Dawson's Integral and the Sampling Theorem

The Sampling Theorem and the Bandpass Theorem

existence and approximate $l^{p}$ and continuous solution of nonlinear integral equations of the hammerstein and volterra types

بسیاری از پدیده ها در جهان ما اساساً غیرخطی هستند، و توسط معادلات غیرخطی ‎‏بیان شد‎‎‏ه اند. از آنجا که ظهور کامپیوترهای رقمی با عملکرد بالا، حل مسایل خطی را آسان تر می کند. با این حال، به طور کلی به دست آوردن جوابهای دقیق از مسایل غیرخطی دشوار است. روش عددی، به طور کلی محاسبه پیچیده مسایل غیرخطی را اداره می کند. با این حال، دادن نقاط به یک منحنی و به دست آوردن منحنی کامل که اغلب پرهزینه و ...

Cauchy Integral Theorem

where we use the notation dxI for (1.4) dxI = dxi1 ∧ dxi2 ∧ ... ∧ dxik for I = {i1, i2, ..., ik} with i1 < i2 < ... < ik. So ΩX is a free module over C ∞(X) generated by dxI . Obviously, Ω k X = 0 for k > n and ⊕ΩX is a graded ring (noncommutative without multiplicative identity) with multiplication defined by the wedge product (1.5) ∧ : (ω1, ω2)→ ω1 ∧ ω2. Note that (1.6) ω1 ∧ ω2 = (−1)12ω2 ∧ ω...

3: the Shannon Sampling Theorem

In this final set of notes, we would like to end where we began. In the first set of notes, we mentioned that mathematics could be used to show how CDs and MP3s can reproduce our music well, even though the file sizes of MP3s are relatively small. In this set of notes, we attempt to describe how this is possible. Surprisingly, the exponential function will play a key role. We assume familiarity...

The Shannon Sampling Theorem and Its Implications

We say that a function f (with values in either R or C) is supported on a set A if it is zero on the complement of this set. The support of f , which we denote by supp(g), is the minimal closed set on which f is supported, equivalently, it is the closure of the set on which f is non-zero. We note that if f ∈ L1(R) is real-valued then the support of f̂ is symmetric around zero (since the real par...