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Published: 30 May 2022
Last updated: 31 May 2022

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1

Leindler, László. "Relations among Fourier coefficients and sum-functions." Acta Mathematica Hungarica 104, no. 1/2 (2004): 171–83. http://dx.doi.org/10.1023/b:amhu.0000034370.78464.82.

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2

Yu, Dansheng, and Songping Zhou. "On relations among Fourier coefficients and sum-functions." Studia Scientiarum Mathematicarum Hungarica 45, no. 3 (September 1, 2008): 301–19. http://dx.doi.org/10.1556/sscmath.2008.1052.

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3

Maidanik, G., and K. J. Becker. "Double-sum technique for performing a Fourier transformation." Journal of the Acoustical Society of America 101, no. 5 (May 1997): 2448–51. http://dx.doi.org/10.1121/1.418487.

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4

Chen, Chang-Pao. "L1-convergence of Fourier series." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 41, no. 3 (December 1986): 376–90. http://dx.doi.org/10.1017/s144678870003384x.

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AbstractFor an integrable function f on T, we introduce a modified partial sum and establish its L1-convergence property. The relation between the sum and L1-convergence classes is also established. As a corollary, a new L1-convergence class is obtained. It is shown that this class covers all known L1-convergence classes.
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5

Albu_Amer, Ansam Ghazi Nsaif. "Mathematical Analysis of Fourier Expansion Using Gauss Partial Sum." IOP Conference Series: Materials Science and Engineering 571 (August 8, 2019): 012033. http://dx.doi.org/10.1088/1757-899x/571/1/012033.

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6

Sankaranarayanan, A. "On a sum involving Fourier coefficients of cusp forms." Lithuanian Mathematical Journal 46, no. 4 (October 2006): 459–74. http://dx.doi.org/10.1007/s10986-006-0042-y.

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7

Alotaibi, Abdullah, and M. Mursaleen. "Applications of Hankel and Regular Matrices in Fourier Series." Abstract and Applied Analysis 2013 (2013): 1–3. http://dx.doi.org/10.1155/2013/947492.

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Recently, Alghamdi and Mursaleen (2013) used the Hankel matrix to determine the necessary and suffcient condition to find the sum of the Walsh-Fourier series. In this paper, we propose to use the Hankel matrix as well as any general nonnegative regular matrix to obtain the necessary and sufficient conditions to sum the derived Fourier series and conjugate Fourier series.
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8

Li, Bing-Zhao, Ran Tao, Tian-Zhou Xu, and Yue Wang. "The Poisson sum formulae associated with the fractional Fourier transform." Signal Processing 89, no. 5 (May 2009): 851–56. http://dx.doi.org/10.1016/j.sigpro.2008.10.030.

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9

McGuire, J. A., W. Beck, X. Wei, and Y. R. Shen. "Fourier-transform sum-frequency surface vibrational spectroscopy with femtosecond pulses." Optics Letters 24, no. 24 (December 15, 1999): 1877. http://dx.doi.org/10.1364/ol.24.001877.

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10

Shparlinski, Igor E. "Bounds on the Fourier coefficients of the weighted sum function." Information Processing Letters 103, no. 3 (July 2007): 83–87. http://dx.doi.org/10.1016/j.ipl.2007.02.011.

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11

Panzone, Pablo A. "Fourier transforms related to ζ(s)." Proceedings of the Edinburgh Mathematical Society 64, no. 2 (April 30, 2021): 200–216. http://dx.doi.org/10.1017/s0013091521000092.

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AbstractUsing some formulas of S. Ramanujan, we compute in closed form the Fourier transform of functions related to Riemann zeta function $\zeta (s)=\sum \nolimits _{n=1}^{\infty } {1}/{n^{s}}$ and other Dirichlet series.
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12

Pavelčík, F. "The heavy-atom method without Patterson. A symmetry sum function." Journal of Applied Crystallography 22, no. 2 (April 1, 1989): 181–82. http://dx.doi.org/10.1107/s0021889888010532.

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13

Yun, Beong In. "Improving Fourier Partial Sum Approximation for Discontinuous Functions Using a Weight Function." Abstract and Applied Analysis 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/1364914.

