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Sinc function
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<h2 id="properties">Properties</h2> <p>The <a href="/facts/Zero_crossing/g70XI9o6">zero crossings</a> of the unnormalized sinc are at non-zero integer multiples of π, while zero crossings of the normalized sinc occur at non-zero integers. </p><p>The local maxima and minima of the unnormalized sinc correspond to its intersections with the <a href="/facts/Cosine/czej7ur0">cosine</a> function. That is, sin(<i>ξ</i>)/<i>ξ</i> = cos(<i>ξ</i>) for all points ξ where the derivative of sin(<i>x</i>)/<i>x</i> is zero and thus a local extremum is reached. This follows from the derivative of the sinc function: d d x sinc ( x ) = { cos ( x ) − sinc ( x ) x , x ≠ 0 0 , x = 0 . {\displaystyle {\frac {d}{dx}}\operatorname {sinc} (x)={\begin{cases}{\dfrac {\cos(x)-\operatorname {sinc} (x)}{x}},&x\neq 0\\0,&x=0\end{cases}}.} </p><p>The first few terms of the infinite series for the x coordinate of the n-th extremum with positive x coordinate are x n = q − q − 1 − 2 3 q − 3 − 13 15 q − 5 − 146 105 q − 7 − ⋯ , {\displaystyle x_{n}=q-q^{-1}-{\frac {2}{3}}q^{-3}-{\frac {13}{15}}q^{-5}-{\frac {146}{105}}q^{-7}-\cdots ,} where q = ( n + 1 2 ) π , {\displaystyle q=\left(n+{\frac {1}{2}}\right)\pi ,} and where odd n lead to a local minimum, and even n to a local maximum. Because of symmetry around the y axis, there exist extrema with x coordinates −<i>xn</i>. In addition, there is an absolute maximum at <i>ξ</i>0 = (0, 1). </p><p>The normalized sinc function has a simple representation as the <a href="/facts/Infinite_product/qaaNh5PM">infinite product</a>: sin ( π x ) π x = ∏ n = 1 ∞ ( 1 − x 2 n 2 ) {\displaystyle {\frac {\sin(\pi x)}{\pi x}}=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}}}\right)} </p> <p>and is related to the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> Γ(<i>x</i>) through <a href="/facts/Euler%27s_reflection_formula/aKUhTbH0">Euler's reflection formula</a>: sin ( π x ) π x = 1 Γ ( 1 + x ) Γ ( 1 − x ) . {\displaystyle {\frac {\sin(\pi x)}{\pi x}}={\frac {1}{\Gamma (1+x)\Gamma (1-x)}}.} </p><p><a href="/facts/Euler/z2fsbjgj">Euler</a> discovered<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> that sin ( x ) x = ∏ n = 1 ∞ cos ( x 2 n ) , {\displaystyle {\frac {\sin(x)}{x}}=\prod _{n=1}^{\infty }\cos \left({\frac {x}{2^{n}}}\right),} and because of the product-to-sum identity<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <p> ∏ n = 1 k cos ( x 2 n ) = 1 2 k − 1 ∑ n = 1 2 k − 1 cos ( n − 1 / 2 2 k − 1 x ) , ∀ k ≥ 1 , {\displaystyle \prod _{n=1}^{k}\cos \left({\frac {x}{2^{n}}}\right)={\frac {1}{2^{k-1}}}\sum _{n=1}^{2^{k-1}}\cos \left({\frac {n-1/2}{2^{k-1}}}x\right),\quad \forall k\geq 1,} Euler's product can be recast as a sum sin ( x ) x = lim N → ∞ 1 N ∑ n = 1 N cos ( n − 1 / 2 N x ) . {\displaystyle {\frac {\sin(x)}{x}}=\lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}\cos \left({\frac {n-1/2}{N}}x\right).} </p><p>The <a href="/facts/Continuous_Fourier_transform/XXhKJtc8">continuous Fourier transform</a> of the normalized sinc (to ordinary frequency) is <a href="/facts/Rectangular_function/cVm8QG4F">rect</a>(<i>f</i>): ∫ − ∞ ∞ sinc ( t ) e − i 2 π f t d t = rect ( f ) , {\displaystyle \int _{-\infty }^{\infty }\operatorname {sinc} (t)\,e^{-i2\pi ft}\,dt=\operatorname {rect} (f),} where the <a href="/facts/Rectangular_function/cVm8QG4F">rectangular function</a> is 1 for argument between −1/2 and 1/2, and zero otherwise. This corresponds to the fact that the <a href="/facts/Sinc_filter/1H6q0F5t">sinc filter</a> is the ideal (<a href="/facts/Brick-wall_filter/1H6q0F5t">brick-wall</a>, meaning rectangular