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Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
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TL;DR: A direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are, so this work proves orthog onality in a different way.
Abstract: Each discrete cosine transform (DCT) uses $N$ real basis vectors whose components are cosines In the DCT-4, for example, the $j$th component of $\boldv_k$ is $\cos (j + \frac{1}{2}) (k + \frac{1}{2}) \frac{\pi}{N}$ These basis vectors are orthogonal and the transform is extremely useful in image processing If the vector $\boldx$ gives the intensities along a row of pixels, its cosine series $\sum c_k \boldv_k$ has the coefficients $c_k=(\boldx,\boldv_k)/N$ They are quickly computed from a Fast Fourier Transform But a direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are
We prove orthogonality in a different way Each DCT basis contains the eigenvectors of a symmetric "second difference" matrix By varying the boundary conditions we get the established transforms DCT-1 through DCT-4 Other combinations lead to four additional cosine transforms The type of boundary condition (Dirichlet or Neumann, centered at a meshpoint or a midpoint) determines the applications that are appropriate for each transform The centering also determines the period: $N-1$ or $N$ in the established transforms, $N-\frac{1}{2}$ or $N+ \frac{1}{2}$ in the other four The key point is that all these "eigenvectors of cosines" come from simple and familiar matrices
689 citations
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TL;DR: In this paper, a harmonic balance technique for modeling unsteady nonlinear e ows in turbomachinery is presented, which exploits the fact that many unstaidy e ow variables are periodic in time.
Abstract: A harmonic balance technique for modeling unsteady nonlinear e ows in turbomachinery is presented. The analysis exploits the fact that many unsteady e ows of interest in turbomachinery are periodic in time. Thus, the unsteady e ow conservation variables may be represented by a Fourier series in time with spatially varying coefe cients. This assumption leads to a harmonic balance form of the Euler or Navier ‐Stokes equations, which, in turn, can be solved efe ciently as a steady problem using conventional computational e uid dynamic (CFD) methods, including pseudotime time marching with local time stepping and multigrid acceleration. Thus, the method is computationally efe cient, at least one to two orders of magnitude faster than conventional nonlinear time-domain CFD simulations. Computational results for unsteady, transonic, viscous e ow in the front stage rotor of a high-pressure compressor demonstrate that even strongly nonlinear e ows can be modeled to engineering accuracy with a small number of terms retained in the Fourier series representation of the e ow. Furthermore, in some cases, e uid nonlinearities are found to be important for surprisingly small blade vibrations.
673 citations
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01 Jan 1958
TL;DR: In this paper, the authors present a monograph on generalised functions, Fourier integrals and Fourier series, intended for readers who, while accepting that a theory where each point is proved is better than one based on conjecture, nevertheless seek a treatment as elementary and free from complications as possible.
Abstract: This monograph on generalised functions, Fourier integrals and Fourier series is intended for readers who, while accepting that a theory where each point is proved is better than one based on conjecture, nevertheless seek a treatment as elementary and free from complications as possible. Little detailed knowledge of particular mathematical techniques is required; the book is suitable for advanced university students, and can be used as the basis of a short undergraduate lecture course. A valuable and original feature of the book is the use of generalised-function theory to derive a simple, widely applicable method of obtaining asymptotic expressions for Fourier transforms and Fourier coefficients.
662 citations
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TL;DR: In this article, the distribution of distribution funetions of one variable has been studied in the context of functions of bounded variation, including Fourier-Stieltjes transform.
Abstract: Distribution Funetions of One Variable. Chap. I. Functions of Bounded Variation m~d Their Fourier-Stieltjes Transforms . 8 i . F u n c t i o n s of b o u n d e d v a r i a t i o n . . . . . . . . . . . . . . . . . 9 2. F u n c t i o n s of c lass (T) . . . . . . . . . . . . . . . . . . . . . 10 3. M i n i m u m e x t r a p o l a t i o n in (T) . . . . . . . . . . . . . . . . . 13 4. A u n i q u e n e s s t h e o r e m in (T~ + Ts) . . . . . . . . . . . . . . . 19 5. D i s t r i b u t i o n f u n c t i o n s a n d t h e i r c h a r a c t e r i s t i c f u n c t i o n s . . . . . . 21 6. A u n i q u e n e s s t h e o r e m . . . . . . . . . . . . . . . . . . . . . 22 7. On t h e a p p r o a c h t o w a r d s x of t h e m o d u l u s of a c h a r a c t e r i s t i c f u n c t i o n 26
653 citations
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TL;DR: A pattern-recognition method, making use of Fourier transformations to extract features which are significant for a pattern, is described and some considerations of the technical realizability of a fast preprocessing system for reading printed text are included.
Abstract: A pattern-recognition method, making use of Fourier transformations to extract features which are significant for a pattern, is described. The ordinary Fourier coefficients are difficult to use as input to categorizers because they contain factors dependent upon size and rotation as well as an arbitrary phase angle. From these Fourier coefficients, however, other more useful features can easily be derived. By using these derived property constants, a distinction can be made between genuine shape constants and constants representing size, location, and orientation. The usefulness of the method has been tested with a computer program that was used to classify 175 samples of handprinted letters, e.g., 7 sets of the 25 letters A to Z. In this test, 98 percent were correctly recognized when a simple nonoptimized decision method was used. The last section contains some considerations of the technical realizability of a fast preprocessing system for reading printed text.
649 citations