Fourier Series and . . . Generalized . . . Generalized Trig . . . Physical Meaning of . . . How to Explain the Scale Invariance Empirical Success of Our Idea Our Idea Leads . . . Generalized Trigonometric Conclusion Acknowledgments Functions in Processing Home Page Discontinuous Signals Title Page ◭◭ ◮◮ Pedro Barragan Olague and Vladik Kreinovich ◭ ◮ Department of Computer Science, University of Texas at El Paso Page 1 of 13 El Paso, TX 79968, USA pabarraganolague@miners.utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Fourier Series and . . . Generalized . . . 1. Fourier Series and Their Limitations: A Brief Generalized Trig . . . Reminder Physical Meaning of . . . • Isaac Newton: a prism decomposes each light into Scale Invariance lights of different colors. Our Idea Our Idea Leads . . . • Monochromatic light is a sinusoid Conclusion x ( t ) = A · sin( ω · t + ϕ ) . Acknowledgments • So, any signal can be represented as Home Page n Title Page � x ( t ) = A i · sin( ω i · t + ϕ i ) . ◭◭ ◮◮ i =1 • This Fourier representation helps in solving many ◭ ◮ physics-related differential equations; however: Page 2 of 13 – if we represent a discontinuous signal as a sum of Go Back sinusoids, Full Screen – we get large oscillations near the discontinuity ( Gibbs phenomenon ). Close Quit
Fourier Series and . . . Generalized . . . 2. Fourier Series and Their Limitations (cont-d) Generalized Trig . . . • We can avoid Gibbs phenomenon is we use a linear Physical Meaning of . . . combination of discontinuous functions. Scale Invariance Our Idea • For example, we can use piecewise constant functions Our Idea Leads . . . such as Haar wavelets. Conclusion • However, the resulting representation is not very com- Acknowledgments putationally efficient for smooth signals. Home Page • We need a representation which is efficient both for Title Page smooth and for discontinuous signals. ◭◭ ◮◮ ◭ ◮ Page 3 of 13 Go Back Full Screen Close Quit
Fourier Series and . . . Generalized . . . 3. Generalized Trigonometric Functions: A Suc- Generalized Trig . . . cessful Semi-Heuristic Approach Physical Meaning of . . . • A sinusoid can be defined as a function which is inverse Scale Invariance dt � to the integral (1 − t 2 ) 1 / 2 , expanded by periodicity. Our Idea Our Idea Leads . . . • It is reasonable to consider periodic extension of an Conclusion inverse function to a more general integral Acknowledgments dt Home Page � (1 − t p ) 1 /q . Title Page ◭◭ ◮◮ • The derivative of this generalized function is no longer everywhere continuous. ◭ ◮ • The farther p and q from the value 2, the larger this Page 4 of 13 discontinuity. Go Back • Empirically, these functions are good approximations Full Screen both for smooth and for discontinuous signals. Close Quit
Fourier Series and . . . Generalized . . . 4. Generalized Trig Functions: Challenge Generalized Trig . . . • However, so far, there have been no convincing theo- Physical Meaning of . . . retical explanation for this success. Scale Invariance Our Idea • In principle, we can think of many generalizations of Our Idea Leads . . . trigonometric functions. Conclusion • It is not clear whey namely this generalization is em- Acknowledgments pirically successful. Home Page • This absence of theoretical explanation prevents the Title Page wider use of this technique: ◭◭ ◮◮ – the users are reluctant to use it, ◭ ◮ – since they are not sure that the empirical success Page 5 of 13 so far is not an artifact. Go Back • In this talk, we provide a physics-motivated theoretical Full Screen explanation for this empirical success. Close Quit
Fourier Series and . . . Generalized . . . 5. Physical Meaning of Sinusoids: Reminder Generalized Trig . . . • Sinusoidal signals are frequently observed in nature, Physical Meaning of . . . because they correspond to simple oscillations. Scale Invariance Our Idea • They correspond to situations in which the potential energy E pot is equal to E pot = 1 Our Idea Leads . . . 2 · c · x 2 . Conclusion • In Newtonian mechanics, the kinetic energy is equal to Acknowledgments E kin = 1 Home Page x ) 2 , so the overall energy is 2 · m · ( ˙ Title Page E = E pot + E kin = 1 2 · c · x 2 + 1 x ) 2 . 2 · m · ( ˙ ◭◭ ◮◮ ◭ ◮ • Sinusoidal oscillations correspond to the idealized case Page 6 of 13 when we can ignore the friction: Go Back 1 2 · c · x 2 + 1 x ) 2 = E 0 = const . 2 · m · ( ˙ Full Screen Close Quit