frequency response) <a href="/facts/Low-pass_filter/QFsUjw7h">low-pass filter</a>. </p><p>This Fourier integral, including the special case ∫ − ∞ ∞ sin ( π x ) π x d x = rect ( 0 ) = 1 {\displaystyle \int _{-\infty }^{\infty }{\frac {\sin(\pi x)}{\pi x}}\,dx=\operatorname {rect} (0)=1} is an <a href="/facts/Improper_integral/1YLuane0">improper integral</a> (see <a href="/facts/Dirichlet_integral/fFo9lHcb">Dirichlet integral</a>) and not a convergent <a href="/facts/Lebesgue_integral/f4AgvC6x">Lebesgue integral</a>, as ∫ − ∞ ∞ | sin ( π x ) π x | d x = + ∞ . {\displaystyle \int _{-\infty }^{\infty }\left|{\frac {\sin(\pi x)}{\pi x}}\right|\,dx=+\infty .} </p><p>The normalized sinc function has properties that make it ideal in relationship to <a href="/facts/Interpolation/8PyetX4f">interpolation</a> of <a href="/facts/Sampling_(signal_processing)/xxck7aLN">sampled</a> <a href="/facts/Bandlimited/LrYmxzlA">bandlimited</a> functions: </p> <ul><li>It is an interpolating function, i.e., sinc(0) = 1, and sinc(<i>k</i>) = 0 for nonzero <a href="/facts/Number/auQSAE2m">integer</a> <i>k</i>.</li> <li>The functions <i>xk</i>(<i>t</i>) = sinc(<i>t</i> − <i>k</i>) (k integer) form an <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> for <a href="/facts/Bandlimited/LrYmxzlA">bandlimited</a> functions in the <a href="/facts/Lp_space/gbad0Rv1">function space</a> <i>L</i>2(R), with highest angular frequency <i>ω</i>H = π (that is, highest cycle frequency <i>f</i>H = 1/2).</li></ul> <p>Other properties of the two sinc functions include: </p> <ul><li>The unnormalized sinc is the zeroth-order spherical <a href="/facts/Bessel_function/hSyvFPqC">Bessel function</a> of the first kind, <i>j</i>0(<i>x</i>). The normalized sinc is <i>j</i>0(π<i>x</i>).</li> <li>where Si(<i>x</i>) is the <a href="/facts/Sine_integral/JB75EqnB">sine integral</a>, ∫ 0 x sin ( θ ) θ d θ = Si ( x ) . {\displaystyle \int _{0}^{x}{\frac {\sin(\theta )}{\theta }}\,d\theta =\operatorname {Si} (x).} </li> <li><i>λ</i> sinc(<i>λx</i>) (not normalized) is one of two linearly independent solutions to the linear <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equation</a> x d 2 y d x 2 + 2 d y d x + λ 2 x y = 0. {\displaystyle x{\frac {d^{2}y}{dx^{2}}}+2{\frac {dy}{dx}}+\lambda ^{2}xy=0.} The other is cos(<i>λx</i>)/<i>x</i>, which is not bounded at <i>x</i> = 0, unlike its sinc function counterpart.</li> <li>Using normalized sinc, ∫ − ∞ ∞ sin 2 ( θ ) θ 2 d θ = π ⇒ ∫ − ∞ ∞ sinc 2 ( x ) d x = 1 , {\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{2}(\theta )}{\theta ^{2}}}\,d\theta =\pi \quad \Rightarrow \quad \int _{-\infty }^{\infty }\operatorname {sinc} ^{2}(x)\,dx=1,} </li> <li> ∫ − ∞ ∞ sin ( θ ) θ d θ = ∫ − ∞ ∞ ( sin ( θ ) θ ) 2 d θ = π . {\displaystyle \int _{-\infty }^{\infty }{\frac {\sin(\theta )}{\theta }}\,d\theta =\int _{-\infty }^{\infty }\left({\frac {\sin(\theta )}{\theta }}\right)^{2}\,d\theta =\pi .} </li> <li> ∫ − ∞ ∞ sin 3 ( θ ) θ 3 d θ = 3 π 4 . {\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{3}(\theta )}{\theta ^{3}}}\,d\theta ={\frac {3\pi }{4}}.} </li> <li> ∫ − ∞ ∞ sin 4 ( θ ) θ 4 d θ = 2 π 3 . {\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{4}(\theta )}{\theta ^{4}}}\,d\theta ={\frac {2\pi }{3}}.} </li> <li>The following improper integral involves the (not normalized) sinc function: ∫ 0 ∞ d x x n + 1 = 1 + 2 ∑ k = 1 ∞ ( − 1 ) k + 1 ( k n ) 2 − 1 = 1 sinc ( π n ) . {\displaystyle \int _{0}^{\infty }{\frac {dx}{x^{n}+1}}=1+2\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{(kn)^{2}-1}}={\frac {1}{\operatorname {sinc} ({\frac {\pi }{n}})}}.