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We introduce a generalized sigmoidal transformation wm(r;x) on a given interval [a,b] with a threshold at x=r∈(a,b). Using wm(r;x), we develop a weighted averaging method in order to improve Fourier partial sum approximation for a function having a jump-discontinuity. The method is based on the decomposition of the target function into the left-hand and the right-hand part extensions. The resultant approximate function is composed of the Fourier partial sums of each part extension. The pointwise convergence of the presented method and its availability for resolving Gibbs phenomenon are proved. The efficiency of the method is shown by some numerical examples.
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14

TANG, HENGCAI. "SHIFTED CONVOLUTION SUM OF AND THE FOURIER COEFFICIENT OF HECKE–MAASS FORMS." Bulletin of the Australian Mathematical Society 92, no. 2 (June 2, 2015): 195–204. http://dx.doi.org/10.1017/s000497271500043x.

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Let $\{{\it\phi}_{j}(z):j\geq 1\}$ be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue $1/4+t_{j}^{2}$. Let ${\it\lambda}_{j}(n)$ be the $n$th Fourier coefficient of ${\it\phi}_{j}$ and $d_{3}(n)$ the divisor function of order three. In this paper, by the circle method and the Voronoi summation formula, the average value of the shifted convolution sum for $d_{3}(n)$ and ${\it\lambda}_{j}(n)$ is considered, leading to the estimate $$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leq X}d_{3}(n){\it\lambda}_{j}(n-1)\ll X^{29/30+{\it\varepsilon}}, & & \displaystyle \nonumber\end{eqnarray}$$ where the implied constant depends only on $t_{j}$ and ${\it\varepsilon}$.
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15

Zhang, Jun-Fang, and Shou-Ping Hou. "The Generalization of the Poisson Sum Formula Associated with the Linear Canonical Transform." Journal of Applied Mathematics 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/102039.

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The generalization of the classical Poisson sum formula, by replacing the ordinary Fourier transform by the canonical transformation, has been derived in the linear canonical transform sense. Firstly, a new sum formula of Chirp-periodic property has been introduced, and then the relationship between this new sum and the original signal is derived. Secondly, the generalization of the classical Poisson sum formula to the linear canonical transform sense has been obtained.
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16

Shiu, Peter. "The whole can be very much less than the sum of its parts." Mathematical Gazette 103, no. 556 (February 14, 2019): 117–27. http://dx.doi.org/10.1017/mag.2019.14.

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This Article is on the discrete Fourier transform (DFT) and the fast Fourier transform (FFT). As we shall see, FFT is a slight misnomer, causing confusion to beginners. The idiosyncratic title will be clarified in §4.Computing machines are highly efficient nowadays, and much of the efficiency is based on the use of the FFT to speed up calculations in ultrahigh precision arithmetic. The algorithm is now an indispensable tool for solving problems that involve a large amount of computation, resulting in many useful and important applications: for example, in signal processing, data compression and photo-images in general, and WiFi, mobile phones, CT scanners and MR imaging in particular.
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17

El-Azhary, I. "Exact analysis of radiation patterns using the expansion of the Fourier sum." IEEE Transactions on Antennas and Propagation 38, no. 12 (1990): 1965–67. http://dx.doi.org/10.1109/8.60987.

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18

Pei, Soo-Chang, and Kuo-Wei Chang. "Integer 2-D Discrete Fourier Transform Pairs and Eigenvectors using Ramanujan’s Sum." IEEE Signal Processing Letters 23, no. 1 (January 2016): 70–74. http://dx.doi.org/10.1109/lsp.2015.2501421.

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19

Fei, Ming-gang, and Tao Qian. "Direct Sum Decomposition of L2(Rn1) into Subspaces Invariant under Fourier Transformation." Journal of Fourier Analysis and Applications 12, no. 2 (April 2006): 145–55. http://dx.doi.org/10.1007/s00041-004-4058-6.

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20

Fu, Xiaoye, and Chun-Kit Lai. "Translational absolute continuity and Fourier frames on a sum of singular measures." Journal of Functional Analysis 274, no. 9 (May 2018): 2477–98. http://dx.doi.org/10.1016/j.jfa.2017.10.021.

