, this enable us to establish a new rigidity property for semigroups of parabolic type. A generalization of Newton's attraction theorem will be discussed. The same analytic method is applied for reconstruction in photoacoustic geometry. We extend Caratheodory's generalization of Montel's fundamental normality test to "wandering" exceptional functions i. Furthermore, we prove that if we have a family of pairs a,b of functions meromorphic in a domain such that a and b uniformly "stay away from each other " , then the families of the functions a resp.
The proofs are based on a "simultaneous rescaling" version of Zalcman's Lemma. We also introduce a somewhat "strange" result about some sharing wandering values assumptions that imply normality. I will present a quantum ergodicity theorem on large regular graphs. This is a result of spatial equidistribution of most eigenfunctions of the discrete Laplacian in the limit of large regular graphs. It is analogous to the quantum ergodicity theorem on Riemannian manifolds, which is concerned with the eigenfunctions of the Laplace-Beltrami operator in the high frequency limit.
I will also talk about pseudo-differential calculus on regular graphs, one of the tools constructed for the proof of the theorem. Mayboroda Minnesota and Guo Luo Caltech. In this talk we will introduce a definition of Generalized Analytic Functions in the sense of Vekua , in elliptic complex numbers. We introduce a notion of r-convexity for subsets of the complex plane.
The proof is based on Meyer's quasicrystals. Recently S. Artstein-Avidan and V.
This is a joint work with Alfredo N. Iusem and Benar F. This conjecture was completely solved in by T. This is a joint work with D. The Nitsche conjecture was solved recently by Iwaniec, Kovalov and Onninen and in the same paper they pose the same problem from Teichmuller domain onto Teiuchmuller domain. We present a solution to this problem. We will discuss the case of surfaces of constant negative curvature; in particular, we will explain how to construct examples of sufficiently weak quasimodes that do not satisfy QUE, and show how they fit into the larger theory.
In , Talagrand established a lower bound on the second-level Fourier coefficients of a monotone Boolean function, in terms of its first-level coefficients. This lower bound and its enhancements were used in various applications to correlation inequalities, noise sensitivity, geometry, percolation, etc. In this talk we present a new proof of Talagrand's inequality, which is somewhat simpler than the original proof, and allows to generalize the result easily to non-monotone functions with influences replacing the first-level coefficients and to more general measures on the discrete cube.
We then apply our proof to obtain a quantitative version of a theorem of Benjamini-Kalai-Schramm on the relation between influences and noise sensitivity. Time permitting, we shall present recent results and open questions, related to an application of Talagrand's lower bound to correlation inequalities. Physics 2 , In particular, three criteria of comparison are obtained for functions on the circle, on the axis, and on the half-axis, as well as one sharp inequality. In the talk we first review the highly anisotropic Hardy spaces].
One cannot assume that a linear functional, uniformly bounded on all atoms, is automatically bounded on spaces that have atomic representations e. Hardy spaces. This theorem is very rich in applications and has been generalized by many authors in various directions weak conformality, differentiability, multidimensional analogs, etc.
We essentially improve the underlying modular technique. We study to which extent the Poisson summation formula determines the Fourier transform. The answer is positive under certain technical smoothness and rate of decay conditions.
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The Euler-Gauss linear transformation formula for the hypergeometric function was extended by Goursat for the case of logarithmic singularities. By replacing the perturbed Bessel differential equation by a monodromic functional equation, and studying this equation separately from the differential equation by an appropriate Laplace-Borel technique, we associate with the latter equation another monodromic relation in the dual complex plane. This enables us to prove a duality theorem and to extend Goursat's formula to much larger classes of functions. Conversely, the embeddings result from the inequalities for moduli of smoothness by limit procedures.
Skip to main content Skip to main Navigation. Search Form. Secondary Menu. Usual Time:. Lectures from past years:. Previous Lectures. Ami Viselter, University of Haifa. Speaker: Dr. Title : Generating functionals on quantum groups. Abstract: We will discuss generating functionals on locally compact quantum groups. Alberto Debernardi, Bar-Ilan University. Title : Weighted norm inequalities for integral transforms with splitting kernels.