} </li></ul> <h2 id="relationship-to-the-dirac-delta-distribution">Relationship to the Dirac delta distribution</h2> <p>The normalized sinc function can be used as a <i><a href="/facts/Dirac_delta_function/V8PjRP8T">nascent delta function</a></i>, meaning that the following <a href="/facts/Weak_convergence_(Hilbert_space)/7aQaCX50">weak limit</a> holds: </p><p> lim a → 0 sin ( π x a ) π x = lim a → 0 1 a sinc ( x a ) = δ ( x ) . {\displaystyle \lim _{a\to 0}{\frac {\sin \left({\frac {\pi x}{a}}\right)}{\pi x}}=\lim _{a\to 0}{\frac {1}{a}}\operatorname {sinc} \left({\frac {x}{a}}\right)=\delta (x).} </p><p>This is not an ordinary limit, since the left side does not converge. Rather, it means that </p><p> lim a → 0 ∫ − ∞ ∞ 1 a sinc ( x a ) φ ( x ) d x = φ ( 0 ) {\displaystyle \lim _{a\to 0}\int _{-\infty }^{\infty }{\frac {1}{a}}\operatorname {sinc} \left({\frac {x}{a}}\right)\varphi (x)\,dx=\varphi (0)} </p><p>for every <a href="/facts/Schwartz_space/B035faw7">Schwartz function</a>, as can be seen from the <a href="/facts/Fourier_inversion_theorem/aM1aMkf0">Fourier inversion theorem</a>. In the above expression, as <i>a</i> → 0, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/π<i>x</i>, regardless of the value of a. </p><p>This complicates the informal picture of <i>δ</i>(<i>x</i>) as being zero for all x except at the point <i>x</i> = 0, and illustrates the problem of thinking of the delta function as a function rather than as a distribution. A similar situation is found in the <a href="/facts/Gibbs_phenomenon/JCXxWvcD">Gibbs phenomenon</a>. </p> <h2 id="summation">Summation</h2> <p>All sums in this section refer to the unnormalized sinc function. </p><p>The sum of sinc(<i>n</i>) over integer n from 1 to ∞ equals π − 1/2: </p><p> ∑ n = 1 ∞ sinc ( n ) = sinc ( 1 ) + sinc ( 2 ) + sinc ( 3 ) + sinc ( 4 ) + ⋯ = π − 1 2 . {\displaystyle \sum _{n=1}^{\infty }\operatorname {sinc} (n)=\operatorname {sinc} (1)+\operatorname {sinc} (2)+\operatorname {sinc} (3)+\operatorname {sinc} (4)+\cdots ={\frac {\pi -1}{2}}.} </p><p>The sum of the squares also equals π − 1/2:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p><p> ∑ n = 1 ∞ sinc 2 ( n ) = sinc 2 ( 1 ) + sinc 2 ( 2 ) + sinc 2 ( 3 ) + sinc 2 ( 4 ) + ⋯ = π − 1 2 . {\displaystyle \sum _{n=1}^{\infty }\operatorname {sinc} ^{2}(n)=\operatorname {sinc} ^{2}(1)+\operatorname {sinc} ^{2}(2)+\operatorname {sinc} ^{2}(3)+\operatorname {sinc} ^{2}(4)+\cdots ={\frac {\pi -1}{2}}.} </p><p>When the signs of the <a href="/facts/Addend/apFWJBFB">addends</a> alternate and begin with +, the sum equals 1/2: ∑ n = 1 ∞ ( − 1 ) n + 1 sinc ( n ) = sinc ( 1 ) − sinc ( 2 ) + sinc ( 3 ) − sinc ( 4 ) + ⋯ = 1 2 . {\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}\,\operatorname {sinc} (n)=\operatorname {sinc} (1)-\operatorname {sinc} (2)+\operatorname {sinc} (3)-\operatorname {sinc} (4)+\cdots ={\frac {1}{2}}.} </p><p>The alternating sums of the squares and cubes also equal 1/2:<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> ∑ n = 1 ∞ ( − 1 ) n + 1 sinc 2 ( n ) = sinc 2 ( 1 ) − sinc 2 ( 2 ) + sinc 2 ( 3 ) − sinc 2 ( 4 ) + ⋯ = 1 2 , {\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}\,\operatorname {sinc} ^{2}(n)=\operatorname {sinc} ^{2}(1)-\operatorname {sinc} ^{2}(2)+\operatorname {sinc} ^{2}(3)-\operatorname {sinc} ^{2}(4)+\cdots ={\frac {1}{2}},} </p><p> ∑ n = 1 ∞ ( − 1 ) n + 1 sinc 3 ( n ) = sinc 3 ( 1 ) − sinc 3 ( 2 ) + sinc 3 ( 3 ) − sinc 3 ( 4 ) + ⋯ = 1 2 . {\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}\,\operatorname {sinc} ^{3}(n)=\operatorname {sinc} ^{3}(1)-\operatorname {sinc} ^{3}(2)+\operatorname {sinc} ^{3}(3)-\operatorname {sinc} ^{3}(4)+\cdots ={\frac {1}{2}}.