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21

Knudby, Søren. "Fourier algebras of parabolic subgroups." MATHEMATICA SCANDINAVICA 120, no. 2 (May 27, 2017): 272. http://dx.doi.org/10.7146/math.scand.a-25624.

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We study the following question: given a locally compact group when does its Fourier algebra coincide with the subalgebra of the Fourier-Stieltjes algebra consisting of functions vanishing at infinity? We provide sufficient conditions for this to be the case.As an application, we show that when $P$ is the minimal parabolic subgroup in one of the classical simple Lie groups of real rank one or the exceptional such group, then the Fourier algebra of $P$ coincides with the subalgebra of the Fourier-Stieltjes algebra of $P$ consisting of functions vanishing at infinity. In particular, the regular representation of $P$ decomposes as a direct sum of irreducible representations although $P$ is not compact.
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22

Petros, Biruk. "Navier-Stokes Three Dimensional Equations Solutions Volume Three." Journal of Mathematics Research 10, no. 4 (July 25, 2018): 128. http://dx.doi.org/10.5539/jmr.v10n4p128.

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Solution of Navier-Stokes equation is found by introducing new method for solving differential equations. This new method is writing periodic scalar function in any dimensions and any dimensional vector fields as the sum of sine and cosine series with proper coefficients. The method is extension of Fourier series representation for one variable function to multi-variable functions and vector fields.Before solving Navier-Stokes equations we introduce a new technique for writing periodic scalar functions or vector fields as the sum of cosine and sine series with proper coefficients. Fourier series representation is background for our new technique.Periodic nature of initial velocity for Navier-Stokes problem helps us write the vector field in the form of cosine and sine series sum which simplify the problem.
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23

Bullen, P. S., and S. N. Mukhopadhyay. "The Integrability of Riemann Summable Trigonometric Series." Canadian Mathematical Bulletin 33, no. 3 (September 1, 1990): 273–81. http://dx.doi.org/10.4153/cmb-1990-045-2.

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AbstractIt is shown that if a trigonometric series is (R, 3), respectively (R, 4), summable then its (R, 3) sum, respectively (R, 4) sum, is James P3—, respectively P4—, integrable and that such series are Fourier series with respect to these integrals.
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24

Young, Wo-Sang. "Littlewood-Paley and Multiplier Theorems for Vilenkin-Fourier Series." Canadian Journal of Mathematics 46, no. 3 (June 1, 1994): 662–72. http://dx.doi.org/10.4153/cjm-1994-036-3.

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AbstractLet S2jf be the 2j-th partial sum of the Vilenkin-Fourier series of f ∊ L1, and set S2-1f = 0. For , we show that the ratiois contained between two bounds (independent of f) . From this we obtain the Marcinkiewicz multiplier theorem for Vilenkin-Fourier series.
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25

PETERSSON, HENRIK. "GENERALIZED ANALYTIC FUNCTIONS ON THE COUNTABLE PRODUCT AND DIRECT SUM OF THE COMPLEX NUMBERS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 04, no. 04 (December 2001): 559–67. http://dx.doi.org/10.1142/s0219025701000590.

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A classical result states that, in n variables, the space of the entire functionals can be identified with the space of exponential type functions via the Fourier–Borel transform. Thus, in this way the spaces of the entire and exponential type functions can be put in duality, the Martineau duality. We give a proof that the entire functionals, on the countable direct product and direct sum of the field of complex numbers, can be identified with exponential type functions in the same way. In other words, we show that the infinite dimensional Fourier–Borel transform defines Martineau dualities analogous to the finite dimensional case.
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26

TANG, HENGCAI. "A SHIFTED CONVOLUTION SUM OF AND THE FOURIER COEFFICIENTS OF HECKE–MAASS FORMS II." Bulletin of the Australian Mathematical Society 101, no. 3 (September 26, 2019): 401–14. http://dx.doi.org/10.1017/s000497271900100x.