Abstract: Given an integral transform on the positive real line, we say that its kernel is splitting if it satisfies upper pointwise estimates given by products of two functions, each of them taken in a different variable. Speaker: Prof. Yuli Eidelman, Tel-Aviv University. Title : On some inverse and nonlocal problems for operator differential equations and numerical methods for their solution.
Samuel Krushkal, Bar-Ilan University. Title : A general coefficient theorem for univalent functions. Title : Chebyshev-type Quadratures for Doubling Weights. Abstract: A Chebyshev-type quadrature for a given weight function is a quadrature formula with equal weights. Jeremy Schiff, BIU. Title : New algorithms for convex interpolation. Abstract: In various settings, from computer graphics to financial mathematics, it is necessary to smoothly interpolate a convex curve from a set of data points.
Jeremy Schiff, Bar-Ilan University. Liflyand, Bar-Ilan University. Title : Hardy type inequalities in the category of Hausdorff operators. Abstract: Classical Hardy's inequalities are concerned with the Hardy operator and its adjoint, the Bellman operator. Debernardi, Bar-Ilan University. Title : Hankel transforms and general monotonicity. Title : Nonlinear resolvent of holomorphic generators. Title : Noise Stability and Majority Functions. Abstract: Two important results in Boolean analysis highlight the role of majority functions in the theory of noise stability.
Title : On a local version of the fifth Busemann-Petty Problem. Title : Multivariable Hardy spaces and the classification of universal dilation algebras. Kuleshov, Moscow State University, Russia. Abstract: We prove that each function of one variable forming a continuous finite sum of ridge functions on a convex body belongs to the VMO space on every compact interval of its domain.
Title : The Fourier transform of a convex function revisited. Abstract: Asymptotic-wise results for the Fourier transform of a function of convex type are proved. Title : Completely monotonic gamma ratio and infinitely divisible H-function of Fox. Abstract: We investigate conditions for the logarithmic complete monotonicity of a quotient of two products of gamma functions, where the argument of each gamma function has a different scaling factor.
Yuki Takahashi, Bar-Ilan University. Title : Dimension of attractors for Iterated Function Systems of linear fractional transformations and the Diophantine property of matrix semigroups.
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Abstract: We consider Iterated Function Systems of linear fractional transformations, and show that the Hausdorff dimension of the attractor is given by the Bowen's pressure formula, if the Iterated Function Systems satisfy the exponential separation condition. Andrei Lerner, Bar-Ilan University. Title : On the weak Muckenhoupt-Wheeden conjecture. Abstract: We construct an example showing the sharpness of certain weighted weak type 1,1 bounds for the Hilbert transform. Naomi Feldheim, Bar-Ilan University. Title : Spectral gap and sign changes of Gaussian stationary processes. Yuval Peres, Microsoft Research.
Title : Trace reconstruction for the deletion channel. Anatoly Golberg, Holon Institute of Technology. Title : Asymptotic dilation of regular homeomorphisms. Vladimir Golubyatnikov, Sobolev institute of mathematics, Novosibirsk, Russia. Title : On cycles in asymmetric models of circular gene networks.
Abstract: We study geometry and combinatorial structures of phase portrait of some nonlinear kinetic dynamical systems as models of circular gene networks in order to find conditions of existence of cycles of these systems. Title : Spectrality of Product Domains. Krasnov, Bar-Ilan University. More precisely, given a differential equation on an algebra A, we are interested in the following two problems: 1. Title : Holomorphic extensions of trace formulas. Abstract: The Chazarain-Poisson summation formula for Riemannian manifolds which generalizes the Poisson Summation formula computes the distribution trace.
Yosef Yomdin, Weizmann Institute. Title : Smooth parametrizations of semi-algebraic, o-minimal, … sets , and their applications in Dynamics, Analysis, and Diophantine geometry and, maybe, in Complex Hyperbolic Geometry. Abstract: Smooth parametrization consists in a subdivision of a mathematical object under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives.
Bochen Liu, Bar-Ilan University. Abstract: Given a measure on a subset of Euclidean spaces. Elza Farkhi, Tel-Aviv University. Abstract: The talk surveys joint works with T. Title : Ternary generalizations of graded algebras and their applications in physics. Boris Solomyak Bar-Ilan University.