} </p> <h2 id="series-expansion">Series expansion</h2> <p>The <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> of the unnormalized sinc function can be obtained from that of the sine (which also yields its value of 1 at <i>x</i> = 0): sin x x = ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n + 1 ) ! = 1 − x 2 3 ! + x 4 5 ! − x 6 7 ! + ⋯ {\displaystyle {\frac {\sin x}{x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n+1)!}}=1-{\frac {x^{2}}{3!}}+{\frac {x^{4}}{5!}}-{\frac {x^{6}}{7!}}+\cdots } </p><p>The series converges for all x. The normalized version follows easily: sin π x π x = 1 − π 2 x 2 3 ! + π 4 x 4 5 ! − π 6 x 6 7 ! + ⋯ {\displaystyle {\frac {\sin \pi x}{\pi x}}=1-{\frac {\pi ^{2}x^{2}}{3!}}+{\frac {\pi ^{4}x^{4}}{5!}}-{\frac {\pi ^{6}x^{6}}{7!}}+\cdots } </p><p><a href="/facts/Leonhard_Euler/z2fsbjgj">Euler</a> famously compared this series to the expansion of the infinite product form to solve the <a href="/facts/Basel_problem/2PManmlC">Basel problem</a>. </p> <h2 id="higher-dimensions">Higher dimensions</h2> <p>The product of 1-D sinc functions readily provides a <a href="/facts/Multivariable_calculus/PEs5SgUF">multivariate</a> sinc function for the square Cartesian grid (<a href="/facts/Lattice_graph/4EHMt1b6">lattice</a>): sincC(<i>x</i>, <i>y</i>) = sinc(<i>x</i>) sinc(<i>y</i>), whose <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> is the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of a square in the frequency space (i.e., the brick wall defined in 2-D space). The sinc function for a non-Cartesian <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice</a> (e.g., <a href="/facts/Hexagonal_lattice/p6Qtm7FB">hexagonal lattice</a>) is a function whose <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> is the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of the <a href="/facts/Brillouin_zone/V8unhAAl">Brillouin zone</a> of that lattice. For example, the sinc function for the hexagonal lattice is a function whose <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> is the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of the unit hexagon in the frequency space. For a non-Cartesian lattice this function can not be obtained by a simple tensor product. However, the explicit formula for the sinc function for the <a href="/facts/Hexagonal_lattice/p6Qtm7FB">hexagonal</a>, <a href="/facts/Body-centered_cubic/92Co7BKM">body-centered cubic</a>, <a href="/facts/Face-centered_cubic/92Co7BKM">face-centered cubic</a> and other higher-dimensional lattices can be explicitly derived<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> using the geometric properties of Brillouin zones and their connection to <a href="/facts/Zonohedron/w16kH9H3">zonotopes</a>. </p><p>For example, a <a href="/facts/Hexagonal_lattice/p6Qtm7FB">hexagonal lattice</a> can be generated by the (integer) <a href="/facts/Linear_span/vPjclEtb">linear span</a> of the vectors u 1 = [ 1 2 3 2 ] and u 2 = [ 1 2 − 3 2 ] . {\displaystyle \mathbf {u} _{1}={\begin{bmatrix}{\frac {1}{2}}\\{\frac {\sqrt {3}}{2}}\end{bmatrix}}\quad {\text{and}}\quad \mathbf {u} _{2}={\begin{bmatrix}{\frac {1}{2}}\\-{\frac {\sqrt {3}}{2}}\end{bmatrix}}.} </p><p>Denoting ξ 1 = 2 3 u 1 , ξ 2 = 2 3 u 2 , ξ 3 = − 2 3 ( u 1 + u 2 ) , x = [ x y ] , {\displaystyle {\boldsymbol {\xi }}_{1}={\tfrac {2}{3}}\mathbf {u} _{1},\quad {\boldsymbol {\xi }}_{2}={\tfrac {2}{3}}\mathbf {u} _{2},\quad {\boldsymbol {\xi }}_{3}=-{\tfrac {2}{3}}(\mathbf {u} _{1}+\mathbf {u} _{2}),\quad \mathbf {x} ={\begin{bmatrix}x\\y\end{bmatrix}},} one can derive<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> the sinc function for this hexagonal lattice as sinc H ( x ) = 1 3 ( cos ( π ξ 1 ⋅ x ) sinc ( ξ 2 ⋅ x ) sinc ( ξ 3 ⋅ x ) + cos ( π ξ 2 ⋅ x ) sinc ( ξ 3 ⋅ x ) sinc ( ξ 1 ⋅ x ) + cos ( π ξ 3 ⋅ x ) sinc ( ξ 1 ⋅ x ) sinc ( ξ 2 ⋅ x ) ) . {\displaystyle {\begin{aligned}\operatorname {sinc} _{\text{H}}(\mathbf {x} )={\tfrac {1}{3}}{\big (}&\cos \left(\pi {\boldsymbol {\xi }}_{1}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{2}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{3}\cdot \mathbf {x} \right)\\&{}+\cos \left(\pi {\boldsymbol {\xi }}_{2}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{3}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{1}\cdot \mathbf {x} \right)\\&{}+\cos \left(\pi {\boldsymbol {\xi }}_{3}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{1}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{2}\cdot \mathbf {x} \right){\big )}.\end{aligned}}} </p><p>This construction can be used to design <a href="/facts/Lanczos_window/uup50eym">Lanczos window</a> for general multidimensional lattices.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p> <h2 id="sinhc">Sinhc</h2> <p>Some authors, by analogy, define the hyperbolic sine cardinal function.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p> s i n h c ( x ) = { sinh ( x ) x , if x ≠ 0 1 , if x = 0 {\displaystyle \mathrm {sinhc} (x)={\begin{cases}{\displaystyle {\frac {\sinh(x)}{x}},}&{\text{if }}x\neq 0\\{\displaystyle 1,}&{\text{if }}x=0\end{cases}}} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Anti-aliasing_filter/PqhNr0FQ">Anti-aliasing filter</a> – Mathematical transformation reducing the damage caused by aliasing</li> <li><a href="/facts/Borwein_integral/xlojGy2b">Borwein integral</a> – Type of mathematical integrals</li> <li><a href="/facts/Dirichlet_integral/fFo9lHcb">Dirichlet integral</a> – Integral of sin(x)/x from 0 to infinity.</li> <li><a href="/facts/Lanczos_resampling/uup50eym">Lanczos resampling</a> – Application of a mathematical formula</li> <li><a href="/facts/List_of_mathematical_functions/bua9ERtP">List of mathematical functions</a></li> <li><a href="/facts/Shannon_wavelet/TsvKFNQq">Shannon wavelet</a></li> <li><a href="/facts/Sinc_filter/1H6q0F5t">Sinc filter</a> – Ideal low-pass filter or averaging filter</li> <li><a href="/facts/Sinc_numerical_methods/HqbChbkm">Sinc numerical methods</a></li> <li><a href="/facts/Trigonometric_functions_of_matrices/Fh3aJjou">Trigonometric functions of matrices</a> – Important functions in solving differential equations</li> <li><a href="/facts/Trigonometric_integral/JB75EqnB">Trigonometric integral</a> – Special function defined by an integral</li> <li><a href="/facts/Whittaker%E2%80%93Shannon_interpolation_formula/t9ETvjKc">Whittaker–Shannon interpolation formula</a> – Signal (re-)construction algorithm</li> <li><a href="/facts/Winkel_tripel_projection/lrp6vHsl">Winkel tripel projection</a> – Pseudoazimuthal compromise map projection (cartography)</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/SincFunction.html">"Sinc Function"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), "Numerical methods", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.. <a href="978-0-521-19225-5" target="_blank">978-0-521-19225-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Singh, R. P.; Sapre, S. D. (2008). Communication Systems, 2E (illustrated ed.). Tata McGraw-Hill Education. p. 15. ISBN 978-0-07-063454-1. Extract of page 15 <a href="978-0-07-063454-1" target="_blank">978-0-07-063454-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Weisstein, Eric W. "Sinc Function". mathworld.wolfram.com. Retrieved 2023-06-07. <a href="https://mathworld.wolfram.com/" target="_blank">https://mathworld.wolfram.com/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Merca, Mircea (2016-03-01). "The cardinal sine function and the Chebyshev–Stirling numbers". 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