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Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$. Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$th Hecke eigenvalue of $g$. Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that $$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$ This corrects and improves the result of the author [‘Shifted convolution sum of $d_{3}$ and the Fourier coefficients of Hecke–Maass forms’, Bull. Aust. Math. Soc.92 (2015), 195–204].
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27

Xiong, Xinhua. "Small values of coefficients of a half Lerch sum." International Journal of Number Theory 13, no. 09 (September 20, 2017): 2461–70. http://dx.doi.org/10.1142/s1793042117501366.

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Andrews, Dyson and Hickerson proved many interesting properties of coefficients for a Ramanujan’s [Formula: see text]-hypergeometric series by relating it to real quadratic field [Formula: see text] and using the arithmetic of [Formula: see text] to solve a conjecture of Andrews on the distributions of its Fourier coefficients. Motivated by Andrews’s conjecture, we discuss an interesting [Formula: see text]-hypergeometric series which comes from a Lerch sum and rank and crank moments for partitions and overpartitions. We give Andrews-like conjectures for its coefficients. We obtain partial results on the distributions of small values of its coefficients toward these conjectures.
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28

Wilson, Hugh R., and Jeounghoon Kim. "A model for motion coherence and transparency." Visual Neuroscience 11, no. 6 (November 1994): 1205–20. http://dx.doi.org/10.1017/s0952523800007008.

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AbstractA recent model for two-dimensional motion processing in MT has demonstrated that perceived direction can be accurately predicted by combining Fourier and non-Fourier component motion signals using a vector sum computation. The vector sum direction is computed by a neural network that weights Fourier and non-Fourier components by the cosine of the component direction relative to that of each pattern unit, after which competitive inhibition extracts the signals of the most active units. It is shown here that a minor modification of the connectivity in this network suffices to predict transitions from motion coherence to transparency under a wide range of circumstances. It is only necessary that the cosine weighting function and competitive inhibition be limited to directions within ± 120 deg of each pattern unit's preferred direction. This network responds by signaling one pattern direction for coherent motion but two distinct directions for transparent motion. Based on this, neural networks with properties of MT and MST neurons can automatically signal motion coherence or transparency. In addition, the model accurately predicts motion repulsion under transparency conditions.
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29

Zhai, Shuai. "Average behavior of Fourier coefficients of cusp forms over sum of two squares." Journal of Number Theory 133, no. 11 (November 2013): 3862–76. http://dx.doi.org/10.1016/j.jnt.2013.05.013.

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30

Magomed-Kasumov, M. G. "Peculiarities of the partial Fourier-Haar sum behavior at dyadic irrational discontinuity points." Siberian Mathematical Journal 54, no. 6 (November 2013): 1059–63. http://dx.doi.org/10.1134/s0037446613060128.

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31

Chandra, Satish. "On $ (J, p_n) $ summability of fourier series." Tamkang Journal of Mathematics 32, no. 3 (September 30, 2001): 225–30. http://dx.doi.org/10.5556/j.tkjm.32.2001.378.

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In this paper we prove the following two theorems for $ | J, p_n | $ summability of fourier series, which generalizes many previous result: Theorem 1. If $$ \Phi (t) = \int_t^{\pi} \frac{\phi (u)}{u} du = o \{ p (1- \frac{1}{t} ) \} ~~~~ (t \to 0) $$ then the Fourier series for $ t = x $ is summable $ (J, p_n) $ to sum $ s $. Theorem 2. If $$ G(t) = \int_t^{\pi} \frac{g(u)}{u} du = o \{ p(1-\frac{1}{t}) \} ~~~~ (t \to 0) $$ then the differentiated Fourier series is summable $ (J, p_n) $ to the value $ C $.
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32

Weniger, Ernst Joachim. "Comment on “Fourier transform of hydrogen-type atomic orbitals”." Canadian Journal of Physics 97, no. 12 (December 2019): 1349–60. http://dx.doi.org/10.1139/cjp-2019-0046.