Shai Dekel, Tel-Aviv University. Title : Hardy spaces over manifolds. Title : CLT for small scale mass distribution of toral Laplace eigenfunctions. Title : Approximations of convex bodies by measure-generated sets. Golberg, Holon Institute of Technology. Title : Mappings with integrally controlled moduli: regularity properties. Nir Lev, Bar-Ilan University. Title : Fourier frames for singular measures and pure type phenomena.
Levin, Bar-Ilan University. Title : On low discrepancy sequences and lattice points problem for parallelepiped. Ami Viselter University of Haifa. Title : Convolution semigroups on quantum groups and non-commutative Dirichlet forms. Abstract: We will discuss convolution semigroups of states on locally compact quantum groups. David Levin, Tel-Aviv University.
Abstract: Iterated Function Systems IFS have been at the heart of fractal geometry almost from its origin, and several generalizations for the notion of IFS have been suggested. Title : Maximal and Riesz potential operators with rough kernel in non-standard function spaces: unabridged version. Abstract: In this talk we will discuss the boundedness of the maximal operator with rough kernel in some non-standard function spaces, e. Ami Viselter, Haifa University.
Title : Around Property T for quantum groups. Kolomoitsev, University of Luebeck, Germany. Title : On the growth of Lebesgue constants for convex polyhedra. Abstract: The talk is devoted to the Lebesgue constants of polyhedral partial sums of the Fourier series. Shustin, Tel-Aviv University. Title : Milnor fibers of real singularities. Abstract: Milnor fibers of isolated hypersurface singularities carry the most important information on the singularity. Amos Nevo, Technion. Title : The non-Euclidean lattice points counting problem. Michael I. Abstract: In this talk we discuss asymptotic relations between sharp constants of approximation theory in a general setting.
Michael Megrelishvili, Bar-Ilan University. Title : Tame dynamical systems. Abstract: Tame dynamical systems were introduced by A. Kerner, Ben-Gurion University. Title : Matrices over local rings. Abstract: Linear algebra over a field have been studied for centuries. Alesker, Tel-Aviv University. Title : Continuous valuations on convex sets and Monge-Ampere operators. Abstract: Finitely additive measures on convex convex sets are called valuations. Title : Exotic Poisson summation formulas. Title : More on Differential Inequalities and Normality. Shahar Nevo, Bar-Ilan University.
Title : Differential inequalities and normality. Abstract: Following Marty's Theorem we present recent results about differential inequalities that imply or not some degree of normality. Title : Asymptotic relations for the Fourier transform of a function of bounded variation.
Abstract: Earlier and recent one-dimensional estimates and asymptotic relations for the cosine and sine Fourier transform of a function of bounded variation are refined in such a way that become applicable for obtaining multidimensional asymptotic relations for the Fourier transform of a function with bounded Hardy variation.
Title : Boundary triples and Weyl functions of symmetric operators. Title : A two-phase mother body and a Muskat problem. Title : Spectrality and tiling by cylindric domains. Lerner, Bar-Ilan University.
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Title : On pointwise domination of Calderon-Zygmund operators by sparse operators. Title : Finite volume scheme for a parabolic equation with a non-monotone diffusion function. Title : Hardy spaces and variants of the div-curl lemma. Abstract: The theory of real Hardy spaces has been applied to the study of partial differential equations in many different contexts. Title : On the Riemann—Hilbert problem for the Beltrami equations in quasidisks. Abstract: For the nondegenerate Beltrami equations in the quasidisks and, in particular, in smooth Jordan domains, we prove the existence of regular solutions of the Riemann—Hilbert problem with coefficients of bounded variation and boundary data that are measurable with respect to the absolute harmonic measure logarithmic capacity.
Abdullayev, Mersin University, Turkey. Title : On sharp inequalities for orthonormal polynomials along a contour. Title : On the Fourier transform of a function of several variables. Leviatan, Tel-Aviv University. Title : Comparing the degrees of unconstrained and constrained approximation. Abstract: It is quite obvious that one should expect that the degree of constrained approximation be worse than the degree of unconstrained approximation. In the talk we. This allows us to investigate the interconnection between mappings of bounded and finite.
