Local Smoothness of Graph Signals

<p>Piecewise monocomponent signal on the graph. The signal is composed of three parts. The support sets <span class="nowrap"><svg height="12.1958pt" id="M151" style="vertical-align:-3.181479pt" version="1.1" viewbox="-0.0498162 -9.01432 17.2075 12.1958" width="17.2075pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M214 21C291 90 365 210 421 347C466 459 502 542 606 654L591 669L564 647C523 612 476 598 440 598C417 598 389 610 354 629C284 664 259 678 218 678C121 678 35 602 35 527C35 453 94 401 174 401C242 401 290 443 323 508L307 517C274 464 229 431 176 431C113 431 76 473 76 525S130 626 194 626C262 626 339 543 435 543C446 543 462 545 471 548L474 545C385 459 329 384 281 262C214 94 191 43 134 -9C167 -16 174 -15 181 -15C246 -15 310 18 385 77C452 128 534 229 634 385C741 552 801 627 834 646H835C835 614 852 597 879 597C902 597 922 615 922 640C922 668 900 687 872 687C802 687 722 573 607 397C470 188 363 51 216 19L214 21Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.116,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg>,</span> <span class="nowrap"><svg height="12.1958pt" id="M152" style="vertical-align:-3.181479pt" version="1.1" viewbox="-0.0498162 -9.01432 17.2075 12.1958" width="17.2075pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M214 21C291 90 365 210 421 347C466 459 502 542 606 654L591 669L564 647C523 612 476 598 440 598C417 598 389 610 354 629C284 664 259 678 218 678C121 678 35 602 35 527C35 453 94 401 174 401C242 401 290 443 323 508L307 517C274 464 229 431 176 431C113 431 76 473 76 525S130 626 194 626C262 626 339 543 435 543C446 543 462 545 471 548L474 545C385 459 329 384 281 262C214 94 191 43 134 -9C167 -16 174 -15 181 -15C246 -15 310 18 385 77C452 128 534 229 634 385C741 552 801 627 834 646H835C835 614 852 597 879 597C902 597 922 615 922 640C922 668 900 687 872 687C802 687 722 573 607 397C470 188 363 51 216 19L214 21Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.116,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"></path></g></svg>,</span> and <svg height="12.3054pt" id="M153" style="vertical-align:-3.29108pt" version="1.1" viewbox="-0.0498162 -9.01432 17.2075 12.3054" width="17.2075pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M214 21C291 90 365 210 421 347C466 459 502 542 606 654L591 669L564 647C523 612 476 598 440 598C417 598 389 610 354 629C284 664 259 678 218 678C121 678 35 602 35 527C35 453 94 401 174 401C242 401 290 443 323 508L307 517C274 464 229 431 176 431C113 431 76 473 76 525S130 626 194 626C262 626 339 543 435 543C446 543 462 545 471 548L474 545C385 459 329 384 281 262C214 94 191 43 134 -9C167 -16 174 -15 181 -15C246 -15 310 18 385 77C452 128 534 229 634 385C741 552 801 627 834 646H835C835 614 852 597 879 597C902 597 922 615 922 640C922 668 900 687 872 687C802 687 722 573 607 397C470 188 363 51 216 19L214 21Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.116,3.132)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"></path></g></svg> are presented with different vertex colors.</p>

Mathematical Problems in Engineering

Local Smoothness of Graph Signals

Figure 3