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Podolsky and Pauling (Phys. Rev. 34, 109 (1929) doi: 10.1103/PhysRev.34.109 ) were the first ones to derive an explicit expression for the Fourier transform of a bound-state hydrogen eigenfunction. Yükçü and Yükçü (Can. J. Phys. 96, 724 (2018) doi: 10.1139/cjp-2017-0728 ), who were apparently unaware of the work of Podolsky and Pauling or of the numerous other earlier references on this Fourier transform, proceeded differently. They expressed a generalized Laguerre polynomial as a finite sum of powers, or equivalently, they expressed a bound-state hydrogen eigenfunction as a finite sum of Slater-type functions. This approach looks very simple, but it leads to comparatively complicated expressions that cannot match the simplicity of the classic result obtained by Podolsky and Pauling. It is, however, possible to reproduce not only Podolsky and Pauling’s formula for the bound-state hydrogen eigenfunction, but to obtain results of similar quality also for the Fourier transforms of other, closely related, functions, such as Sturmians, Lambda functions, or Guseinov’s functions, by expanding generalized Laguerre polynomials in terms of so-called reduced Bessel functions.
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33

WILLIAMS, KENNETH S. "FOURIER SERIES OF A CLASS OF ETA QUOTIENTS." International Journal of Number Theory 08, no. 04 (May 16, 2012): 993–1004. http://dx.doi.org/10.1142/s1793042112500595.

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The sum of divisors function σ(m) is defined by [Formula: see text] Let [Formula: see text] denote the upper half of the complex plane. Let η(z)[Formula: see text] be the Dedekind eta function. A class [Formula: see text] of eta quotients is given for which the Fourier series of each member of [Formula: see text] can be given explicitly. One example is [Formula: see text] where [Formula: see text]
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34

Zhuo, Zhi-Hai. "Poisson Summation Formulae Associated with the Special Affine Fourier Transform and Offset Hilbert Transform." Mathematical Problems in Engineering 2017 (2017): 1–5. http://dx.doi.org/10.1155/2017/1354129.

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This paper investigates the generalized pattern of Poisson summation formulae from the special affine Fourier transform (SAFT) and offset Hilbert transform (OHT) points of view. Several novel summation formulae are derived accordingly. Firstly, the relationship between SAFT (or OHT) and Fourier transform (FT) is obtained. Then, the generalized Poisson sum formulae are obtained based on above relationships. The novel results can be regarded as the generalizations of the classical results in several transform domains such as FT, fractional Fourier transform, and the linear canonical transform.
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35

Gustafsson, Mats. "Time-domain approach to the forward scattering sum rule." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2124 (May 26, 2010): 3579–92. http://dx.doi.org/10.1098/rspa.2009.0680.

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The forward scattering sum rule relates the extinction cross section integrated over all wavelengths with the polarizability dyadics. It is useful for deriving bounds on the interaction between scatterers and electromagnetic fields, antenna bandwidth and directivity and energy transmission through sub-wavelength apertures. The sum rule is valid for linearly polarized plane waves impinging on linear, passive and time translational invariant scattering objects in free space. Here, a time-domain approach is used to clarify the derivation and the used assumptions. The time-domain forward scattered field defines an impulse response. Energy conservation shows that this impulse response is the kernel of a passive convolution operator, which implies that the Fourier transform of the impulse response is a Herglotz function. The forward scattering sum rule is finally constructed from integral identities for Herglotz functions.
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36

Xi, Ping. "A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)." Forum Mathematicum 30, no. 4 (July 1, 2018): 1013–27. http://dx.doi.org/10.1515/forum-2017-0236.

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Abstract In this paper, we estimate the shifted convolution sum \sum_{n\geqslant 1}\lambda_{1}(1,n)\lambda_{2}(n+h)V\Big{(}\frac{n}{X}\Big{)}, where V is a smooth function with support in {[1,2]} , {1\leqslant|h|\leqslant X} , and {\lambda_{1}(1,n)} and {\lambda_{2}(n)} are the n-th Fourier coefficients of {\mathrm{SL}(3,\mathbf{Z})} and {\mathrm{SL}(2,\mathbf{Z})} Hecke–Maass cusp forms, respectively. We prove an upper bound {O(X^{\frac{21}{22}+\varepsilon})} , updating a recent result of Munshi.
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37

Tewary, V. K., R. H. Wagoner, and J. P. Hirth. "Elastic Green's function for a composite solid with a planar interface." Journal of Materials Research 4, no. 1 (February 1989): 113–23. http://dx.doi.org/10.1557/jmr.1989.0113.

