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Analytic function in mathematics
analysis, a lacunary function or series is an analytic function containing many zero coefficients in its basis decomposition. A series can be lacunary in the
Lacunary_function
Number not in the image of a given function
In complex analysis, a subfield of mathematics, a lacunary value or gap of a complex-valued function defined on a subset of the complex plane is a complex
Lacunary_value
Extension of the domain of an analytic function (mathematics)
{L}}_{c}(z^{c^{m}}),\forall |z|<1.} For any positive natural numbers c, the lacunary series function diverges at z = 1 {\displaystyle z=1} . We consider the question
Analytic_continuation
continuation of lacunary power series. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on
Hadamard_gap_theorem
Function that is holomorphic on the whole complex plane
called a lacunary value of the function. The possibility of a lacunary value is illustrated by the exponential function, which never takes on the value
Entire_function
Statistical model
violate condition (*). An example found by Marcus and Shepp is a random lacunary Fourier series X t = ∑ n = 1 ∞ c n ( ξ n cos λ n t + η n sin λ n t
Gaussian_process
Mathematical theorem
continuation of lacunary power series. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on
Fabry_gap_theorem
Topics referred to by the same term
band Lacunary function, an analytic function in mathematics Lacunarity, a mathematical measure of the extent that a pattern contains gaps Lacunary polynomial
Lacuna
Concept in complex analysis
holomorphy. A particular example is the open unit disk. Define the lacunary function ∑ n = 0 ∞ z 2 n {\displaystyle \textstyle \sum _{n=0}^{\infty }z^{2^{n}}}
Domain_of_holomorphy
Fundamental theorem in probability theory and statistics
the original on 2017-10-31. Gaposhkin, V. F. (1966). "Lacunary series and independent functions". Russian Mathematical Surveys. 21 (6): 1–82. Bibcode:1966RuMaS
Central_limit_theorem
Mathematical function
… , X n ) = P lacunary ( X 1 , … , X n ) + X 1 ⋯ X n ⋅ Q ( X 1 , … , X n ) . {\displaystyle P(X_{1},\ldots ,X_{n})=P_{\text{lacunary}}(X_{1},\ldots
Elementary symmetric polynomial
Elementary_symmetric_polynomial
Topics referred to by the same term
theorem may refer to: The Hadamard gap theorem on lacunary functions The Fabry gap theorem on lacunary functions Pólya's gap theorem on power series The Weierstrass
Gap_theorem_(disambiguation)
In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a polynomial that has far fewer terms than its degree and number of variables
Sparse_polynomial
Theorem about the range of an analytic function
single point, this point is called a lacunary value of the function. Great Picard's Theorem: If an analytic function f {\textstyle f} has an essential singularity
Picard_theorem
Point where a mathematical object behaves irregularly
such a set include the lacunary series and the prime zeta function. Branch points are generally the result of a multi-valued function, such as z {\displaystyle
Singularity_(mathematics)
Romanian-American mathematician (1935–2025)
showed that every lacunary sequence of integers is in fact a "bad universal sequence" in L 1 {\displaystyle L_{1}} . Thus lacunary sequences are ‘canonical’
Alexandra_Bellow
Series representing modular forms
ISSN 0264-9381. S2CID 250864942. Mertens, Michael H.; Rolen, Larry (2015). "Lacunary recurrences for Eisenstein series". Research in Number Theory. 1 9. arXiv:1504
Eisenstein_series
Canadian–American mathematician
1090/S0002-9939-1950-0036346-7. Hirschman, I. I.; Jenkins, J. A. (1950). "On lacunary Dirichlet series". Proceedings of the American Mathematical Society. 1
James_Allister_Jenkins
Sumerian myth
ultimately locates him, resulting in a mitigation of Dumuzi's fate. The lacunary text allows for several interpretations, contingent on how it is read.
Descent of Inanna into the Underworld
Descent_of_Inanna_into_the_Underworld
American mathematician
Figà-Talamanca, Alessandro; Rider, Daniel (1966). "A theorem of Littlewood and lacunary series for compact groups". Pacific Journal of Mathematics. 16 (3): 505–514
Daniel_Rider
American mathematician
Reinhold-Larsson, Karin; Wierdl, Máté (1996). "The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers and related matters". Ergodic
Roger_Jones_(mathematician)
Unsolved problem in mathematics
02.002. S2CID 26415692. —— (May 2018). "The lonely runner problem for lacunary sequences". Discrete Mathematics. 341 (5): 1301–1306. doi:10.1016/j.disc
Lonely_runner_conjecture
Polish-French mathematician (1899–1983)
harmonic analysis, with an emphasis on Dirichlet series, lacunary series, and entire functions. Rather than a Bourbakist, he is perhaps more accurately
Szolem_Mandelbrojt
Open conjecture in multiplicative Diophantine approximation
Eduard Stefanescu subsequently sharpened dispersion estimates for dilated lacunary sequences and obtained corresponding improvements in inhomogeneous multiplicative
Littlewood_conjecture
Icelandic mathematician (1927–2023)
1090/S0002-9947-1957-0095428-5. hdl:1721.1/26691. MR 0095428. Helgason, Sigurdur (1958). "Lacunary Fourier series on noncommutative groups". Proceedings of the American Mathematical
Sigurður Helgason (mathematician)
Sigurður_Helgason_(mathematician)
Polynomial whose coefficients are all 1 or −1
Rodgers, Brad (2017). "On the distribution of Rudin–Shapiro polynomials and lacunary walks on S U ( 2 ) {\displaystyle SU(2)} ". Advances in Mathematics. 320:
Littlewood_polynomial
American mathematician
doi:10.1090/S0002-9939-1950-0032705-7. ——; Jenkins, J. A. (1950). "On lacunary Dirichlet series". Proceedings of the American Mathematical Society. 1
Isidore_Isaac_Hirschman_Jr.
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