Fourier Series and . . . Generalized . . . 6. Physical Meaning of Sinusoids (cont-d) Generalized Trig . . . • Once we know the coordinate x , we can determine ˙ x Physical Meaning of . . . as Scale Invariance √ 2 E 0 − c · x 2 x = dx Our Idea ˙ dt = . √ m Our Idea Leads . . . • This equation can be simplified if we separate the vari- Conclusion ables: Acknowledgments dx √ m · Home Page √ 2 E 0 − c · x 2 = dt. Title Page • In appropriately selected units of time and x , we have ◭◭ ◮◮ dx dx � dt = 1 − x 2 , thus t = 1 − x 2 . ◭ ◮ √ √ Page 7 of 13 • The desired dependence x ( t ) of x on t is the inverse Go Back function. Full Screen • As we have mentioned, this is exactly the sinusoid. Close Quit
Fourier Series and . . . Generalized . . . 7. Scale Invariance Generalized Trig . . . • The formula for the potential energy E pot = 1 2 · c · x 2 is Physical Meaning of . . . Scale Invariance scale-invariant in the following sense: Our Idea – if we change the measuring unit for x to a one which Our Idea Leads . . . is λ times smaller x ′ = λ · x , Conclusion – then, by appropriately re-scaling the unit for mea- Acknowledgments suring energy, i.e., by taking E ′ = λ 2 · E , Home Page – we will have the exact same dependence between Title Page E ′ and x ′ in the new units: E ′ = 1 2 · c · ( x ′ ) 2 . ◭◭ ◮◮ • Similarly, the dependence E kin = 1 x ) 2 is also ◭ ◮ 2 · c · ( ˙ Page 8 of 13 scale-invariant. Go Back Full Screen Close Quit
Fourier Series and . . . Generalized . . . 8. Our Idea Generalized Trig . . . • Physical laws should not depend on the choice of mea- Physical Meaning of . . . suring units. Scale Invariance Our Idea • Thus, scale-invariance is an important physical princi- Our Idea Leads . . . ple. Conclusion • Scale-invariance does not necessarily mean that Acknowledgments E pot ( x ) = c · x 2 : e.g., E pot = x 3 is also scale-invariant. Home Page • Let us therefore consider a general case in which both Title Page E pot ( x ) and E kin ( ˙ x ) are scale-invariant. ◭◭ ◮◮ ◭ ◮ Page 9 of 13 Go Back Full Screen Close Quit
Fourier Series and . . . Generalized . . . 9. Our Idea Leads Exactly to Generalized Generalized Trig . . . Trigonometric Functions Physical Meaning of . . . • Scale-invariance of the dependence E pot ( x ) means that: Scale Invariance Our Idea – for every parameter λ describing re-scaling of the Our Idea Leads . . . coordinate x , Conclusion – there exists an appropriate re-scaling µ ( λ ) of energy Acknowledgments – that preserves this dependence, i.e., for which E = E pot ( x ) implies that E ′ = E pot ( x ′ ), where Home Page E ′ = µ ( λ ) · E and x ′ = λ · x. Title Page ◭◭ ◮◮ • Here, µ ( λ ) · E = E pot ( λ · x ), i.e., ◭ ◮ µ ( λ ) · E pot ( x ) = E pot ( λ · x ) . Page 10 of 13 • It is known that all continuous solutions of this func- tional equation have the form E pot ( x ) = c · x p . Go Back Full Screen • Similarly, scale-invariance of the expression for kinetic x ) q . energy implies that E kin ( ˙ x ) = m · ( ˙ Close Quit
Fourier Series and . . . Generalized . . . 10. Our Idea (cont-d) Generalized Trig . . . x ) = c · x p + m · ( ˙ x ) q . • Thus, E = E pot ( x ) + E kin ( ˙ Physical Meaning of . . . Scale Invariance • In the no-friction approximation, energy is preserved: Our Idea E = const . Our Idea Leads . . . Conclusion • By selecting appropriate units for E , x , and t (hence x ), we get a simplified expression 1 = x p + ( ˙ Acknowledgments x ) q . for ˙ Home Page x = dx x ) q = 1 − x p , hence ˙ dt = (1 − x p ) 1 /q , • In this case, ( ˙ Title Page ◭◭ ◮◮ dx � dx dt = (1 − x p ) 1 /q , and t ( x ) = (1 − x p ) 1 /q . ◭ ◮ Page 11 of 13 • The desired dependence x ( t ) is the inverse function to this integral t ( x ). Go Back • This is exactly the above-described generalized Full Screen trigonometric function. Close Quit
Fourier Series and . . . Generalized . . . 11. Conclusion Generalized Trig . . . • We have shown that: Physical Meaning of . . . Scale Invariance – a seemingly arbitrary generalization of sinusoids Our Idea – can be naturally derived from a physically mean- Our Idea Leads . . . ingful model, Conclusion – and the only functions obtained from this model Acknowledgments are indeed the generalized trigonometric functions. Home Page • This derivation provides a theoretical explanation of Title Page the empirical success of these functions: ◭◭ ◮◮ – while there are many mathematically possible gen- ◭ ◮ eralizations of sinusoids, Page 12 of 13 – these functions are the only one which are consis- Go Back tent with the corresponding physical model. Full Screen Close Quit
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