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The elastic plane-strain Green's function is calculated for a general anisotropic composite solid with a plane interface and a line load parallel to the composite interface. The interface may be between two different solids or between different orientations of the same solid such as a grain boundary. The equations of elastic equilibrium are solved by the Fourier transform method. Analytical expressions are obtained for the Green's function in real as well as Fourier space. These expressions should be useful for calculations of elastic properties of a composite solid containing defects. Two sum rules are also derived for matrices which constitute the Green's function and the stress tensor. These sum rules can serve as numerical checks in detailed computer simulation calculations.
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38

Sameer, A. S., M. Yusuf, and U. L. Ukafor. "AN APPLICATION OF TIME INDEPENDENT FOURIER AMPLITUDE MODEL ON FORECASTING THE UNITED STATE POPULATION." FUDMA JOURNAL OF SCIENCES 6, no. 1 (March 31, 2022): 54–59. http://dx.doi.org/10.33003/fjs-2022-0601-881.

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This study applied the Time Independent Fourier Amplitude Model Approach to forecast the Population of the United States of America from 1790 to 2020 and beyond on a 10-year interval using Number Crunches Statistical software (NCSS). Results obtained using this methodology was compared with the results obtained in the other models: Malthusian, Logistics, and Logistics (Least Squares) Model. These models were compared using the goodness of fit (the coefficient of determination (R2) and the sum of square error (SSE)), the Akaike information criterion (AIC), Bayesian information criterion (BIC), Mean Absolute Deviation (MAD), Mean Error (ME), and Mean Sum of square Error (MSSE), Results displays that the Time Independent Fourier Amplitude Model and also is a suitable model for predicting the United States population
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39

Zhang, Zhihua. "Fourier Expansions with Polynomial Terms for Random Processes." Journal of Function Spaces 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/763075.

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Based on calculus of random processes, we present a kind of Fourier expansions with simple polynomial terms via our decomposition method of random processes. Using our method, the expectations and variances of the corresponding coefficients decay fast and partial sum approximations attain the best approximation order. Moreover, since we remove boundary effect in our decomposition of random process, these coefficients can discover the instinct frequency information of this random process. Therefore, our method has an obvious advantage over traditional Fourier expansion. These results are also new for deterministic functions.
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40

Guadalupe, José J., Mario Pérez, and Juan L. Varona. "Commutators and Analytic Dependence of Fourier-Bessel Series on (0, ∞)." Canadian Mathematical Bulletin 42, no. 2 (June 1, 1999): 198–208. http://dx.doi.org/10.4153/cmb-1999-024-1.

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AbstractIn this paper we study the boundedness of the commutators [b, Sn] where b is a BMO function and Sn denotes the n-th partial sum of the Fourier-Bessel series on (0, ∞). Perturbing the measure by exp(2b) we obtain that certain operators related to Sn depend analytically on the functional parameter b.
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41

LIN, SHUEEI-MUH. "ANALYTICAL SOLUTIONS OF BIO-HEAT CONDUCTION ON SKIN IN FOURIER AND NON-FOURIER MODELS." Journal of Mechanics in Medicine and Biology 13, no. 04 (July 7, 2013): 1350063. http://dx.doi.org/10.1142/s0219519413500632.

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In general, the transport of thermal energy in living tissue is a complex process. The analysis of the heat conduction of skin tissue is helpful for understanding of the bio-thermo-mechanical behavior of skin tissue. So far, three kinds of conduction law — (1) the Fourier model, (2) the C-V model and (3) dual-phase-lag (DPL) model — are often investigated in bio-thermal transfer process. In this study, the mathematical model of heat conduction of the skin tissue subjected to a general transient heating at the skin surface was established. The analytical solutions of these three conduction models are presented. In addition, the measure of thermal injury of the skin tissue subjected to a harmonic heating was investigated. It was found that the phenomenon of Fourier model is greatly different to those of the C-V and DPL models. Moreover, the effects of the phase lags, the heating frequency, and the heat quantity on the temperature variation and the index of thermal injury were significant. In sum, the analytical method can be used to solve the conduction problem of any one-layer tissue.
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42

Pilar Rubio Montero, M., and Eduardo García-Toraño. "Use of discrete Fourier transform to sum spectra in measurements with long counting times." Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 577, no. 3 (July 2007): 715–18. http://dx.doi.org/10.1016/j.nima.2007.04.135.