Vladimir Rovenski, University of Haifa. Title : Integral formulae for codimension-one foliated Finsler spaces. Tobias Hartnick, Technion. Title : Diffraction theory for aperiodic point sets in Lie groups. Panagiotis Mavroudis, University of Crete, Greece. Title : Extremal and approximation problems for positive definite functions. Title : Tiling by translates of a function. Liflyand Bar-Ilan University. Title : A tale of two Hardy spaces.
Abstract: New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. Krasnov Bar-Ilan University. Title : Quantifying isolated singularity in DEs. Cwikel, Technion. Title : Some new partial answers to a 52 year old interpolation question. A fairly recent survey which discusses this question is available at arXiv Title : Hardy spaces on the Klein-Dirac quadric and multidimensional annulus: applications to Interpolation, Moment problems, and Cubature.
Krushkal, Bar-Ilan University. Title : Wavelets on fractals. Title : On the Zariski Cancellation Problem. Or, in other words, whether varieties with isomorphic cylinders should be isomorphic. This occurs to be true for affine. In this case, the answer. The birational counterpart of the special Zariski Cancellation Problem asks whether stable rationality implies rationality.
The answer. We will survey on the subject, both on some classical results and on a very recent development, reporting in particular on a joint. Title : Bounded approximation and radial interpolation in the unit disc and related questions. Abstract: The talk is devoted to some bounded approximation and interpolation problems and theorems in. Fatou, W.
Rudin, L. Carleson, L. Zalcman, and other authors. Among other results, a new theorem due to S. Gardiner on radial interpolation will be presented. Maz'ya, University of Liverpool and University of Linkoeping. Title : Criteria for the Poincare-Hardy inequalities. The attention is focused on conditions both necessary and sufficient, as well as on their sharp corollaries. Yakovenko, Weizmann Institute. Title : Infinitesimal Hilbert 16th problem. Abstract: I will describe the current state of affairs in both the original Hilbert 16th problem.
Functions of this class admit explicit albeit very. This result lies at the core of the solution of the infinitesimal Hilbert problem, achieved with. Trigub, Donetsk National University, Ukraine. Title : On summability methods for Fourier series and Fourier integrals. Abstract: In the problem of summability at a point at which the derivative of indefinite integral exists for Fourier series and Fourier integrals of integrable functions a new sufficient condition is obtained.
Shvartsman, Technion. This is a joint work with Nahum Zobin. Title : Bernoulli convolution measures and their Fourier transforms. Eremenko, Purdue University. Title : Zeros of solutions of linear differential equations. Abstract: This is a joint work with Walter Bergweiler. Title : On some properties of linear spaces and linear operators in the case of quaternionic scalars. Abstract: In recent years the study of quaternionic linear spaces has been widely developed by mathematicians and has been widely used by physicists.
Title : Sets of bounded discrepancy for multi-dimensional irrational rotation. Abstract: Hecke, Ostrowski and Kesten characterized the intervals on the circle for which the ergodic sums of their indicator function, under an irrational rotation, stay at a bounded distance from their integral with respect to the Lebesgue measure on the circle.
Balanov, University of Texas at Dallas. Title : Three faces of equivariant degree. Abstract: Topological methods based on the usage of degree theory have proved themselves to be an important tool for qualitative studying of solutions to nonlinear differential systems including such problems as existence, uniqueness, multiplicity, bifurcation, etc. The talk is addressed to a general audience, without any special knowledge of the subject. Title : Riesz sequences and arithmetic progressions. Title : Certain problems in Fourier Analysis.
Abstract: The following problems or a part of them will be discussed. Sodin, Tel-Aviv University. Title : Entire functions of exponential type represented by pseudo-random and random Taylor series. Abstract: We study the influence the angular distribution of zeroes of the Taylor series with pseudo-random and random coefficients, and show that the distribution of zeroes is governed by certain autocorrelations of the coefficients. Agranovsky, Bar-Ilan University.
Title : Ruled common nodal surfaces. Abstract: Nodal sets are zero loci of Laplace eigenfunctions e. These angular solutions are a product of trigonometric functions , here represented as a complex exponential , and associated Legendre polynomials:. Such an expansion is valid in the ball. In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum .