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43

Cheon, Gyeong Woo, Peter L. Gehlbach, and Jin U. Kang. "Ghost Reduction in CP-SSOCT Having Multiple References Using Fourier-Domain Shift and Sum." IEEE Photonics Technology Letters 28, no. 18 (September 15, 2016): 1972–75. http://dx.doi.org/10.1109/lpt.2016.2580588.

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44

McGuire, John A., and Y. R. Shen. "Signal and noise in Fourier-transform sum-frequency surface vibrational spectroscopy with femtosecond lasers." Journal of the Optical Society of America B 23, no. 2 (February 1, 2006): 363. http://dx.doi.org/10.1364/josab.23.000363.

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Jayashree, V., and Shaila Subbaraman. "Identification of twill grey fabric defects using DC suppressed Fourier power spectrum sum features." Textile Research Journal 82, no. 14 (April 20, 2012): 1485–97. http://dx.doi.org/10.1177/0040517511404593.

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Wei, Bin. "Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ)." International Journal of Number Theory 11, no. 01 (November 24, 2014): 39–49. http://dx.doi.org/10.1142/s1793042115500025.

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Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n>0λf(n)ϕ(n/X)e(αn), where [Formula: see text]. It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 < X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions.
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Pau, Chou-Pong, and Isiah M. Warner. "Evaluation of a Fourier-Transform-Based Pattern-Recognition Algorithm for Two-Dimensional Fluorescence Data." Applied Spectroscopy 41, no. 3 (March 1987): 496–502. http://dx.doi.org/10.1366/0003702874448904.

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A pattern-recognition algorithm for two-dimensional fluorescence data previously reported is critically evaluated. The three spectral matching criteria—sum of the absolute value of the imaginary coefficients of the frequency-domain correlation function, sum of the absolute value of the real-negative coefficients of the frequency-domain correlation function, and the intervector distance between the abbreviated Fourier transforms of two spectra—are calculated. Spectra simulated with a computer as well as data acquired with a video fluorometer are examined. Results indicate that all three parameters are sensitive to changes in peak position, peak width, relative peak height, and intensity of background noises.
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Ansari, Alireza. "On the Fourier transform of the products of M-Wright functions." Boletim da Sociedade Paranaense de Matemática 33, no. 1 (May 21, 2014): 245. http://dx.doi.org/10.5269/bspm.v33i1.22914.

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In this note, by applying the Bromwich's integral for the inverse Mellin transform we find a new integral representation for the M-Wright function $$ M_\alpha(x)=\sum _{k=0}^{\infty }\frac{(-x)^{k} }{k!\Gamma (-\alpha k+1-\alpha )},\quad \alpha=\frac{1}{2n+1}, n\in \mathbb{N},$$ and state the Fourier transform of this function. Also, using the new integral representations for the products of the M-Wright functions, we get the Fourier transform of it.
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Zhao, Wei, Shuai Feng, Yadan Wang, Yuanguo Wang, Zhihui Han, and Hu Peng. "A joint delay-and-sum and Fourier beamforming method for high frame rate ultrasound imaging." Computer Modeling in Engineering & Sciences 123, no. 1 (2020): 427–40. http://dx.doi.org/10.32604/cmes.2020.09387.

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Oktay Sh. Mukhtarov and Kadriye Aydemir. "MINIMIZATION PRINCIPLE AND GENERALIZED FOURIER SERIES FOR DISCONTINUOUS STURM-LIOUVILLE SYSTEMS IN DIRECT SUM SPACES." Journal of Applied Analysis & Computation 8, no. 5 (2018): 1511–23. http://dx.doi.org/10.11948/2018.1511.

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