The spherical harmonics are eigenfunctions of the square of the orbital angular momentum. Laplace's spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and the generator of rotations about the azimuthal axis:. These operators commute, and are densely defined self-adjoint operators on the weighted Hilbert space of functions f square-integrable with respect to the normal distribution as the weight function on R 3 :. Furthermore, L 2 is a positive operator. If Y is a joint eigenfunction of L 2 and L z , then by definition.
Furthermore, since. This polynomial is easily seen to be harmonic. Several different normalizations are in common use for the Laplace spherical harmonic functions. In acoustics  , the Laplace spherical harmonics are generally defined as this is the convention used in this article. This normalization is used in quantum mechanics because it ensures that probability is normalized, i. The disciplines of geodesy  and spectral analysis use. The magnetics  community, in contrast, uses Schmidt semi-normalized harmonics.
In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre polynomials , or to append it to the definition of the spherical harmonic functions. There is no requirement to use the Condon—Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators.
The geodesy  and magnetics communities never include the Condon—Shortley phase factor in their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials. A real basis of spherical harmonics can be defined in terms of their complex analogues by setting.
The Condon-Shortley phase convention is used here for consistency. The corresponding inverse equations are. The real spherical harmonics are sometimes known as tesseral spherical harmonics. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as. The same sine and cosine factors can be also seen in the following subsection that deals with the cartesian representation. As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics.
This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. Here, it is important to note that the real functions span the same space as the complex ones would. Essentially all the properties of the spherical harmonics can be derived from this generating function. They are, moreover, a standardized set with a fixed scale or normalization. It may be verified that this agrees with the function listed here and here. The spherical harmonics have deep and consequential properties under the operations of spatial inversion parity and rotation.
The spherical harmonics have definite parity. That is, they are either even or odd with respect to inversion about the origin. The statement of the parity of spherical harmonics is then. That is,. However, this is not the standard way of expressing this property. In the standard way one writes,. The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. Many facts about spherical harmonics such as the addition theorem that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry.
The Laplace spherical harmonics form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions. On the unit sphere, any square-integrable function can thus be expanded as a linear combination of these:. This expansion holds in the sense of mean-square convergence — convergence in L 2 of the sphere — which is to say that. This is justified rigorously by basic Hilbert space theory. For the case of orthonormalized harmonics, this gives:. The convergence of the series holds again in the same sense, but the benefit of the real expansion is that for real functions f the expansion coefficients become real.
The total power of a function f is defined in the signal processing literature as the integral of the function squared, divided by the area of its domain. Using the orthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization of Parseval's theorem here, the theorem is stated for Schmidt semi-normalized harmonics, the relationship is slightly different for orthonormal harmonics :. In a similar manner, one can define the cross-power of two functions as.
If the functions f and g have a zero mean i. It is common that the cross- power spectrum is well approximated by a power law of the form. The general technique is to use the theory of Sobolev spaces. Specifically, if. In particular, the Sobolev embedding theorem implies that f is infinitely differentiable provided that. A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. The addition theorem states . This expression is valid for both real and complex harmonics.
From this perspective, one has the following generalization to higher dimensions. Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics . The Clebsch—Gordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves.
A variety of techniques are available for doing essentially the same calculation, including the Wigner 3-jm symbol , the Racah coefficients , and the Slater integrals. Abstractly, the Clebsch—Gordan coefficients express the tensor product of two irreducible representations of the rotation group as a sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities.
When the spherical harmonic order m is zero upper-left in the figure , the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. Such spherical harmonics are a special case of zonal spherical functions. For the other cases, the functions checker the sphere, and they are referred to as tesseral. Analytic expressions for the first few orthonormalized Laplace spherical harmonics that use the Condon-Shortley phase convention:.
The classical spherical harmonics are defined as functions on the unit sphere S 2 inside three-dimensional Euclidean space. Spherical harmonics can be generalized to higher-dimensional Euclidean space R n as follows. An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of separation of variables , by solving the Sturm-Liouville problem for the spherical Laplacian.
The end result of such a procedure is .
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